Research Article -Trigonometric and -Hyperbolic Functions in …downloads.hindawi.com/journals/aaa/2016/3249439.pdf · on the theory of di erential equations in the complex domain

Research Article119901-Trigonometric and 119901-Hyperbolic Functions inComplex Domain

Petr Girg and Lukaacuteš Kotrla

Department of Mathematics and NTIS Faculty of Applied Sciences University of West BohemiaUniverzitnı 8 306 14 Plzen Czech Republic

Correspondence should be addressed to Petr Girg pgirgkmazcucz

Received 28 July 2015 Accepted 1 December 2015

Academic Editor Paul W Eloe

Copyright copy 2016 P Girg and L KotrlaThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study extension of 119901-trigonometric functions sin119901 and cos119901 and of 119901-hyperbolic functions sinh119901 and cosh119901 to complex domainOur aim is to answer the question under what conditions on 119901 these functions satisfy well-known relations for usual trigonometricand hyperbolic functions such as for example sin(119911) = minus119894sdotsinh(119894sdot119911) In particular we prove in the paper that for119901 = 6 10 14 the119901-trigonometric and 119901-hyperbolic functions satisfy very analogous relations as their classical counterparts Our methods are basedon the theory of differential equations in the complex domain using the Maclaurin series for 119901-trigonometric and 119901-hyperbolicfunctions

1 Introduction

The 119901-trigonometric functions are generalizations of regulartrigonometric functions sine and cosine and arise from thestudy of the eigenvalue problem for the one-dimensional 119901-Laplacian

In recent years the 119901-trigonometric functions wereintensively studied from various points of views by manyresearchers see for example monograph [1] for systematicsurvey and further references The purpose of this paper istwofold We begin with a short survey of results from [2 3]Then we extend the ideas from [3] to define correspondinggeneralization of hyperbolic functions and study relations of119901-trigonometric and 119901-hyperbolic functions on a disc in thecomplex domain

More precisely our goal is to generalize the hyperbolicfunctions such that the relations

sin 119911 = minus119894 sdot sinh (119894 sdot 119911) cos 119911 = cosh (119894 sdot 119911)

(1)

cos 119911 = sin1015840119911

cosh 119911 = sinh1015840119911(2)

cos2119911 + sin2119911 = 1

cosh2119911 minus sinh2119911 = 1(3)

where 119911 isin C have their counterparts for generalized 119901-trigonometric and 119901-hyperbolic functions It turns out thatthis goal can be achieved only for even integer 119901 gt 2

The 119901-trigonometric functions in the real domain R

originate naturally from the study of the nonlinear eigenvalueproblem

minus (10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901minus2

1199061015840)1015840

minus 120582 |119906|119901minus2

119906 = 0 in (0 120587119901)

119906 (0) = 119906 (120587119901) = 0

(4)

where 119901 gt 1 120582 isin R is a parameter and

120587119901 = 2int

1

0

1

(1 minus 119904119901)1119901

d119904 = 2120587

119901 sin (120587119901) (5)

It was shown in Elbert [4] that all eigenfunctions of (4) can beexpressed in terms of solutions of the initial-value problem

minus (10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901minus2

1199061015840)1015840

minus (119901 minus 1) |119906|119901minus2

119906 = 0 119909 isin R

119906 (0) = 0

1199061015840(0) = 1

(6)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2016 Article ID 3249439 18 pageshttpdxdoiorg10115520163249439

2 Abstract and Applied Analysis

Indeed (6) has the unique solution inR see for example[5 LemmaA1] [6 Section 3] and [4] Denoting the solutionof (6) by sin119901119909 the set of all eigenvalues 120582119896 isin R andeigenfunctions 119906119896 isin 119882

1119901

0(0 120587119901) of (4) can be written as

120582119896 = (119901 minus 1) 119896119901

119906119896 (119909) = sin119901 (119896 sdot 119909)

where 119896 isin N

(7)

A piecewise construction of the solution of (6) wasprovided in [4] At first one sets

arcsin119901119909def= int

119909

0

1

(1 minus 119904119901)1119901

d119904 119909 isin [0 1] (8)

Then the restriction of sin119901119909 on [0 1205871199012] is the inversefunction to arcsin119901119909 For 119909 isin (1205871199012 120587119901] sin119901119909 satisfiessin119901119909 = sin119901(120587119901 minus 119909) where clearly 120587119901 minus 119909 isin [0 1205871199012) andsin119901119909 = minussin119901(minus119909) for 119909 isin [minus120587119901 0) Finally sin119901119909 is a 2120587119901-periodic function on R

We also extend arcsin119901119909 from (8) to [minus1 1] as an oddfunction Then it is the inverse function to the restriction ofsin119901119909 to [minus1205871199012 1205871199012] and we have

sin119901 (arcsin119901119909) = 119909 forall119909 isin [minus1 1] (9)

Finally let us define cos119901119909def= sin1015840

119901119909 for all 119909 isin R

Then the functions sin119901119909 and cos119901119909 satisfy the so-called 119901-trigonometric identity

10038161003816100381610038161003816cos119901119909

10038161003816100381610038161003816

119901

+10038161003816100381610038161003816sin119901119909

10038161003816100381610038161003816

119901

= 1 (10)

for all 119909 isin R see for example [4ndash6]Note that there is an alternative definition of ldquocos119901119909rdquo

in [7] andor [8] which leads to different ldquo119901-trigonometricrdquoidentity Yet another alternative generalization of trigono-metric and hyperbolic functions motivated by geometricalpoint of view was introduced in [9] Studies of relationsbetween their respective generalizations of 119901-trigonometricand 119901-hyperbolic functions were suggested in [7] and in [9]respectively

Remark 1 In the paper we use Gaussrsquo hypergeometric func-tion21198651(119886 119887 119888 119911) where 119886 119887 119888 isin C are parameters and 119911 isin C

is variable (for definition see [10 1511 p 556]) to expressintegrals of the type

int

119909

0

1

(1 plusmn 119904119901)1119901

d119904

int

119911

0

1

(1 plusmn 119904119901)1119901

d119904(11)

for 119901 gt 1 119909 isin R and 119911 isin C (by 1199111119901 we understand theprincipal branch thereof) Indeed

11991121198651 (

1

1199011

119901 1 +

1

119901 ∓119911119901) = int

119911

0

1

(1 plusmn 119904119901)1119901

d119904 (12)

for |119911| lt 1 (by comparing respective series expansions) Bythe uniqueness of analytic extension the equation is valid for119911 isin C 119909 + 119894119910 119909 gt 1 and 119910 = 0 (for analytic continuationof 21198651 see eg [10 1531 p 558] and [11 Theorem 221 p65])

In the definition of 120587119901 (ie (5)) and in (8) we need toevaluate integral

int

1

0

1

(1 minus 119904119901)1119901

d119904 (13)

By [11 Theorem 222 p 66] this is possible sinceR[119888 minus 119886 minus119887] = 1 + 1119901 minus 1119901 minus 1119901 = 1 minus 1119901 gt 0 for 119901 gt 1

Notation 1 This paper combines real variable and complexvariable approach to the 119901-trigonometric and 119901-hyperbolicfunctions Each of these approaches has its own natural wayof how to define the functions sin119901 and sinh119901 Thus we needto distinguish between real and complex definitions By sin119901119909and sinh119901119909 we mean functions defined by the real variableapproach and by sin119901119911 and sinh119901119911wemean functions definedby the complex variable approach throughout the paper

2 Real Analyticity Results for sin119901119909 and cos119901119909

It is well known that the 119901-trigonometric functions are notreal analytic functions in general see for example [12 13]Very detailed study of the degree of smoothness of therestriction of sin119901119909 to (minus1205871199012 1205871199012) was performed in [2]including the following two results The first one concernsldquogenericrdquo 119901 gt 1

Proposition 2 (see [2] Theorem 32 on p 105) Let 119901 isin R

2119898 119898 isin N 119901 gt 1 Then

sin119901119909 isin 119862lceil119901rceil(minus120587119901

2120587119901

2) (14)

but

sin119901119909 notin 119862lceil119901rceil+1

(minus120587119901

2120587119901

2) (15)

Here lceil119901rceil def= min119896 isin N 119896 ge 119901

The second result treats only the even integers 119901 gt 2 anddiffers significantly from the previous case in an unexpectedway

Proposition 3 (see [2]Theorem 31 on p 105) Let 119901 = 2(119898+1) 119898 isin N Then

sin119901119909 isin 119862infin(minus120587119901

2120587119901

2) (16)

Thus the Maclaurin series of sin119901119909

119872119901 (119909)def=

infin

sum

119899=1

1

119899sin(119899)119901(0) sdot 119909

119899 (17)

is well defined for 119901 = 2(119898 + 1) 119898 isin N Moreoverthe following result establishes an explicit expression for theradius of convergence of119872119901

Abstract and Applied Analysis 3

Proposition 4 (see [2] Theorem 33 on p 106) Let 119901 =

2(119898+1) for119898 isin N Then theMaclaurin series119872119901(119909) of sin119901119909converges on (minus1205871199012 1205871199012)

Thus for 119901 = 2(119898 + 1) 119898 isin N we can computeapproximate values of sin119901119909 using Maclaurin series It turnsout that the most effective method of computing coefficientsin (17) is to use formal inversion of the Maclaurin series of

arcsin119901119909 = int119909

0

1

(1 minus 119904119901)1119901

d119904

= 119909 sdot21198651 (

1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896119901 + 1) 119896Γ (1119901)119909119896119901+1

(18)

where 119901 = 2(119898 + 1) for some 119898 isin N The procedureof inverting power series is well known see for example[14] This task can be easily performed using computeralgebra systems In Pseudocode 1 we provide an example ofcomputing the partial sum of 1198724 up to terms of order 32in Mathematica v 90 In this way we can easily get partialsums of 119872119901 and get approximations of sin119901 for any 119901 =

2(119898+1) 119898 isin N up to terms of orders of hundreds Note thatthis formal inverse can be applied also for119901 = 2119898+1 119898 isin NThequestion iswhat is themathematical sense of the resultingformal series Let

119879119901 (119909)def=

infin

sum

119899=1

119886119899 sdot 119909119899 (19)

denote the series that is the formal inverse of (18) for 119901 =

2119898 + 1 119898 isin N For 119901 = 2119898 + 1 119898 isin N let us also define

119872119901 (119909)def=

infin

sum

119899=1

1

119899( lim119909rarr0+

sin(119899)119901119909) sdot 119909

119899 (20)

which is a formal Maclaurin series of some unknown func-tion It turns out that this unknow function is not sin119901119909 asthe following result holds


2119898+1119898 isin NThen the formalMaclaurin series119872119901 convergeson (minus1205871199012 1205871199012) Moreover the formal Maclaurin series 119872119901converges towards sin119901119909 on [0 1205871199012) but does not convergetowards sin119901119909 on (minus1205871199012 0)

In Appendix A we prove that 119879119901 and119872119901 are identical

Theorem 6 Let 119901 = 2119898 + 1119898 isin N Then

119886119899 =1

119899( lim119909rarr0+

sin(119899)119901119909) forall119899 isin N (21)

and 119879119901(119909) = 119872119901(119909) for all 119909 isin (minus1205871199012 1205871199012)

It turns out that the pattern of zero coefficients of119872119901 isthe same as in theMaclaurin series of arcsin119901119909 compare (18)

Theorem 7 Let 119901 gt 2 be an integer Then 119886119894 = 0 for all 119894 isin N

such that 119894 minus 1 is not divisible by 119901

Theproof is technical and thus postponed to Appendix BIt is based on the formal inversion of (18) Note that thestructure of powers in (18) does not allow any substitutionthat will transform it into a power series of new variablewithout zero coefficients This makes the proof technicallycomplicated

Using Theorem 7 we can omit zero entries and rewritethe series119872119901

119872119901 (119909) =

infin

sum

119897=0

120572119897 sdot 119909119897119901+1

(22)

where 120572119897 can be obtained by formal inversion of the Maclau-rin series of arcsin119901119909 in (18) In particular

1205720 = 1

1205721 = minus1

119901 (119901 + 1)

1205722 = minus1199012minus 2119901 minus 1

21199012 (119901 + 1) (2119901 + 1)

(23)

3 Extension of sin119901119911 and cos119901119911 to the ComplexDomain for Integer 119901 gt 1

The conclusion of this theorem follows from the discussionin [2]

Theorem 8 Let 119901 = 2(119898 + 1) 119898 isin N Then the Maclaurinseries of sin119901119909 converges on the open disc

119861119901def= 119911 isin C |119911| lt

120587119901

2 (24)

Proof In fact the Maclaurin series suminfin119897=0120572119897 sdot 119909119897sdot119901+1 converges

towards the values of sin119901119909 on (minus1205871199012 1205871199012) absolutely for119901 = 2(119898 + 1)119898 isin N

For 119901 = 2(119898 + 1) 119898 isin N the expressions with powersin the initial-value problem (6) can be written without theabsolute values Thus the resulting initial-value problem

(1199061015840)119901minus2

11990610158401015840+ 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(25)

makes sense also in the complex domain (the derivativesare understood in the sense of the derivative with respect to


In[1] = Series[sHypergeometric2F1[14 14 54 sand4]

s 0 32]

( computes the Maclaurin series of arcsin 4 )

Out[1] = 119904 +1199045

20+51199049

288+1511990413

1664+19511990417

34816+22111990421

57344+4641119904

25

1638400+16575119904

29

7602176+ 119874(119904

33)

In[2] = InverseSeries[]( computes the inverse series )

Out[2] = 119904 minus1199045

20minus71199049

1440minus46311990413

374400minus211741119904

17

509184000minus104361161119904

21

641571840000minus

897899621311990425

128314368000000minus

799573586746311990429

248783228928000000+ 119874(119904

33)

(which is M 4 up to terms of order 32)

Pseudocode 1Mathematica v 90 code

complex variable) Let us observe that using the substitution1199061015840= V we get the following first-order system

1199061015840= V

V1015840 = minus119906119901minus1

V119901minus2

119906 (0) = 0

V (0) = 1

(26)

By [15 Theorem 91 p 76] there exists 120575119901 gt 0 such thatproblems (26) and hence (25) have the unique solution onthe open disc |119911| lt 120575119901

Nowwewill consider initial-value problems (25) and (26)also for 119901 = 2119898 + 1119898 isin N

Theorem 9 Let 119901 = 2119898+1 119898 isin N The unique solution 119906(119911)of (25) restricted to open disc 119861119901 is the Maclaurin series119872119901

Proof Let 119906(119911) = suminfin119896=1

119887119896119911119896 be the unique solution of (25) in

any point of the open disc |119911| lt 120575119901 Observe that the solution119906(119911) = sum

infin

119896=1119887119896119911119896 solves also the real-valued Cauchy problem

(6) for 119909 gt 0 (where there is no need for | sdot |) On the otherhand sin119901119909 is the unique solution of the real-valued Cauchyproblem (6) Since119872119901 given by (20) converges towards sin119901119909in [0 1205871199012) we find that119872119901 satisfies (6) in [0 1205871199012) Thus119872119901(119909) = 119906(119909 + 119894 sdot 119910) for 119909 isin [0 120575119901) and 119910 = 0 In particulartaking a sequence of points 119911119899 = 120575119901(119899 + 1) + 0 sdot 119894 119899 isin Nwe have119872119901(119911119899) = 119906(119911119899) thus we infer from Proposition A2that

119887119896 =1

119896( lim119909rarr0+

sin(119896)119901119909) (27)

119872119901 has radius of convergence 1205871199012 by Proposition 5 and sodoes 119906(119911) Thus 120575119901 = 1205871199012

Theorems 8 and 9 enable us to extend the range ofdefinition of the function sin119901119909 to the complex open disc 119861119901by 119872119901 for 119901 = 2(119898 + 1) and for 119901 = 2119898 + 1 119898 isin NrespectivelyThus we can consider 119901 = 119898+2 in the followingdefinition Note that all the powers of 119911 are of positive-integerorder 119897 sdot 119901 + 1 and the function sin119901119911 is an analytic complexfunction on 119861119901 and thus is single-valued

Definition 10 Let 119901 = 119898 + 2 119898 isin N and 119911 isin 119861119901 Then

sin119901119911def=

infin

sum

119897=0

120572119897 sdot 119911119897sdot119901+1

cos119901119911def= sin1015840119901119911 =

dd119911

sin119901119911

(28)

where the derivative dd119911 is considered in the sense ofcomplex variables

The following fundamental results were proved in [3](providing explicit value for 120575119901)


2(119898 + 1) 119898 isin N then the unique solution of the initial-valueproblem (25) on 119861119901 is the function sin119901119911

Proposition 12 (see [3]Theorem 31 on p 229) Let119901 = 2119898+1119898 isin NThen the unique solution 119906(119911) of the complex initial-value problem (25) differs from the solution sin119901119909 of the Cauchyproblem (6) for 119911 = 119909 isin (minus1205871199012 0)

In [3] it was shown that there is no hope for solutions of(25) to be entire functions for 119901 = 119898 + 2 119898 isin N This resultfollows from the complex analogy of the 119901-trigonometric(10)

Lemma 13 Let 119901 = 119898 + 1119898 isin N and 119903 gt 0 be such that thesolution 119906 of (25) is holomorphic on a disc 119863119903 = 119911 isin C |119911| lt

119903 Then 119906 satisfies the complex 119901-trigonometric identity

(1199061015840(119911))119901

+ (119906 (119911))119901= 1 (29)

on the disc 119863119903

Proof Multiplying (25) by 1199061015840 and integrating from 0 to 119911 isin119863119903 we obtain

(1199061015840(119911))119901

minus (1199061015840(0))119901

+ (119906 (119911))119901minus (119906 (0))

119901= 0 (30)

Now using the initial conditions of (25) we get (29) which isthe first integral of (25) and we can think of it as complex 119901-trigonometric identity for holomorphic solutions of (25) for119901 = 119898 + 1 119898 isin N


Now we state the very classical result from complexanalysis

Proposition 14 (see [16] Theorem 1220 on p 433) Let 119891and 119892 be entire functions and for some positive integer 119899 satisfyidentity

119891119899+ 119892119899= 1 (31)

(i) If 119899 = 2 then there is an entire function ℎ such that119891 = cos ∘ ℎ and 119892 = sin ∘ ℎ

(ii) If 119899 gt 2 then 119891 and 119892 are each constant

The following interesting connection between complexanalysis (including the classical reference [16 Theorem1220]) and 119901-trigonometric functions was studied in [3] Weshould point out that it was an interesting internet discussion[17] that called our attention towards this connection Itseems to us that this connection was overlooked by the ldquo119901-trigonometric communityrdquoThus we provide itsmore preciseproof here

Theorem 15 The solution 119906 of complex initial-value problem(25) cannot be entire function for any 119901 = 119898 + 2 119898 isin N

Proof Assume by contradiction that the solution 119906 of (25)is entire function Then we can choose 119903 gt 0 arbitrarilylarge in Lemma 13 Thus 119906 and 1199061015840 must satisfy (29) at anypoint 119911 isin C Note that 1199061015840 is an entire function too Thusby Proposition 14 119906 and 1199061015840 are constant which contradicts1199061015840(0) = 1 This concludes the proof

In particular the solution of (25) is 119906(119911) = sin119901119911 with1199061015840(119911) = cos119901119911 Thus (29) becomes

cos119901119901119911 + sin119901

119901119911 = 1 (32)

and we see that sin119901 and cos119901 cannot be entire functions for119901 = 119898 + 2119898 isin N

4 Generalized Hyperbolometric Functionargsinh

119901119909 and Generalized Hyperbolic

Function sinh119901119909 in the Real Domain forReal 119901 gt 1

In analogy to119901 = 2 we define sinh119901119909 for119901 gt 1 as the solutionto the initial-value problem

minus (10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901minus2

1199061015840)1015840

+ (119901 minus 1) |119906|119901minus2

119906 = 0 119909 isin R

119906 (0) = 0

1199061015840(0) = 1

(33)

The uniqueness of the solution of this problem can be provedin the sameway as in the case of (6) using the first integral (see

[4]) Note that the first integral of the real-valued initial-valueproblem (33) is the real 119901-hyperbolic identity

1 + |119906|119901=10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

(34)

for 119901 gt 1 compare [4] Thus |1199061015840| ge 1 on the domain ofdefinition of solution to (33) Since 1199061015840(0) = 1 and 1199061015840 mustbe absolutely continuous we find that 1199061015840 gt 0 on the domainof definition of solution to (33) and the real 119901-hyperbolicidentity can be rewritten in equivalent form

1199061015840= (1 + |119906|

119901)1119901 (35)

which is a separable first-order ODE in R By the standardintegration procedure we obtain inverse function of thesolution 119906 (cf [4])

Therefore it is natural to define

argsinh119901119909

def= int

119909

0

1

(1 + |119904|119901)1119901

d119904 119909 isin R (36)

for any 119901 gt 1 in the real domain (cf eg [18ndash21]) Notethat the integral on the right-hand side can be evaluated interms of the analytic extension of Gaussrsquos

21198651 hypergeometric

function to C 119904 + 119894119905 119904 gt 1 119905 = 0 (see eg [10 sect 1531 p558] and [11Theorem 221 p 65]) thus (taking into accountthat integrand in (36) is even)

argsinh119901119909

=

119909 sdot21198651 (

1

1199011

119901 1 +

1

119901 minus119909119901) 119909 isin [0 +infin)

minusargsinh119901(minus119909) 119909 isin (minusinfin 0)

(37)

Since argsinh119901 R rarr R is strictly increasing function

on R its inverse exists and it is in fact sinh119901119909 by the samereasoning as in [4] (cf eg [20])

5 Generalized Hyperbolic Functionssinh119901119911 and cosh119901119911 in Complex Domain forInteger 119901 gt 1

In the previous section we introduced real-valued general-ization of sinh119909 called sinh119901119909 Our aim is to extend thisfunction to complex domain It is important to observe thatfor 119901 = 2 the following relations between complex functionssin 119911 and sinh 119911 are known

sin 119911 sinh 11991111990610158401015840+ 119906 = 0 119906

10158401015840minus 119906 = 0

119906 (0) = 0 119906 (0) = 0

1199061015840(0) = 1 119906

1015840(0) = 1

sin 119911 = minus119894 sdot sinh (119894 sdot 119911)

(38)

where 119911 isin C and the equations are understood in the senseof ordinary differential equations in the complex domain


Since the function | sdot | Crarr [0 +infin) (complex modulus)is not analytic at 0 isin C we cannot work with (6) and (33) butwe need to consider (25) and

(1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(39)

in our discussion in the complex domain Thus the directanalogy of the classical relations summarized in the tableabove for 119901 = 2 is stated in the following table

sin119901119911 sinh119901119911(1199061015840)119901minus2

11990610158401015840+ 119906119901minus1

= 0 (1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0 119906 (0) = 0

1199061015840(0) = 1 119906

1015840(0) = 1

sin119901119911= minus119894 sdot sinh119901 (119894 sdot 119911)

(40)

where 119911 belongs to some complex disc centred at 0 isin C withradius small enough such that both complex initial-valueproblems are solvable However it turns out (see below) thatif we define sinh119901119911 as the solution (39) then the ldquo119901-analogiesrdquoof (2)-(3) are satisfied but the ldquo119901-analogyrdquo of the identity (1)that is

sin119901119911 = minus119894 sdot sinh119901 (119894 sdot 119911) (41)

is not satisfied in general Our aim is to provide conditionswhen (41) holds as well

Let us formalize the above-stated ideas Denote

119863119901def= 119911 isin C |119911| lt 120574119901 (42)

an open disc in C where 120574119901 gt 0 is given radius At first weprove unique solvability of (39) in119863119901

Lemma 16 Let119901 = 119898+2 119898 isin NThen there exists a complexdisc119863119901 such that the initial-value problem in complex domain(39) has a unique solution on119863119901

Proof Using the substitution 1199061015840 = V we get the followingfirst-order system

1199061015840= V

V1015840 =119906119901minus1

V119901minus2

119906 (0) = 0

V (0) = 1

(43)

By [15 Theorem 91] on page 76 the statement of the lemmafollows

Nowwe can define sinh119901 119863119901 rarr C for any integer 119901 gt 2

Definition 17 Let 119901 = 119898 + 2 119898 isin N The complex functionsinh119901119911 is defined on 119863119901 as the unique solution of the initial-

value problem (39) and cosh119901119911def= sinh1015840

119901119911 for all 119911 isin 119863119901


119903 Then 119906 satisfies the complex ldquo119901-hyperbolicrdquo identity

(1199061015840(119911))119901

minus (119906 (119911))119901= 1 (44)

on the disc 119863119903

The proof of Lemma 18 is analogous to the proof ofLemma 13 and thus it is omitted

Remark 19 Les us note that the real-valued identity

(119906 (119909)1015840)119901

minus (119906 (119909))119901= 1 (45)

for general 119901 gt 0 already appeared in [22] where theformal Maclaurin power series expansion of the solution tothis identity was treated Interesting recurrence formula forthe coefficients of the Maclaurin power series can be foundthere It will be very interesting to use the following relationsbetween sin119901119911 and sinh119901119911 to find the analogous recurrenceformulas for sin119901119911

Now we are ready to state main results of Section 5

Theorem 20 Let 119901 = 4119897 + 2 119897 isin N Then


cos119901119911 = cosh119901 (119894 sdot 119911) (47)

for all 119911 isin 119861119901 Moreover

sinh119901119911 =infin

sum

119896=0

(minus1)119896sdot 120572119896 sdot 119911

119896119901+1 (48)

On the other hand we have also the following surprisingresult

Theorem 21 Let 119901 = 4119897 119897 isin N Then

sin119901119911 = minus119894 sdot sin119901 (119894 sdot 119911)

cos119901119911 = cos119901 (119894 sdot 119911)(49)

for all 119911 isin 119861119901

The statement of the previous theorem is closely relatedto similar result for 119901-hyperbolic functions


sinh119901119911 = minus119894 sdot sinh119901 (119894 sdot 119911) (50)

cosh119901119911 = cosh119901 (119894 sdot 119911) (51)

for all 119911 isin 119863119901


Proof of Theorem 20 Let 119901 = 4119897 + 2 119897 isin N and 119906(119911) = sinh119901119911be the unique solution of the initial-value problem (39) on119863119901

We show that119908(119911) = minus119894 sdot 119906(119894 sdot 119911) satisfies (25) on119863119901 cap119861119901Due to uniqueness of solution of (25) the identity (46) musthold on119863119901 cap 119861119901

Indeed plugging into the left-hand side of (25) we get

(1199081015840(119911))119901minus2

11990810158401015840(119911) + 119908 (119911)

119901minus1

= (dd119911119908 (119911))

119901minus2 d2

d1199112119908 (119911) + 119908 (119911)

119901minus1

= [dd119911(minus119894 sdot 119906 (119894 sdot 119911))]

119901minus2 d2

d1199112(minus119894 sdot 119906 (119894 sdot 119911))

+ (minus119894 sdot 119906 (119894 sdot 119911))119901minus1

= 119894 sdot [d

d (119894 sdot 119911)119906 (119894 sdot 119911)]

119901minus2 d2

d (119894 sdot 119911)2119906 (119894 sdot 119911)


= 119894 sdot [dd119904119906 (119904)]

119901minus2 d2

d1199042119906 (119904) minus 119894 sdot (minus119894)

119901minus2sdot 119906 (119904)119901minus1

= 119894 sdot ([dd119904119906 (119904)]

119901minus2 d2

d1199042119906 (119904) minus (minus119894)

119901minus2sdot 119906 (119904)119901minus1)

= 119894 sdot ([dd119904119906 (119904)]

119901minus2 d2

d1199042119906 (119904) minus 119906 (119904)

119901minus1) = 0

(52)

Note that for 119901 = 4119897 + 2 119897 isin N (minus119894)119901minus2 = 1 The last equalitythen follows from (39) The right-hand side of (25) is zeroSo we have verified that 119908(119911) = minus119894 sdot sinh119901(119894 sdot 119911) satisfies thedifferential equation in (39)The initial conditions of (25) arealso satisfied by 119908(119911) = minus119894 sdot sinh119901(119894 sdot 119911) since 119906(119911) = sinh119901119911satisfies the initial conditions of (39)

Now it remains to show that 119863119901 = 119861119901 and hence sin119901119911 =minus119894sdotsinh119901(119894sdot119911) on119861119901 To this end let us write sinh119901119911 = sum

infin

119899=1119888119899 sdot

119911119899 for yet unknown 119888119899 isin C Theninfin

sum

119899=1

119886119899 sdot 119911119899= sin119901119911 = minus119894 sdot sinh119901 (119894 sdot 119911) =

infin

sum

119899=1

119888119899 sdot 119894119899minus1

sdot 119911119899 (53)

on119863119901 cap 119861119901 From here

119888119899 =119886119899

119894119899minus1= 1198943119899+1

sdot 119886119899 (54)

Since |1198943119899+1| = 1119863119901 = 119861119901Now taking into account that 119886119899 = 0 for 119899minus1 not divisible

by 119901 we immediately get 119888119899 = 0 for 119899 minus 1 not divisible by 119901Now using our notation 120572119896 = 119886119896119901+1 119896 isin N we find that

119888119896119901+1 = (119894)3(119896119901+1)+1

120572119896 = (minus1)119896120572119896 (55)

which establishes (48)Equation (47) follows directly from cosh119901119911 = sinh1015840

119901119911 and

(46)

Proof of Theorem 21 Let 119901 = 4119897 119897 isin N and 119906(119911) = sin119901119911Now plugging119908(119911) = minus119894sdot119906(119894sdot119911) into the left-hand side of (25)we proceed in the sameway as in the proof ofTheorem20Themost important difference is that for 119901 = 4119897 119897 isin N (minus119894)119901minus2 =minus1 Then all following steps are analogous to those in proofof Theorem 20 with several obvious changes

Proof of Theorem 22 The proof is almost identical to theproof of Theorem 20 with obvious changes (cf the proof ofTheorem 21)

6 Real Restrictions of the Complex ValuedSolutions of (25) and (39) and TheirMaximal Domains of Extension as RealInitial-Value Problems

Let us denote the restriction of the complex valued solutionof (25) and (39) to the real axis by s119901(119909) and by sh119901(119909)respectively Since the equation in (25) and (39) containsonly integer powers of the solution and its derivatives allcoefficients in the equation are real and the initial conditionsin (25) and (39) are real the value of sin119901119911 and sinh119901119911 mustbe a real number for 119911 = 119909 + 119894119910 with minus1205871199012 lt 119909 lt 1205871199012

and minus120574119901 lt 119909 lt 120574119901 and 119910 = 0 respectively Hence s119901(119909) andsh119901(119909) attain only real values

We start with the slightly more complicated case whichis sh119901(119909) Moreover since the solution of (39) is an analyticfunction it has the series representation

sinh119901119911 =infin

sum

119896=1

119888119896119911119896 119911 isin 119863119901 (56)

where 119888119896 isin R 119896 isin N (note that sinh119901119911must be a real numberfor any 119911 = 119909 + 119894119910 with minus120574119901 lt 119909 lt 120574119901 and 119910 = 0)

Now we show that sh119901(119909) solves (33) (in the sense ofdifferential equations in real domain) for 119901 = 2(119898 + 1) 119898 isin

N and does not solve (33) (in the sense of differentialequations in real domain) for 119901 = 2119898 + 1 119898 isin N and119909 lt 0 For this purpose we use an interesting consequenceof omitting of the modulus function

Theorem 23 Let 119901 = 2119898+1 119898 isin N and 119909 isin (minus1205871199012 1205871199012)Then

sh119901 (119909) =

sinh119901119909 119909 isin [0120587119901

2)

sin119901119909 119909 isin (minus120587119901

2 0)

(57)

Proof For 119909 isin [0 120574119901) the statement of Theorem followsdirectly from the definition of real function sinh119901119909 and thefacts that sinh119901119909 ge 0 and sinh1015840

119901119909 ge 0 on [0 120574119901)


By the definition the function sin119901119909 is the uniquesolution of (6) that is

minus (10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901minus2

1199061015840)1015840


119906 = 0

119906 (0) = 0

1199061015840(0) = 1

(58)

Assume that 119909 isin (minus1205871199012 0) Then sin119901119909 lt 0 and sin1015840119901119909 gt 0

Hence we can rewrite (6) as

(1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(59)

which is formally (39) but here considered in real domainBy Lemma 16 (39) has the unique solution on 119863119901 Its

restriction to (minus120574119901 0)cap(minus1205871199012 0) clearly satisfies (59) Hence

sin119901119909 =infin

sum

119896=1

119888119896119909119896= sh119901 (119909)

119909 isin (minus120574119901 0) cap (minus120587119901

2 0)

(60)

Moreover sin119901119909 = suminfin

119896=0120572119896 sdot 119909 sdot |119909|

119896119901 which is generalizedMaclaurin series of sin119901119909 (see [2 Remark 66 p 125])convergent on (minus1205871199012 1205871199012) For (minus1205871199012 0) we obtain

infin

sum

119896=0

120572119896 sdot 119909 sdot |119909|119896119901=

infin

sum

119896=0

120572119896 sdot (minus1)119896119901sdot 119909119896119901+1

= 119866 (119909) (61)

Hence theMaclaurin series119866(119909) converges on (minus1205871199012 1205871199012)(but not towards sin119901119909 for 119909 gt 0) From (60) we get

infin

sum

119896=1

119888119896119909119896= 119866 (119909) on (minus120574119901 0) (62)

and using Proposition A2 we obtain 120574119901 = 1205871199012

Corollary 24 Let 119901 = 2119898 + 1119898 isin N Then sh119901(119909) does notsolve (33) for 119909 isin (minus1205871199012 0)

Proof Since sin119901119909 = sinh119901119909 for 119909 = 0 the statementof Corollary follows directly from Theorem 23 and theuniqueness of solution of (33)

Theorem 25 Let 119901 = 2(119898 + 1) 119898 isin N Then sh119901(119909) solves(33) for 119909 isin (minus120574119901 120574119901) In particular 120574119901 = 1205871199012 for 119901 = 4119898+ 2119898 isin N

Proof Since 119901 is even we can drop the absolute values in (33)obtaining (59) which is formally (39) but here considered inreal domain Since sinh119901(119911) solves (39) on 119863119901 its restrictionsh119901(119909) to (minus120574119901 120574119901)must solve (59) on (minus120574119901 120574119901)

For 119901 = 4119898 + 2 119898 isin N we get 120574119901 = 1205871199012 by (46) inTheorem 20


s119901 (119909) =

sin119901119909 119909 isin [0120587119901

2)

sinh119901119909 119909 isin (minus120587119901

2 0)

(63)

Proof The proof follows the same steps as the proof ofTheorem 23 with obvious modifications

Now we will consider (25) and (39) as real-valued prob-lems and find their maximal domains of extension Let

harrs119901(119909)

andharr

sh119901(119909) denote solutions with maximal domains of (25)

and (39) respectively We also defineharrc119901(119909)

def= (dd119909)

harrs119901(119909)

andharr

ch119901(119909)def= (dd119909)

harr

sh119901(119909)


harrs119901119909 =

sinh119901119909 119909 isin (minusinfin 0)

sin119901119909 119909 isin [0120587119901

2)

harr

sh119901119909 =

sin119901119909 119909 isin (minus120587119901

2 0)

sinh119901119909 119909 isin [0 +infin)

(64)

Theorem 28 Let 119901 = 2(119898 + 1)119898 isin N Then

harrs119901119909 = sin119901119909 119909 isin R

harr

sh119901119909 = sinh119901119909 119909 isin R

(65)

Proof of Theorems 27 and 28 The solutions with maximaldomain of extension are known for (6) and (33) The proofuses uniqueness of the solutions of real initial-value problems(6) and (33) and initial-value problems (25) and (39) con-sidered in real domain and the fact that (6) and (33) can berewritten as (25) and (39) depending on 119906(119909) ≶ 0 1199061015840(119909) ≶ 0and the parity of the positive integer119901Themain ideas of howto combine these ingredients are contained in the proof ofTheorem 23

It easily follows from Theorems 27 and 28 thatharrc119901(119909) is

defined on (minusinfin 1205871199012) for 119901 = 2119898 + 1 and on R for 119901 =

2(119898+1) 119898 isin N Similarlyharr

ch119901(119909) is defined on (minus1205871199012 +infin)

for 119901 = 2119898 + 1 and on R for 119901 = 2(119898 + 1) 119898 isin NMoreover functions

harrs119901(119909) and

harrc119901(119909) satisfy the complex 119901-

trigonometric identity (29) that is

(harrs119901 (119909))

119901

+ (harrc119901 (119909))

119901

= 1 (66)


In[1] = pip[p ] =

Integrate[1(1 - sandp)and(1p) s 0 1

Assumptions -gt p gt 1]

( assigns definition to function pip which returns pi p2 )

Out[1] =120587csc(120587119901)

119901

In[2] = s3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == - 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -5 pip[3]][[1]])

( assigns definition to auxiliar function s3 )

In[3] = sh3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -pip[3] 5][[1]])

( assigns definition to auxiliar function sh3 )

In[4] = sin3[x ] = (u[x]

NDSolve[u[x] == (Abs[v[x]])and(12) Sign[v[x]]

v[x] == -2 Abs[u[x]]u[x]

u[0] == 0 v[0] == 1

uv x -4 pip[3] 4 pip[3]][[1]])

( assigns definition to auxiliar function sin3 )

Pseudocode 2 Mathematica v 90 code Code for pip[p ] computes 1205871199012 for given argument 119901 Code for s3[x ] computesharr

1199043(119909) for

119909 isin (minus5 12058732) ⊊ (minusinfin 12058732) code for sh3[x ] computesharr

sh3(119909) for 119909 isin (12058732 5) ⊊ (12058732 +infin) and code for sin3[x ] computes real-valuedfunction sin3(119909) for 119909 isin (minus21205873 21205873) ⊊ R

Analogously functionsharr

sh119901(119909) andharr

ch119901(119909) satisfy the complex119901-hyperbolic identity (44) that is

(harr

ch119901 (119909))119901

minus (harr

sh119901 (119909))119901

= 1 (67)

7 Visualizations

In this section we provide visualizations of theoretical resultsfrom previous sections To generate graphical output weneed to approximate special functions from previous sectionsnumerically Note that the standard numerical methods(available in Mathematica or Matlab) can handle onlyinitial-value problems on real intervals Thus in our numeri-cal calculations we need to consider initial-value problems inreal domain This is not a problem for (6) and (33) For (25)and (39) we calculate either the partial sum of the Maclaurinseries of solutions or we calculate functions

harrs119901(119909) and

harr

sh119901(119909) which come from real initial-value problems In ourgraphical outputs the solutions of real initial-value problemsare numerically approximated by the NDSolve command ofMathematica version 90 For the convenience of the readerwe provide some source code In Pseudocode 2 we list source

code for approximation of functionsharrs3(119909)harr

sh3(119909) and sin3119909Figure 1 compares graphs of sin3119909 and

harrs3(119909) for 119909 isin

(minus12058732 12058732) Figure 2 compares graphs ofharrs3(119909) and

328(119909) for 119909 isin (minus12058732 12058732) Here 328(119909) is partial sumof1198723(119909) up to the order 28 which is

328 (119909) = 119909 minus1199094

12minus1199097

252minus83 11990910

90 720minus1 817 119909

13

7 076 160

minus199 691 119909

16

2 377 589 760minus12 324 719 119909

19

406 567 848 960

minus22 008 573 061 119909

22

1 878 343 462 195 200

minus107 355 387 043 119909

25

22 540 121 546 342 400

minus89 152 153 354 993 119909

28

44 304 862 911 490 621 440

(68)

Since the difference |harrs3(119909) minus 328(119909)| varies in order of

several magnitudes throughout the radius of convergence12058732 we use logarithmic scale on the vertical axis

We can also compute the functionsharrs119901(119909) and

harr

sh119901(119909) byinverting formulas (8) and (36) for arcsin119901119909 and argsinh119901119909respectively It turns out that this approach provides moreprecision than solving differential equation and enables

computing values ofharrc119901(119909) and

harr

ch119901(119909) using identity (66)and identity (67) respectively We provide sample code for

computingharr

sh3(119909) for 119909 isin (12058732 5) ⊊ (12058732 +infin) in


12058734 12058732minus12058732 minus12058734

minus15

minus10

minus05

05

10

15

(a)

12058734 12058732minus12058732 minus12058734

01

02

minus02

minus01

(b)

Figure 1 (a) Real function sin3119909 (short-dashed line) versusharrs3(119909) (dashed line) (b) Plot of sin3119909 minus

harrs3(119909)

12058734 12058732minus12058732 minus12058734

05

10

minus10

minus05

(a)

0 12058734minus12058734 12058732

10minus15

10minus10

10minus5

10minus1

(b)

Figure 2 (a)harrs3(119909) versus 328 (the partial sum of1198723(119909) up to the order 28 which is (68)) (b) The logarithmic plot of |

harrs3(119909) minus 1198723(119909)|

Pseudocode 3 Analogously we wrote a code for computingharr

sh4(119909)harr

sh30(119909)harr

sh31(119909)harrs3(119909)harrs4(119909)harrs30(119909) and

harrs31(119909)

In the same way as we defined real function argsinh119901119909 by(36) we can define complex valued function argsinh119901119911 by

argsinh119901119911

def= int

119911

0

1

(1 + 119904119901)1119901

d119904

= 11991121198651 (

1

1199011

119901 1 +

1

119901 minus 119911119901) 119911 isin D ⊊ C

(69)

for any 119901 = 119898 + 2 Note that the integrand has poles at 119911satisfying 1 + 119911119901 = 0 Thus the function argsinh119901119911 is not anentire function In particular for 119901 = 2119898 + 1 there is a poleat 119911 = minus1 In Figure 3 we compare graphs of argsinh119901119909 for119901 = 2 3 31 and 119909 isin (minus3 3) ⊊ R with the restriction of thecomplex valued function argsinh119901119911 for 119901 = 2 3 31 where119911 = 119909 + 119894119910 where 119910 = 0 119909 isin (minus3 3) for 119901 = 2 and 119909 isin (minus1 3)for119901 = 3 31 andwith argsinh119901119909 = argsinh119901119911 for119901 = 2 4 30119909 isin (minus3 3) 119910 = 0 and 119911 = 119909 + 119894119910

In Figures 4 5 and 6 we compare graphs of real-valuedfunctions sinh119909 sinh4119909 sinh30119909 cosh119909 cosh4119909 cosh30119909


minus2

minus1

1

2

1 2 3minus2 minus1minus3

(a)


1

2

minus1

(b)


1

2

minus2

minus1

(c)

Figure 3 (a) argsinh119901119909(= 119909 21198651(1119901 1119901 1 + 1119901 minus|119909|119901)) for 119901 = 2 3 31 and 119909 isin (minus3 3) (b) argsinh119901119911(= 119911 21198651(1119901 1119901 1 + 1119901 minus119911

119901)) for

119901 = 2 3 31 Here 119911 = 119909 + 119894119910 where 119910 = 0 119909 isin (minus3 3) for 119901 = 2 and 119909 isin (minus1 3) for 119901 = 3 31 (c) argsinh119901119909 = argsinh119901119911 for 119901 = 2 4 30119909 isin (minus3 3) 119910 = 0 and 119911 = 119909 + 119894119910

In[2] = auxAgSh3[s NumberQ] =

s Hypergeometric2F1[13 13 1 + 13 -sand3]

( assigns definition to auxiliar function auxAgSh3 )

In[3] = sh3inv[x NumericQ] =

(s FindRoot[auxAgSh3[s] - x s 0 -1 20])

( assigns definition to function sh3inv )

In[4] = Plot[

sh3inv[x] x -pip[3] 2

PlotStyle -gt Thick PlotRange -gt -2 7]

( plots the graph )

Pseudocode 3 Mathematica v 90 code Function sh3inv returns values ofharr

sh3(119909) for 119909 isin (12058732 5) ⊊ (12058732 +infin) using inversion of theformula (36) for argsinh3119909 This approach provides more precision than solving differential equation


1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)

Figure 4 (a) sinh119909 (short-dashed line) sinh4119909 (dashed line) and sinh30119909 (solid line) (b) cosh119909 (short-dashed line) cosh4119909 (dashed line)and cosh30119909 (solid line)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)


(see Figure 4) sinh119909 sinh3119909 sinh31119909 cosh119909 cosh3119909

cosh31119909 (see Figure 5)harr

sh3(119909)harr

sh31(119909)harr

ch3(119909) andharr

ch31(119909)(see Figure 6)

Figure 7 illustrates the relation between sin3119909 sinh3119909harrs3(119909) and

harr

sh3(119909) which are due toTheorem 27It follows from identity (66) and identity (67) that

the pairs of functions (harrs119901(119909)harrc119901(119909)) and (

harr

sh119901(119909)harr

ch119901(119909))respectively are parametrizations of Lame curves restrictedto the first and fourth quadrant see [23 Book V Chapter V

pp 384ndash407] and [24] Since the Lame curves are frequentlyused in geometrical modeling we provide graphical compar-ison of the Lame curves and phase portraits of initial-valueproblems (25) and (39) in real domain on Figures 8 and 9respectively

8 Conclusion

We have discussed real and complex approaches of how todefine generalized trigonometric and hyperbolic functions


1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

1

2

3

minus3

minus2

minus1

(b)

Figure 6 (a) sinh119909 (short-dashed line)harr

sh3(119909) (dashed line) andharr

sh31(119909) (solid line) (b) cosh119909 (short-dashed line)harr

ch3(119909) (dashed line)

andharr

ch31(119909) (solid line) Note thatharr

sh119901(119909) andharr

ch119901(119909) are defined on (minus1205871199012 +infin) for integer 119901 gt 1 odd

2 4minus4 minus2

minus3

minus2

minus1

1

2

3

(a)

42minus4 minus2

minus3

minus2

minus1

1

2

3

(b)

Figure 7 (a) sin3119909 sinh3119909 andharrs3(119909) (b) sin3119909 sinh3119909 and

harr

sh3(119909)

The real approach is motivated by minimization of Rayleighquotient (see eg [1 Equation (34) p 51] and referencestherein)

int120587119901

0

10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

d119909

int120587119901

0|119906|119901 d119909

(70)

in 1198821119901

0(0 120587119901) This leads to (4) with 120582 = 119901 minus 1 and

to initial-value problem (6) in turn Thus from this pointof view the solution of (6) and its derivative can be seenas natural generalizations of the functions sine and cosineUnfortunately presence of absolute value in (6) does notallow for extension to complex domain for general 119901 gt 1


p = 2

u

minus4

minus2

0

2

4

p = 3

p = 4 p = 5

p = 10 p = 11

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

u

minus2

minus4

0

2

4

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

40 2minus2minus4

u998400

minus2 0 2 4minus4

u998400

Figure 8 Bold lines dependence of 119906 =harrs119901(119909) on 119906

1015840=harrc119901(119909) Both bold and thin lines dependence of restriction to real axes of derivative

of solution of (29) on the restriction to real axes of the solution itself (Lame curves)


p = 2

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 3

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 5

u

minus2

minus4

0

2

4

minus2 0 2 4minus4

u998400

p = 11

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

u

p = 10

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

Figure 9 Bold lines dependence of 119906 =harr

sh119901(119909) on 1199061015840=harr

ch119901(119909) Both bold and thin lines dependence of restriction to real axes of derivativeof solution of (44) on the restriction to real axes of the solution itself (Lame curves)


Table 1 Summary of results according to discussed functions and their domain

119901 isin Function Domain IVP Results

(1 +infin) sin119901119909 R (6) Propositions 2ndash5Theorems 6 7 8 23 and 26ndash28

(1 +infin) sinh119901119909 R (33) Theorems 23 and 26ndash28

N1 sin119901119911 119861119901 (25) Propositions 11 and 12 Lemma 13Theorems 6 7 8 9 15 20 and 21

N1 sinh119901119911 119863119901 (39) Lemmas 16 and 18Theorems 20 and 22

Table 2 Interrelations between 119901-trigonometric and 119901-hyperbolicfunctions Note that the relations must be symmetric

sin119901119909 sinh119901119909 sin119901119911 sinh119901119911sin119901119909 mdash (57) (63)ndash(64) mdash mdashsinh119901119909 sym mdash mdash mdashsin119901119911 sym sym (49) (46)sinh119901119911 sym sym sym (50)

It was shown in [2] that functions sin119901119909 are real analyticfunctions for any even integer 119901 gt 2 Moreover there is noneed to write absolute value in (6) for 119909 isin [minus1205871199012 1205871199012]

provided 119901 gt 2 is an even integerIt turns out that the relation between the real and complex

approach is not as smooth as in the classical case 119901 = 2 Thuswe summarize our results in Tables 1 and 2

We also discussed the Lame curves which are importantcurves in geometrical modeling We hope this will stimu-late interest in 119901-trigonometric and 119901-hyperbolic functionsamong the geometric-modeling community

Appendix

A Proof of Theorem 6

We will use the following result to proveTheorem 6

Proposition A1 (see [15] Theorem 24b on p 97) Let theformal power series 119865 def

= 1198861119909 + 11988621199092+ sdot sdot sdot 1198861 = 0 have a

positive radius of convergence The inversion 119865minus1 of 119865 then alsohas positive radius of convergence

Let us note that the term reversion of series is used in [15]instead of inversion of series (see [15] p 46)

Proposition A2 (see [25] Theorem 166 on p 352) If thesum of two power series in the variable 119911 minus 1199110 coincides on aset of points 119864 for which 1199110 is a limit point and 1199110 notin 119864 thenidentical powers of 119911 minus 1199110 have identical coefficients that isthere is a unique power series in the variable 119911 minus 1199110 which hasgiven sum on the set 119864

Proof of Theorem 6 Let us remember that 119879119901(119909) is given by(19) which is the formal inverse of arcsin119901119909 and 119872119901(119909) isgiven by formula (20) The idea is to prove that there exists120575119901 gt 0 small enough such that both series 119879119901(119909) and119872119901(119909)have the sum equal to uniquely defined value sin119901119909 at any 119909 isin[0 120575119901) Then 119879119901(119909) = 119872119901(119909) on [0 120575119901) and the assumptionsof Proposition A2 are satisfied on 119911119899 = 120575119901(119899 + 1) It followsthat119879119901(119909)has identical coefficients as119872119901(119909)has and so119879119901(119909)also converges on (minus1205871199012 1205871199012)

By Propositions 4 and 5 119872119901(119909) converges to sin119901119909 for119901 gt 2 even on (minus1205871199012 1205871199012) and for 119901 gt 1 odd on [0 1205871199012)respectively It remains to show that there exists 120590119901 gt 0 suchthat

119879119901 (119909) = sin119901119909 (A1)

on [0 120590119901) Since 119879119901(119909) is defined as formal inverse ofarcsin119901119909 (A1) holds on domain of convergence of 119879119901(119909)Since for 119909 isin [0 1]

arcsin119901119909 = 119909 sdot 21198651 (1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896 sdot 119901 + 1) sdot 119896 sdot Γ (1119901)119909119896119901+1

(A2)

where right-hand side series has radius of convergence equalto 1 hence the existence of 120590119901 le 1205871199012 is provided byProposition A1 and 120575119901 = 120590119901

B Proof of Theorem 7

By Theorem 6119872119901 = 119879119901 Hence we can prove the statementof Theorem 7 for 119879119901 instead of119872119901

Assume by contradiction that there exists 119886119899 = 0 for some119899 such that 119899 minus 1 is not divisible by 119901 For this purpose letus denote by 119887119895 the 119895th coefficient of the Maclaurin series ofarcsin119901 corresponding to 119895th power From (18) we get

119887119895 =

Γ (119897 + 1119901)

(119897119901 + 1) sdot 119897 sdot Γ (1119901)if 119895 = 119897 sdot 119901 + 1 for some 119897 isin N cup 0

0 otherwise(B1)


Since 119879119901 is the formal inverse series of

arcsin119901119909 =infin

sum

119895=1

119887119895 sdot 119909119895 (B2)

the coefficients 119886119899 can be computed from the formula

119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1

(minus1)1198981+1198982+sdotsdotsdot+119898119899minus1 sdot

119899 (119899 + 1) sdot sdot (119899 minus 1 + 1198981 + 1198982 + 1198983 + sdot sdot sdot + 119898119894 + sdot sdot sdot + 119898119899minus1)

119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983

sdot sdot sdot (119887119894+1

1198871)

119898119894

sdot sdot sdot (119887119899

1198871)

119898119899minus1

(B3)

where the summation is taken over all 1198981 1198982 1198983 isin N cup

0 such that

1198981 + 21198982 + 31198983 + sdot sdot sdot + 119894119898119894 sdot sdot sdot (119899 minus 1)119898119899minus1 = 119899 minus 1 (B4)

and if 119898119894 = 0 then the corresponding term (119887119894+11198871)119898119894 is

dropped from the product on the last line of (B3)Let us note that this procedure is fully described in [14]

p 411ndash413 and it requires that 1198871 = 0 Note that

1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)

by (B1)Now let us fix1198981 1198982 1198983 119898119899minus1 satisfying (B4) If 119887119894 =

0 and119898119894 = 0 for at least one 119894 = 1 2 3 119899minus1 the summandof sum (B3) corresponding to1198981 1198982 1198983 equals 0 Takinginto account (B1) 119887119894 = 0whenever 119894 = 119897119901+1 for any 119897 isin Ncup0This leads us to conclusion that nonzero terms in (B3) can beformed only from 119898119894rsquos where 119894 is divisible by 119901 Thus (B4)implies that the following equation must be satisfied

119901 sdot 119898119901 + 2119901 sdot 1198982119901 + 3119901 sdot 1198983119901 + sdot sdot sdot + 119897 sdot 119901 sdot 119898119897sdot119901 + sdot sdot sdot

+ 119896 sdot 119901 sdot 119898119896sdot119901 = 119899 minus 1

(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)

But here the left-hand side is a multiple of 119901 while the right-hand side 119899 minus 1 is not divisible by 119901 by our assumption Thisis a contradiction

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

Both the authors were partially supported by a joint exchangeprogram between the Czech Republic and Germany by

the Ministry of Education Youth and Sports of the CzechRepublic under Grant no 7AMB14DE005 (exchange pro-gram ldquoMobilityrdquo) and by the Federal Ministry of Educationand Research of Germany under Grant no 57063847 (DAADProgram ldquoPPPrdquo) Both the authors were partially supportedby the Grant Agency of the Czech Republic Project no 13-00863S

References

[1] J Lang and D Edmunds Eigenvalues Embeddings and Gen-eralised Trigonometric Functions vol 2016 of Lecture Notes inMathematics Springer Heidelberg Germany 2011

[2] P Girg and L Kotrla ldquoDifferentiability properties of p-trigonometric functionsrdquo in Proceedings of the Conferenceon Variational and Topological Methods Theory ApplicationsNumerical Simulations and Open Problems vol 21 of ElectronicJournal of Differential Equations Conference pp 101ndash127 SanMarcos Tex USA 2014

[3] P Girg and L Kotrla ldquoGeneralized trigonometric functions incomplex domainrdquo Mathematica Bohemica vol 140 no 2 pp223ndash239 2015

[4] A Elbert ldquoA half-linear second order differential equationrdquo inQualitative Theory of Differential Equations Vol I II (Szeged1979) vol 30 of Colloquia Mathematica Societatis Janos Bolyaipp 153ndash180 North-Holland Amsterdam The Netherlands1981

[5] M del Pino P Drabek and R Manasevich ldquoThe Fredholmalternative at the first eigenvalue for the one-dimensional p-Laplacian-Laplacianrdquo Journal of Differential Equations vol 151no 2 pp 386ndash419 1999

[6] M del Pino M Elgueta and R Manasevich ldquoA homotopicdeformation along p of a Leray-Schauder degree result andexistence for (|1199061015840|119901minus21205831015840) + 119891(119905 120583) = 0 120583(0) = 120583(119879) = 0 119901 gt 1rdquoJournal of Differential Equations vol 80 no 1 pp 1ndash13 1989

[7] P Lindqvist ldquoSome remarkable sine and cosine functionsrdquoRicerche di Matematica vol 44 no 2 pp 269ndash290 1995

[8] P Lindqvist and J Peetre ldquoTwo remarkable identities calledtwos for inverses to some Abelian integralsrdquo The AmericanMathematical Monthly vol 108 no 5 pp 403ndash410 2001

[9] W E Wood ldquoSquigonometryrdquo Mathematics Magazine vol 84no 4 pp 257ndash265 2011

[10] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs andMathematical Tables vol


55 of National Bureau of Standards Applied Mathemat-ics ForSale by the Superintendent of Documents US GovernmentPrinting Office Washington DC USA 1964

[11] G E Andrews R Askey and R Roy Special Functions vol 71of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1999

[12] J Benedikt P Girg and P Takac ldquoOn the Fredholm alternativefor the p-Laplacian at higher eigenvalues (in one dimension)rdquoNonlinear Analysis Theory Methods amp Applications vol 72 no6 pp 3091ndash3107 2010

[13] D E Edmunds P Gurka and J Lang ldquoProperties of generalizedtrigonometric functionsrdquo Journal of ApproximationTheory vol164 no 1 pp 47ndash56 2012

[14] P M Morse and H Feshbach Methods of Theoretical Physicsvol 2 McGraw-Hill New York NY USA 1953

[15] P Henrici Applied and Computational Complex Analysisvol 2 of Special Functions-Integral Transforms- Asymptotics-Continued Fractions John Wiley amp Sons New York NY USA1991 Reprint of the 1977 original

[16] R B BurckelAn Introduction to Classical ComplexAnalysis vol82 part 1 of Pure and Applied Mathematics Academic PressHarcourt Brace Jovanovich New York NY USA 1979

[17] AHorwitz P Vojta R IsraelH P Boas andBDubuqueEntireSolutions of f2+g2=110158401015840 The Math Forum Drexel UniversityPhiladelphia Pa USA 1996 httpmathforumorgkbmessagejspamessageID=21242

[18] A Baricz B A Bhayo and T K Pogany ldquoFunctional inequal-ities for generalized inverse trigonometric and hyperbolicfunctionsrdquo Journal of Mathematical Analysis and Applicationsvol 417 no 1 pp 244ndash259 2014

[19] D B Karp and E G Prilepkina ldquoParameter convexity andconcavity of generalized trigonometric functionsrdquo Journal ofMathematical Analysis and Applications vol 421 no 1 pp 370ndash382 2015

[20] R Klen M Vuorinen and X Zhang ldquoInequalities for thegeneralized trigonometric and hyperbolic functionsrdquo Journal ofMathematical Analysis and Applications vol 409 no 1 pp 521ndash529 2014

[21] C-Y Yang ldquoInequalities on generalized trigonometric andhyperbolic functionsrdquo Journal of Mathematical Analysis andApplications vol 419 no 2 pp 775ndash782 2014

[22] J Peetre ldquoThe differential equation 1199101015840119901 minus 119910119875 = plusmn(119901 gt 0)rdquoRicerche di Matematica vol 43 no 1 pp 91ndash128 1994

[23] G LoriaCurve Piane Speciali Algebriche e Trascendenti vol 1 ofCurve Algebriche of Teoria e Storia Ulrico Hoepli Milan Itlay1930 (Italian)

[24] G Lame Examen des Differentes Methodes Employees pourResoudre les Problemes de Geometrie Reimpression Fac-SimileLibrairie Scientifique A Hermann Libraire de S M Le Roi deSuede et de Norwege Paris France 1903 (French)

[25] A I MarkushevichTheory of Functions of a Complex Variablevol 1ndash3 of Translated and Edited by Richard A SilvermanChelsea Publishing New York NY USA 1977

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Indeed (6) has the unique solution inR see for example[5 LemmaA1] [6 Section 3] and [4] Denoting the solutionof (6) by sin119901119909 the set of all eigenvalues 120582119896 isin R andeigenfunctions 119906119896 isin 119882

1119901

0(0 120587119901) of (4) can be written as

120582119896 = (119901 minus 1) 119896119901

119906119896 (119909) = sin119901 (119896 sdot 119909)

where 119896 isin N

(7)

A piecewise construction of the solution of (6) wasprovided in [4] At first one sets

arcsin119901119909def= int

119909

0

1

(1 minus 119904119901)1119901

d119904 119909 isin [0 1] (8)

Then the restriction of sin119901119909 on [0 1205871199012] is the inversefunction to arcsin119901119909 For 119909 isin (1205871199012 120587119901] sin119901119909 satisfiessin119901119909 = sin119901(120587119901 minus 119909) where clearly 120587119901 minus 119909 isin [0 1205871199012) andsin119901119909 = minussin119901(minus119909) for 119909 isin [minus120587119901 0) Finally sin119901119909 is a 2120587119901-periodic function on R

We also extend arcsin119901119909 from (8) to [minus1 1] as an oddfunction Then it is the inverse function to the restriction ofsin119901119909 to [minus1205871199012 1205871199012] and we have

sin119901 (arcsin119901119909) = 119909 forall119909 isin [minus1 1] (9)

Finally let us define cos119901119909def= sin1015840

119901119909 for all 119909 isin R

Then the functions sin119901119909 and cos119901119909 satisfy the so-called 119901-trigonometric identity

10038161003816100381610038161003816cos119901119909

10038161003816100381610038161003816

119901

+10038161003816100381610038161003816sin119901119909

10038161003816100381610038161003816

119901

= 1 (10)

for all 119909 isin R see for example [4ndash6]Note that there is an alternative definition of ldquocos119901119909rdquo

in [7] andor [8] which leads to different ldquo119901-trigonometricrdquoidentity Yet another alternative generalization of trigono-metric and hyperbolic functions motivated by geometricalpoint of view was introduced in [9] Studies of relationsbetween their respective generalizations of 119901-trigonometricand 119901-hyperbolic functions were suggested in [7] and in [9]respectively

Remark 1 In the paper we use Gaussrsquo hypergeometric func-tion21198651(119886 119887 119888 119911) where 119886 119887 119888 isin C are parameters and 119911 isin C

is variable (for definition see [10 1511 p 556]) to expressintegrals of the type

int

119909

0

1

(1 plusmn 119904119901)1119901

d119904

int

119911

0

1

(1 plusmn 119904119901)1119901

d119904(11)

for 119901 gt 1 119909 isin R and 119911 isin C (by 1199111119901 we understand theprincipal branch thereof) Indeed

11991121198651 (

1

1199011

119901 1 +

1

119901 ∓119911119901) = int

119911

0

1

(1 plusmn 119904119901)1119901

d119904 (12)

for |119911| lt 1 (by comparing respective series expansions) Bythe uniqueness of analytic extension the equation is valid for119911 isin C 119909 + 119894119910 119909 gt 1 and 119910 = 0 (for analytic continuationof 21198651 see eg [10 1531 p 558] and [11 Theorem 221 p65])

In the definition of 120587119901 (ie (5)) and in (8) we need toevaluate integral

int

1

0

1

(1 minus 119904119901)1119901

d119904 (13)

By [11 Theorem 222 p 66] this is possible sinceR[119888 minus 119886 minus119887] = 1 + 1119901 minus 1119901 minus 1119901 = 1 minus 1119901 gt 0 for 119901 gt 1

Notation 1 This paper combines real variable and complexvariable approach to the 119901-trigonometric and 119901-hyperbolicfunctions Each of these approaches has its own natural wayof how to define the functions sin119901 and sinh119901 Thus we needto distinguish between real and complex definitions By sin119901119909and sinh119901119909 we mean functions defined by the real variableapproach and by sin119901119911 and sinh119901119911wemean functions definedby the complex variable approach throughout the paper

2 Real Analyticity Results for sin119901119909 and cos119901119909

It is well known that the 119901-trigonometric functions are notreal analytic functions in general see for example [12 13]Very detailed study of the degree of smoothness of therestriction of sin119901119909 to (minus1205871199012 1205871199012) was performed in [2]including the following two results The first one concernsldquogenericrdquo 119901 gt 1

Proposition 2 (see [2] Theorem 32 on p 105) Let 119901 isin R

2119898 119898 isin N 119901 gt 1 Then

sin119901119909 isin 119862lceil119901rceil(minus120587119901

2120587119901

2) (14)

but

sin119901119909 notin 119862lceil119901rceil+1

(minus120587119901

2120587119901

2) (15)

Here lceil119901rceil def= min119896 isin N 119896 ge 119901

The second result treats only the even integers 119901 gt 2 anddiffers significantly from the previous case in an unexpectedway

Proposition 3 (see [2]Theorem 31 on p 105) Let 119901 = 2(119898+1) 119898 isin N Then

sin119901119909 isin 119862infin(minus120587119901

2120587119901

2) (16)

Thus the Maclaurin series of sin119901119909

119872119901 (119909)def=

infin

sum

119899=1

1

119899sin(119899)119901(0) sdot 119909

119899 (17)

is well defined for 119901 = 2(119898 + 1) 119898 isin N Moreoverthe following result establishes an explicit expression for theradius of convergence of119872119901





arcsin119901119909 = int119909

0

1

(1 minus 119904119901)1119901

d119904

= 119909 sdot21198651 (

1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896119901 + 1) 119896Γ (1119901)119909119896119901+1

(18)



119879119901 (119909)def=

infin

sum

119899=1

119886119899 sdot 119909119899 (19)



119872119901 (119909)def=

infin

sum

119899=1

1

119899( lim119909rarr0+

sin(119899)119901119909) sdot 119909

119899 (20)






119886119899 =1

119899( lim119909rarr0+

sin(119899)119901119909) forall119899 isin N (21)







119872119901 (119909) =

infin

sum

119897=0

120572119897 sdot 119909119897119901+1

(22)


1205720 = 1

1205721 = minus1

119901 (119901 + 1)


21199012 (119901 + 1) (2119901 + 1)

(23)




119861119901def= 119911 isin C |119911| lt

120587119901

2 (24)




(1199061015840)119901minus2

11990610158401015840+ 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(25)




s 0 32]


Out[1] = 119904 +1199045

20+51199049

288+1511990413

1664+19511990417

34816+22111990421

57344+4641119904

25

1638400+16575119904

29

7602176+ 119874(119904

33)


Out[2] = 119904 minus1199045

20minus71199049

1440minus46311990413

374400minus211741119904

17

509184000minus104361161119904

21

641571840000minus

897899621311990425

128314368000000minus

799573586746311990429

248783228928000000+ 119874(119904

33)




1199061015840= V

V1015840 = minus119906119901minus1

V119901minus2

119906 (0) = 0

V (0) = 1

(26)







infin



119887119896 =1

119896( lim119909rarr0+

sin(119896)119901119909) (27)




sin119901119911def=

infin

sum

119897=0

120572119897 sdot 119911119897sdot119901+1

cos119901119911def= sin1015840119901119911 =

dd119911

sin119901119911

(28)









(1199061015840(119911))119901

+ (119906 (119911))119901= 1 (29)

on the disc 119863119903


(1199061015840(119911))119901

minus (1199061015840(0))119901

+ (119906 (119911))119901minus (119906 (0))

119901= 0 (30)





119891119899+ 119892119899= 1 (31)







cos119901119901119911 + sin119901

119901119911 = 1 (32)






minus (10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901minus2

1199061015840)1015840

+ (119901 minus 1) |119906|119901minus2

119906 = 0 119909 isin R

119906 (0) = 0

1199061015840(0) = 1

(33)



1 + |119906|119901=10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

(34)


1199061015840= (1 + |119906|

119901)1119901 (35)



argsinh119901119909

def= int

119909

0

1

(1 + |119904|119901)1119901

d119904 119909 isin R (36)




argsinh119901119909

=

119909 sdot21198651 (

1

1199011

119901 1 +

1

119901 minus119909119901) 119909 isin [0 +infin)


(37)





sin 119911 sinh 11991111990610158401015840+ 119906 = 0 119906

10158401015840minus 119906 = 0

119906 (0) = 0 119906 (0) = 0

1199061015840(0) = 1 119906

1015840(0) = 1


(38)




(1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(39)


sin119901119911 sinh119901119911(1199061015840)119901minus2

11990610158401015840+ 119906119901minus1

= 0 (1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0 119906 (0) = 0

1199061015840(0) = 1 119906

1015840(0) = 1


(40)





119863119901def= 119911 isin C |119911| lt 120574119901 (42)




1199061015840= V

V1015840 =119906119901minus1

V119901minus2

119906 (0) = 0

V (0) = 1

(43)





119901119911 for all 119911 isin 119863119901



(1199061015840(119911))119901

minus (119906 (119911))119901= 1 (44)

on the disc 119863119903



(119906 (119909)1015840)119901

minus (119906 (119909))119901= 1 (45)





cos119901119911 = cosh119901 (119894 sdot 119911) (47)


sinh119901119911 =infin

sum

119896=0

(minus1)119896sdot 120572119896 sdot 119911

119896119901+1 (48)




cos119901119911 = cos119901 (119894 sdot 119911)(49)

for all 119911 isin 119861119901




cosh119901119911 = cosh119901 (119894 sdot 119911) (51)

for all 119911 isin 119863119901





(1199081015840(119911))119901minus2

11990810158401015840(119911) + 119908 (119911)

119901minus1

= (dd119911119908 (119911))

119901minus2 d2

d1199112119908 (119911) + 119908 (119911)

119901minus1


119901minus2 d2

d1199112(minus119894 sdot 119906 (119894 sdot 119911))


= 119894 sdot [d

d (119894 sdot 119911)119906 (119894 sdot 119911)]

119901minus2 d2

d (119894 sdot 119911)2119906 (119894 sdot 119911)


= 119894 sdot [dd119904119906 (119904)]

119901minus2 d2


119901minus2sdot 119906 (119904)119901minus1

= 119894 sdot ([dd119904119906 (119904)]

119901minus2 d2

d1199042119906 (119904) minus (minus119894)

119901minus2sdot 119906 (119904)119901minus1)

= 119894 sdot ([dd119904119906 (119904)]

119901minus2 d2

d1199042119906 (119904) minus 119906 (119904)

119901minus1) = 0

(52)



infin

119899=1119888119899 sdot


sum

119899=1


infin

sum

119899=1

119888119899 sdot 119894119899minus1

sdot 119911119899 (53)

on119863119901 cap 119861119901 From here

119888119899 =119886119899

119894119899minus1= 1198943119899+1

sdot 119886119899 (54)



119888119896119901+1 = (119894)3(119896119901+1)+1

120572119896 = (minus1)119896120572119896 (55)


119901119911 and

(46)







sinh119901119911 =infin

sum

119896=1

119888119896119911119896 119911 isin 119863119901 (56)





sh119901 (119909) =

sinh119901119909 119909 isin [0120587119901

2)

sin119901119909 119909 isin (minus120587119901

2 0)

(57)


119901119909 ge 0 on [0 120574119901)



minus (10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901minus2

1199061015840)1015840


119906 = 0

119906 (0) = 0

1199061015840(0) = 1

(58)



(1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(59)



sin119901119909 =infin

sum

119896=1

119888119896119909119896= sh119901 (119909)


2 0)

(60)


119896=0120572119896 sdot 119909 sdot |119909|


infin

sum

119896=0

120572119896 sdot 119909 sdot |119909|119896119901=

infin

sum

119896=0

120572119896 sdot (minus1)119896119901sdot 119909119896119901+1

= 119866 (119909) (61)


infin

sum

119896=1

119888119896119909119896= 119866 (119909) on (minus120574119901 0) (62)








s119901 (119909) =

sin119901119909 119909 isin [0120587119901

2)

sinh119901119909 119909 isin (minus120587119901

2 0)

(63)



harrs119901(119909)

andharr



def= (dd119909)

harrs119901(119909)

andharr

ch119901(119909)def= (dd119909)

harr

sh119901(119909)


harrs119901119909 =


sin119901119909 119909 isin [0120587119901

2)

harr

sh119901119909 =

sin119901119909 119909 isin (minus120587119901

2 0)

sinh119901119909 119909 isin [0 +infin)

(64)


harrs119901119909 = sin119901119909 119909 isin R

harr

sh119901119909 = sinh119901119909 119909 isin R

(65)







harrs119901(119909) and



(harrs119901 (119909))

119901

+ (harrc119901 (119909))

119901

= 1 (66)


In[1] = pip[p ] =




Out[1] =120587csc(120587119901)

119901

In[2] = s3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == - 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -5 pip[3]][[1]])


In[3] = sh3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -pip[3] 5][[1]])


In[4] = sin3[x ] = (u[x]


v[x] == -2 Abs[u[x]]u[x]

u[0] == 0 v[0] == 1

uv x -4 pip[3] 4 pip[3]][[1]])



1199043(119909) for




sh119901(119909) andharr


(harr

ch119901 (119909))119901

minus (harr

sh119901 (119909))119901

= 1 (67)

7 Visualizations


harrs119901(119909) and

harr







328 (119909) = 119909 minus1199094

12minus1199097

252minus83 11990910

90 720minus1 817 119909

13

7 076 160

minus199 691 119909

16

2 377 589 760minus12 324 719 119909

19

406 567 848 960

minus22 008 573 061 119909

22

1 878 343 462 195 200

minus107 355 387 043 119909

25

22 540 121 546 342 400

minus89 152 153 354 993 119909

28

44 304 862 911 490 621 440

(68)




harr



harr


computingharr



12058734 12058732minus12058732 minus12058734

minus15

minus10

minus05

05

10

15

(a)

12058734 12058732minus12058732 minus12058734

01

02

minus02

minus01

(b)


harrs3(119909)

12058734 12058732minus12058732 minus12058734

05

10

minus10

minus05

(a)

0 12058734minus12058734 12058732

10minus15

10minus10

10minus5

10minus1

(b)


harrs3(119909) minus 1198723(119909)|


sh4(119909)harr

sh30(119909)harr


harrs31(119909)


argsinh119901119911

def= int

119911

0

1

(1 + 119904119901)1119901

d119904

= 11991121198651 (

1

1199011

119901 1 +

1

119901 minus 119911119901) 119911 isin D ⊊ C

(69)




minus2

minus1

1

2


(a)


1

2

minus1

(b)


1

2

minus2

minus1

(c)


119901)) for








In[4] = Plot[



( plots the graph )




1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)


1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)




sh3(119909)harr

sh31(119909)harr

ch3(119909) andharr



harr



harr

sh119901(119909)harr



8 Conclusion



1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

1

2

3

minus3

minus2

minus1

(b)





andharr


sh119901(119909) andharr


2 4minus4 minus2

minus3

minus2

minus1

1

2

3

(a)

42minus4 minus2

minus3

minus2

minus1

1

2

3

(b)


harr

sh3(119909)


int120587119901

0

10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

d119909

int120587119901

0|119906|119901 d119909

(70)

in 1198821119901




p = 2

u

minus4

minus2

0

2

4

p = 3

p = 4 p = 5

p = 10 p = 11

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

u

minus2

minus4

0

2

4

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

40 2minus2minus4

u998400

minus2 0 2 4minus4

u998400





p = 2

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 3

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 5

u

minus2

minus4

0

2

4

minus2 0 2 4minus4

u998400

p = 11

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

u

p = 10

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400


sh119901(119909) on 1199061015840=harr















Appendix










119879119901 (119909) = sin119901119909 (A1)


arcsin119901119909 = 119909 sdot 21198651 (1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896 sdot 119901 + 1) sdot 119896 sdot Γ (1119901)119909119896119901+1

(A2)





119887119895 =

Γ (119897 + 1119901)


0 otherwise(B1)




sum

119895=1

119887119895 sdot 119909119895 (B2)


119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1



119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983


1198871)

119898119894


1198871)

119898119899minus1

(B3)


0 such that





1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)





(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)


Competing Interests


Acknowledgments



References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













arcsin119901119909 = int119909

0

1

(1 minus 119904119901)1119901

d119904

= 119909 sdot21198651 (

1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896119901 + 1) 119896Γ (1119901)119909119896119901+1

(18)



119879119901 (119909)def=

infin

sum

119899=1

119886119899 sdot 119909119899 (19)



119872119901 (119909)def=

infin

sum

119899=1

1

119899( lim119909rarr0+

sin(119899)119901119909) sdot 119909

119899 (20)






119886119899 =1

119899( lim119909rarr0+

sin(119899)119901119909) forall119899 isin N (21)







119872119901 (119909) =

infin

sum

119897=0

120572119897 sdot 119909119897119901+1

(22)


1205720 = 1

1205721 = minus1

119901 (119901 + 1)


21199012 (119901 + 1) (2119901 + 1)

(23)




119861119901def= 119911 isin C |119911| lt

120587119901

2 (24)




(1199061015840)119901minus2

11990610158401015840+ 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(25)




s 0 32]


Out[1] = 119904 +1199045

20+51199049

288+1511990413

1664+19511990417

34816+22111990421

57344+4641119904

25

1638400+16575119904

29

7602176+ 119874(119904

33)


Out[2] = 119904 minus1199045

20minus71199049

1440minus46311990413

374400minus211741119904

17

509184000minus104361161119904

21

641571840000minus

897899621311990425

128314368000000minus

799573586746311990429

248783228928000000+ 119874(119904

33)




1199061015840= V

V1015840 = minus119906119901minus1

V119901minus2

119906 (0) = 0

V (0) = 1

(26)







infin



119887119896 =1

119896( lim119909rarr0+

sin(119896)119901119909) (27)




sin119901119911def=

infin

sum

119897=0

120572119897 sdot 119911119897sdot119901+1

cos119901119911def= sin1015840119901119911 =

dd119911

sin119901119911

(28)









(1199061015840(119911))119901

+ (119906 (119911))119901= 1 (29)

on the disc 119863119903


(1199061015840(119911))119901

minus (1199061015840(0))119901

+ (119906 (119911))119901minus (119906 (0))

119901= 0 (30)





119891119899+ 119892119899= 1 (31)







cos119901119901119911 + sin119901

119901119911 = 1 (32)






minus (10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901minus2

1199061015840)1015840

+ (119901 minus 1) |119906|119901minus2

119906 = 0 119909 isin R

119906 (0) = 0

1199061015840(0) = 1

(33)



1 + |119906|119901=10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

(34)


1199061015840= (1 + |119906|

119901)1119901 (35)



argsinh119901119909

def= int

119909

0

1

(1 + |119904|119901)1119901

d119904 119909 isin R (36)




argsinh119901119909

=

119909 sdot21198651 (

1

1199011

119901 1 +

1

119901 minus119909119901) 119909 isin [0 +infin)


(37)





sin 119911 sinh 11991111990610158401015840+ 119906 = 0 119906

10158401015840minus 119906 = 0

119906 (0) = 0 119906 (0) = 0

1199061015840(0) = 1 119906

1015840(0) = 1


(38)




(1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(39)


sin119901119911 sinh119901119911(1199061015840)119901minus2

11990610158401015840+ 119906119901minus1

= 0 (1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0 119906 (0) = 0

1199061015840(0) = 1 119906

1015840(0) = 1


(40)





119863119901def= 119911 isin C |119911| lt 120574119901 (42)




1199061015840= V

V1015840 =119906119901minus1

V119901minus2

119906 (0) = 0

V (0) = 1

(43)





119901119911 for all 119911 isin 119863119901



(1199061015840(119911))119901

minus (119906 (119911))119901= 1 (44)

on the disc 119863119903



(119906 (119909)1015840)119901

minus (119906 (119909))119901= 1 (45)





cos119901119911 = cosh119901 (119894 sdot 119911) (47)


sinh119901119911 =infin

sum

119896=0

(minus1)119896sdot 120572119896 sdot 119911

119896119901+1 (48)




cos119901119911 = cos119901 (119894 sdot 119911)(49)

for all 119911 isin 119861119901




cosh119901119911 = cosh119901 (119894 sdot 119911) (51)

for all 119911 isin 119863119901





(1199081015840(119911))119901minus2

11990810158401015840(119911) + 119908 (119911)

119901minus1

= (dd119911119908 (119911))

119901minus2 d2

d1199112119908 (119911) + 119908 (119911)

119901minus1


119901minus2 d2

d1199112(minus119894 sdot 119906 (119894 sdot 119911))


= 119894 sdot [d

d (119894 sdot 119911)119906 (119894 sdot 119911)]

119901minus2 d2

d (119894 sdot 119911)2119906 (119894 sdot 119911)


= 119894 sdot [dd119904119906 (119904)]

119901minus2 d2


119901minus2sdot 119906 (119904)119901minus1

= 119894 sdot ([dd119904119906 (119904)]

119901minus2 d2

d1199042119906 (119904) minus (minus119894)

119901minus2sdot 119906 (119904)119901minus1)

= 119894 sdot ([dd119904119906 (119904)]

119901minus2 d2

d1199042119906 (119904) minus 119906 (119904)

119901minus1) = 0

(52)



infin

119899=1119888119899 sdot


sum

119899=1


infin

sum

119899=1

119888119899 sdot 119894119899minus1

sdot 119911119899 (53)

on119863119901 cap 119861119901 From here

119888119899 =119886119899

119894119899minus1= 1198943119899+1

sdot 119886119899 (54)



119888119896119901+1 = (119894)3(119896119901+1)+1

120572119896 = (minus1)119896120572119896 (55)


119901119911 and

(46)







sinh119901119911 =infin

sum

119896=1

119888119896119911119896 119911 isin 119863119901 (56)





sh119901 (119909) =

sinh119901119909 119909 isin [0120587119901

2)

sin119901119909 119909 isin (minus120587119901

2 0)

(57)


119901119909 ge 0 on [0 120574119901)



minus (10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901minus2

1199061015840)1015840


119906 = 0

119906 (0) = 0

1199061015840(0) = 1

(58)



(1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(59)



sin119901119909 =infin

sum

119896=1

119888119896119909119896= sh119901 (119909)


2 0)

(60)


119896=0120572119896 sdot 119909 sdot |119909|


infin

sum

119896=0

120572119896 sdot 119909 sdot |119909|119896119901=

infin

sum

119896=0

120572119896 sdot (minus1)119896119901sdot 119909119896119901+1

= 119866 (119909) (61)


infin

sum

119896=1

119888119896119909119896= 119866 (119909) on (minus120574119901 0) (62)








s119901 (119909) =

sin119901119909 119909 isin [0120587119901

2)

sinh119901119909 119909 isin (minus120587119901

2 0)

(63)



harrs119901(119909)

andharr



def= (dd119909)

harrs119901(119909)

andharr

ch119901(119909)def= (dd119909)

harr

sh119901(119909)


harrs119901119909 =


sin119901119909 119909 isin [0120587119901

2)

harr

sh119901119909 =

sin119901119909 119909 isin (minus120587119901

2 0)

sinh119901119909 119909 isin [0 +infin)

(64)


harrs119901119909 = sin119901119909 119909 isin R

harr

sh119901119909 = sinh119901119909 119909 isin R

(65)







harrs119901(119909) and



(harrs119901 (119909))

119901

+ (harrc119901 (119909))

119901

= 1 (66)


In[1] = pip[p ] =




Out[1] =120587csc(120587119901)

119901

In[2] = s3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == - 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -5 pip[3]][[1]])


In[3] = sh3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -pip[3] 5][[1]])


In[4] = sin3[x ] = (u[x]


v[x] == -2 Abs[u[x]]u[x]

u[0] == 0 v[0] == 1

uv x -4 pip[3] 4 pip[3]][[1]])



1199043(119909) for




sh119901(119909) andharr


(harr

ch119901 (119909))119901

minus (harr

sh119901 (119909))119901

= 1 (67)

7 Visualizations


harrs119901(119909) and

harr







328 (119909) = 119909 minus1199094

12minus1199097

252minus83 11990910

90 720minus1 817 119909

13

7 076 160

minus199 691 119909

16

2 377 589 760minus12 324 719 119909

19

406 567 848 960

minus22 008 573 061 119909

22

1 878 343 462 195 200

minus107 355 387 043 119909

25

22 540 121 546 342 400

minus89 152 153 354 993 119909

28

44 304 862 911 490 621 440

(68)




harr



harr


computingharr



12058734 12058732minus12058732 minus12058734

minus15

minus10

minus05

05

10

15

(a)

12058734 12058732minus12058732 minus12058734

01

02

minus02

minus01

(b)


harrs3(119909)

12058734 12058732minus12058732 minus12058734

05

10

minus10

minus05

(a)

0 12058734minus12058734 12058732

10minus15

10minus10

10minus5

10minus1

(b)


harrs3(119909) minus 1198723(119909)|


sh4(119909)harr

sh30(119909)harr


harrs31(119909)


argsinh119901119911

def= int

119911

0

1

(1 + 119904119901)1119901

d119904

= 11991121198651 (

1

1199011

119901 1 +

1

119901 minus 119911119901) 119911 isin D ⊊ C

(69)




minus2

minus1

1

2


(a)


1

2

minus1

(b)


1

2

minus2

minus1

(c)


119901)) for








In[4] = Plot[



( plots the graph )




1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)


1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)




sh3(119909)harr

sh31(119909)harr

ch3(119909) andharr



harr



harr

sh119901(119909)harr



8 Conclusion



1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

1

2

3

minus3

minus2

minus1

(b)





andharr


sh119901(119909) andharr


2 4minus4 minus2

minus3

minus2

minus1

1

2

3

(a)

42minus4 minus2

minus3

minus2

minus1

1

2

3

(b)


harr

sh3(119909)


int120587119901

0

10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

d119909

int120587119901

0|119906|119901 d119909

(70)

in 1198821119901




p = 2

u

minus4

minus2

0

2

4

p = 3

p = 4 p = 5

p = 10 p = 11

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

u

minus2

minus4

0

2

4

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

40 2minus2minus4

u998400

minus2 0 2 4minus4

u998400





p = 2

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 3

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 5

u

minus2

minus4

0

2

4

minus2 0 2 4minus4

u998400

p = 11

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

u

p = 10

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400


sh119901(119909) on 1199061015840=harr















Appendix










119879119901 (119909) = sin119901119909 (A1)


arcsin119901119909 = 119909 sdot 21198651 (1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896 sdot 119901 + 1) sdot 119896 sdot Γ (1119901)119909119896119901+1

(A2)





119887119895 =

Γ (119897 + 1119901)


0 otherwise(B1)




sum

119895=1

119887119895 sdot 119909119895 (B2)


119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1



119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983


1198871)

119898119894


1198871)

119898119899minus1

(B3)


0 such that





1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)





(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)


Competing Interests


Acknowledgments



References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











s 0 32]


Out[1] = 119904 +1199045

20+51199049

288+1511990413

1664+19511990417

34816+22111990421

57344+4641119904

25

1638400+16575119904

29

7602176+ 119874(119904

33)


Out[2] = 119904 minus1199045

20minus71199049

1440minus46311990413

374400minus211741119904

17

509184000minus104361161119904

21

641571840000minus

897899621311990425

128314368000000minus

799573586746311990429

248783228928000000+ 119874(119904

33)




1199061015840= V

V1015840 = minus119906119901minus1

V119901minus2

119906 (0) = 0

V (0) = 1

(26)







infin



119887119896 =1

119896( lim119909rarr0+

sin(119896)119901119909) (27)




sin119901119911def=

infin

sum

119897=0

120572119897 sdot 119911119897sdot119901+1

cos119901119911def= sin1015840119901119911 =

dd119911

sin119901119911

(28)









(1199061015840(119911))119901

+ (119906 (119911))119901= 1 (29)

on the disc 119863119903


(1199061015840(119911))119901

minus (1199061015840(0))119901

+ (119906 (119911))119901minus (119906 (0))

119901= 0 (30)





119891119899+ 119892119899= 1 (31)







cos119901119901119911 + sin119901

119901119911 = 1 (32)






minus (10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901minus2

1199061015840)1015840

+ (119901 minus 1) |119906|119901minus2

119906 = 0 119909 isin R

119906 (0) = 0

1199061015840(0) = 1

(33)



1 + |119906|119901=10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

(34)


1199061015840= (1 + |119906|

119901)1119901 (35)



argsinh119901119909

def= int

119909

0

1

(1 + |119904|119901)1119901

d119904 119909 isin R (36)




argsinh119901119909

=

119909 sdot21198651 (

1

1199011

119901 1 +

1

119901 minus119909119901) 119909 isin [0 +infin)


(37)





sin 119911 sinh 11991111990610158401015840+ 119906 = 0 119906

10158401015840minus 119906 = 0

119906 (0) = 0 119906 (0) = 0

1199061015840(0) = 1 119906

1015840(0) = 1


(38)




(1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(39)


sin119901119911 sinh119901119911(1199061015840)119901minus2

11990610158401015840+ 119906119901minus1

= 0 (1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0 119906 (0) = 0

1199061015840(0) = 1 119906

1015840(0) = 1


(40)





119863119901def= 119911 isin C |119911| lt 120574119901 (42)




1199061015840= V

V1015840 =119906119901minus1

V119901minus2

119906 (0) = 0

V (0) = 1

(43)





119901119911 for all 119911 isin 119863119901



(1199061015840(119911))119901

minus (119906 (119911))119901= 1 (44)

on the disc 119863119903



(119906 (119909)1015840)119901

minus (119906 (119909))119901= 1 (45)





cos119901119911 = cosh119901 (119894 sdot 119911) (47)


sinh119901119911 =infin

sum

119896=0

(minus1)119896sdot 120572119896 sdot 119911

119896119901+1 (48)




cos119901119911 = cos119901 (119894 sdot 119911)(49)

for all 119911 isin 119861119901




cosh119901119911 = cosh119901 (119894 sdot 119911) (51)

for all 119911 isin 119863119901





(1199081015840(119911))119901minus2

11990810158401015840(119911) + 119908 (119911)

119901minus1

= (dd119911119908 (119911))

119901minus2 d2

d1199112119908 (119911) + 119908 (119911)

119901minus1


119901minus2 d2

d1199112(minus119894 sdot 119906 (119894 sdot 119911))


= 119894 sdot [d

d (119894 sdot 119911)119906 (119894 sdot 119911)]

119901minus2 d2

d (119894 sdot 119911)2119906 (119894 sdot 119911)


= 119894 sdot [dd119904119906 (119904)]

119901minus2 d2


119901minus2sdot 119906 (119904)119901minus1

= 119894 sdot ([dd119904119906 (119904)]

119901minus2 d2

d1199042119906 (119904) minus (minus119894)

119901minus2sdot 119906 (119904)119901minus1)

= 119894 sdot ([dd119904119906 (119904)]

119901minus2 d2

d1199042119906 (119904) minus 119906 (119904)

119901minus1) = 0

(52)



infin

119899=1119888119899 sdot


sum

119899=1


infin

sum

119899=1

119888119899 sdot 119894119899minus1

sdot 119911119899 (53)

on119863119901 cap 119861119901 From here

119888119899 =119886119899

119894119899minus1= 1198943119899+1

sdot 119886119899 (54)



119888119896119901+1 = (119894)3(119896119901+1)+1

120572119896 = (minus1)119896120572119896 (55)


119901119911 and

(46)







sinh119901119911 =infin

sum

119896=1

119888119896119911119896 119911 isin 119863119901 (56)





sh119901 (119909) =

sinh119901119909 119909 isin [0120587119901

2)

sin119901119909 119909 isin (minus120587119901

2 0)

(57)


119901119909 ge 0 on [0 120574119901)



minus (10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901minus2

1199061015840)1015840


119906 = 0

119906 (0) = 0

1199061015840(0) = 1

(58)



(1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(59)



sin119901119909 =infin

sum

119896=1

119888119896119909119896= sh119901 (119909)


2 0)

(60)


119896=0120572119896 sdot 119909 sdot |119909|


infin

sum

119896=0

120572119896 sdot 119909 sdot |119909|119896119901=

infin

sum

119896=0

120572119896 sdot (minus1)119896119901sdot 119909119896119901+1

= 119866 (119909) (61)


infin

sum

119896=1

119888119896119909119896= 119866 (119909) on (minus120574119901 0) (62)








s119901 (119909) =

sin119901119909 119909 isin [0120587119901

2)

sinh119901119909 119909 isin (minus120587119901

2 0)

(63)



harrs119901(119909)

andharr



def= (dd119909)

harrs119901(119909)

andharr

ch119901(119909)def= (dd119909)

harr

sh119901(119909)


harrs119901119909 =


sin119901119909 119909 isin [0120587119901

2)

harr

sh119901119909 =

sin119901119909 119909 isin (minus120587119901

2 0)

sinh119901119909 119909 isin [0 +infin)

(64)


harrs119901119909 = sin119901119909 119909 isin R

harr

sh119901119909 = sinh119901119909 119909 isin R

(65)







harrs119901(119909) and



(harrs119901 (119909))

119901

+ (harrc119901 (119909))

119901

= 1 (66)


In[1] = pip[p ] =




Out[1] =120587csc(120587119901)

119901

In[2] = s3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == - 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -5 pip[3]][[1]])


In[3] = sh3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -pip[3] 5][[1]])


In[4] = sin3[x ] = (u[x]


v[x] == -2 Abs[u[x]]u[x]

u[0] == 0 v[0] == 1

uv x -4 pip[3] 4 pip[3]][[1]])



1199043(119909) for




sh119901(119909) andharr


(harr

ch119901 (119909))119901

minus (harr

sh119901 (119909))119901

= 1 (67)

7 Visualizations


harrs119901(119909) and

harr







328 (119909) = 119909 minus1199094

12minus1199097

252minus83 11990910

90 720minus1 817 119909

13

7 076 160

minus199 691 119909

16

2 377 589 760minus12 324 719 119909

19

406 567 848 960

minus22 008 573 061 119909

22

1 878 343 462 195 200

minus107 355 387 043 119909

25

22 540 121 546 342 400

minus89 152 153 354 993 119909

28

44 304 862 911 490 621 440

(68)




harr



harr


computingharr



12058734 12058732minus12058732 minus12058734

minus15

minus10

minus05

05

10

15

(a)

12058734 12058732minus12058732 minus12058734

01

02

minus02

minus01

(b)


harrs3(119909)

12058734 12058732minus12058732 minus12058734

05

10

minus10

minus05

(a)

0 12058734minus12058734 12058732

10minus15

10minus10

10minus5

10minus1

(b)


harrs3(119909) minus 1198723(119909)|


sh4(119909)harr

sh30(119909)harr


harrs31(119909)


argsinh119901119911

def= int

119911

0

1

(1 + 119904119901)1119901

d119904

= 11991121198651 (

1

1199011

119901 1 +

1

119901 minus 119911119901) 119911 isin D ⊊ C

(69)




minus2

minus1

1

2


(a)


1

2

minus1

(b)


1

2

minus2

minus1

(c)


119901)) for








In[4] = Plot[



( plots the graph )




1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)


1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)




sh3(119909)harr

sh31(119909)harr

ch3(119909) andharr



harr



harr

sh119901(119909)harr



8 Conclusion



1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

1

2

3

minus3

minus2

minus1

(b)





andharr


sh119901(119909) andharr


2 4minus4 minus2

minus3

minus2

minus1

1

2

3

(a)

42minus4 minus2

minus3

minus2

minus1

1

2

3

(b)


harr

sh3(119909)


int120587119901

0

10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

d119909

int120587119901

0|119906|119901 d119909

(70)

in 1198821119901




p = 2

u

minus4

minus2

0

2

4

p = 3

p = 4 p = 5

p = 10 p = 11

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

u

minus2

minus4

0

2

4

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

40 2minus2minus4

u998400

minus2 0 2 4minus4

u998400





p = 2

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 3

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 5

u

minus2

minus4

0

2

4

minus2 0 2 4minus4

u998400

p = 11

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

u

p = 10

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400


sh119901(119909) on 1199061015840=harr















Appendix










119879119901 (119909) = sin119901119909 (A1)


arcsin119901119909 = 119909 sdot 21198651 (1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896 sdot 119901 + 1) sdot 119896 sdot Γ (1119901)119909119896119901+1

(A2)





119887119895 =

Γ (119897 + 1119901)


0 otherwise(B1)




sum

119895=1

119887119895 sdot 119909119895 (B2)


119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1



119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983


1198871)

119898119894


1198871)

119898119899minus1

(B3)


0 such that





1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)





(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)


Competing Interests


Acknowledgments



References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119891119899+ 119892119899= 1 (31)







cos119901119901119911 + sin119901

119901119911 = 1 (32)






minus (10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901minus2

1199061015840)1015840

+ (119901 minus 1) |119906|119901minus2

119906 = 0 119909 isin R

119906 (0) = 0

1199061015840(0) = 1

(33)



1 + |119906|119901=10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

(34)


1199061015840= (1 + |119906|

119901)1119901 (35)



argsinh119901119909

def= int

119909

0

1

(1 + |119904|119901)1119901

d119904 119909 isin R (36)




argsinh119901119909

=

119909 sdot21198651 (

1

1199011

119901 1 +

1

119901 minus119909119901) 119909 isin [0 +infin)


(37)





sin 119911 sinh 11991111990610158401015840+ 119906 = 0 119906

10158401015840minus 119906 = 0

119906 (0) = 0 119906 (0) = 0

1199061015840(0) = 1 119906

1015840(0) = 1


(38)




(1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(39)


sin119901119911 sinh119901119911(1199061015840)119901minus2

11990610158401015840+ 119906119901minus1

= 0 (1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0 119906 (0) = 0

1199061015840(0) = 1 119906

1015840(0) = 1


(40)





119863119901def= 119911 isin C |119911| lt 120574119901 (42)




1199061015840= V

V1015840 =119906119901minus1

V119901minus2

119906 (0) = 0

V (0) = 1

(43)





119901119911 for all 119911 isin 119863119901



(1199061015840(119911))119901

minus (119906 (119911))119901= 1 (44)

on the disc 119863119903



(119906 (119909)1015840)119901

minus (119906 (119909))119901= 1 (45)





cos119901119911 = cosh119901 (119894 sdot 119911) (47)


sinh119901119911 =infin

sum

119896=0

(minus1)119896sdot 120572119896 sdot 119911

119896119901+1 (48)




cos119901119911 = cos119901 (119894 sdot 119911)(49)

for all 119911 isin 119861119901




cosh119901119911 = cosh119901 (119894 sdot 119911) (51)

for all 119911 isin 119863119901





(1199081015840(119911))119901minus2

11990810158401015840(119911) + 119908 (119911)

119901minus1

= (dd119911119908 (119911))

119901minus2 d2

d1199112119908 (119911) + 119908 (119911)

119901minus1


119901minus2 d2

d1199112(minus119894 sdot 119906 (119894 sdot 119911))


= 119894 sdot [d

d (119894 sdot 119911)119906 (119894 sdot 119911)]

119901minus2 d2

d (119894 sdot 119911)2119906 (119894 sdot 119911)


= 119894 sdot [dd119904119906 (119904)]

119901minus2 d2


119901minus2sdot 119906 (119904)119901minus1

= 119894 sdot ([dd119904119906 (119904)]

119901minus2 d2

d1199042119906 (119904) minus (minus119894)

119901minus2sdot 119906 (119904)119901minus1)

= 119894 sdot ([dd119904119906 (119904)]

119901minus2 d2

d1199042119906 (119904) minus 119906 (119904)

119901minus1) = 0

(52)



infin

119899=1119888119899 sdot


sum

119899=1


infin

sum

119899=1

119888119899 sdot 119894119899minus1

sdot 119911119899 (53)

on119863119901 cap 119861119901 From here

119888119899 =119886119899

119894119899minus1= 1198943119899+1

sdot 119886119899 (54)



119888119896119901+1 = (119894)3(119896119901+1)+1

120572119896 = (minus1)119896120572119896 (55)


119901119911 and

(46)







sinh119901119911 =infin

sum

119896=1

119888119896119911119896 119911 isin 119863119901 (56)





sh119901 (119909) =

sinh119901119909 119909 isin [0120587119901

2)

sin119901119909 119909 isin (minus120587119901

2 0)

(57)


119901119909 ge 0 on [0 120574119901)



minus (10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901minus2

1199061015840)1015840


119906 = 0

119906 (0) = 0

1199061015840(0) = 1

(58)



(1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(59)



sin119901119909 =infin

sum

119896=1

119888119896119909119896= sh119901 (119909)


2 0)

(60)


119896=0120572119896 sdot 119909 sdot |119909|


infin

sum

119896=0

120572119896 sdot 119909 sdot |119909|119896119901=

infin

sum

119896=0

120572119896 sdot (minus1)119896119901sdot 119909119896119901+1

= 119866 (119909) (61)


infin

sum

119896=1

119888119896119909119896= 119866 (119909) on (minus120574119901 0) (62)








s119901 (119909) =

sin119901119909 119909 isin [0120587119901

2)

sinh119901119909 119909 isin (minus120587119901

2 0)

(63)



harrs119901(119909)

andharr



def= (dd119909)

harrs119901(119909)

andharr

ch119901(119909)def= (dd119909)

harr

sh119901(119909)


harrs119901119909 =


sin119901119909 119909 isin [0120587119901

2)

harr

sh119901119909 =

sin119901119909 119909 isin (minus120587119901

2 0)

sinh119901119909 119909 isin [0 +infin)

(64)


harrs119901119909 = sin119901119909 119909 isin R

harr

sh119901119909 = sinh119901119909 119909 isin R

(65)







harrs119901(119909) and



(harrs119901 (119909))

119901

+ (harrc119901 (119909))

119901

= 1 (66)


In[1] = pip[p ] =




Out[1] =120587csc(120587119901)

119901

In[2] = s3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == - 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -5 pip[3]][[1]])


In[3] = sh3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -pip[3] 5][[1]])


In[4] = sin3[x ] = (u[x]


v[x] == -2 Abs[u[x]]u[x]

u[0] == 0 v[0] == 1

uv x -4 pip[3] 4 pip[3]][[1]])



1199043(119909) for




sh119901(119909) andharr


(harr

ch119901 (119909))119901

minus (harr

sh119901 (119909))119901

= 1 (67)

7 Visualizations


harrs119901(119909) and

harr







328 (119909) = 119909 minus1199094

12minus1199097

252minus83 11990910

90 720minus1 817 119909

13

7 076 160

minus199 691 119909

16

2 377 589 760minus12 324 719 119909

19

406 567 848 960

minus22 008 573 061 119909

22

1 878 343 462 195 200

minus107 355 387 043 119909

25

22 540 121 546 342 400

minus89 152 153 354 993 119909

28

44 304 862 911 490 621 440

(68)




harr



harr


computingharr



12058734 12058732minus12058732 minus12058734

minus15

minus10

minus05

05

10

15

(a)

12058734 12058732minus12058732 minus12058734

01

02

minus02

minus01

(b)


harrs3(119909)

12058734 12058732minus12058732 minus12058734

05

10

minus10

minus05

(a)

0 12058734minus12058734 12058732

10minus15

10minus10

10minus5

10minus1

(b)


harrs3(119909) minus 1198723(119909)|


sh4(119909)harr

sh30(119909)harr


harrs31(119909)


argsinh119901119911

def= int

119911

0

1

(1 + 119904119901)1119901

d119904

= 11991121198651 (

1

1199011

119901 1 +

1

119901 minus 119911119901) 119911 isin D ⊊ C

(69)




minus2

minus1

1

2


(a)


1

2

minus1

(b)


1

2

minus2

minus1

(c)


119901)) for








In[4] = Plot[



( plots the graph )




1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)


1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)




sh3(119909)harr

sh31(119909)harr

ch3(119909) andharr



harr



harr

sh119901(119909)harr



8 Conclusion



1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

1

2

3

minus3

minus2

minus1

(b)





andharr


sh119901(119909) andharr


2 4minus4 minus2

minus3

minus2

minus1

1

2

3

(a)

42minus4 minus2

minus3

minus2

minus1

1

2

3

(b)


harr

sh3(119909)


int120587119901

0

10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

d119909

int120587119901

0|119906|119901 d119909

(70)

in 1198821119901




p = 2

u

minus4

minus2

0

2

4

p = 3

p = 4 p = 5

p = 10 p = 11

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

u

minus2

minus4

0

2

4

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

40 2minus2minus4

u998400

minus2 0 2 4minus4

u998400





p = 2

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 3

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 5

u

minus2

minus4

0

2

4

minus2 0 2 4minus4

u998400

p = 11

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

u

p = 10

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400


sh119901(119909) on 1199061015840=harr















Appendix










119879119901 (119909) = sin119901119909 (A1)


arcsin119901119909 = 119909 sdot 21198651 (1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896 sdot 119901 + 1) sdot 119896 sdot Γ (1119901)119909119896119901+1

(A2)





119887119895 =

Γ (119897 + 1119901)


0 otherwise(B1)




sum

119895=1

119887119895 sdot 119909119895 (B2)


119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1



119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983


1198871)

119898119894


1198871)

119898119899minus1

(B3)


0 such that





1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)





(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)


Competing Interests


Acknowledgments



References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(39)


sin119901119911 sinh119901119911(1199061015840)119901minus2

11990610158401015840+ 119906119901minus1

= 0 (1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0 119906 (0) = 0

1199061015840(0) = 1 119906

1015840(0) = 1


(40)





119863119901def= 119911 isin C |119911| lt 120574119901 (42)




1199061015840= V

V1015840 =119906119901minus1

V119901minus2

119906 (0) = 0

V (0) = 1

(43)





119901119911 for all 119911 isin 119863119901



(1199061015840(119911))119901

minus (119906 (119911))119901= 1 (44)

on the disc 119863119903



(119906 (119909)1015840)119901

minus (119906 (119909))119901= 1 (45)





cos119901119911 = cosh119901 (119894 sdot 119911) (47)


sinh119901119911 =infin

sum

119896=0

(minus1)119896sdot 120572119896 sdot 119911

119896119901+1 (48)




cos119901119911 = cos119901 (119894 sdot 119911)(49)

for all 119911 isin 119861119901




cosh119901119911 = cosh119901 (119894 sdot 119911) (51)

for all 119911 isin 119863119901





(1199081015840(119911))119901minus2

11990810158401015840(119911) + 119908 (119911)

119901minus1

= (dd119911119908 (119911))

119901minus2 d2

d1199112119908 (119911) + 119908 (119911)

119901minus1


119901minus2 d2

d1199112(minus119894 sdot 119906 (119894 sdot 119911))


= 119894 sdot [d

d (119894 sdot 119911)119906 (119894 sdot 119911)]

119901minus2 d2

d (119894 sdot 119911)2119906 (119894 sdot 119911)


= 119894 sdot [dd119904119906 (119904)]

119901minus2 d2


119901minus2sdot 119906 (119904)119901minus1

= 119894 sdot ([dd119904119906 (119904)]

119901minus2 d2

d1199042119906 (119904) minus (minus119894)

119901minus2sdot 119906 (119904)119901minus1)

= 119894 sdot ([dd119904119906 (119904)]

119901minus2 d2

d1199042119906 (119904) minus 119906 (119904)

119901minus1) = 0

(52)



infin

119899=1119888119899 sdot


sum

119899=1


infin

sum

119899=1

119888119899 sdot 119894119899minus1

sdot 119911119899 (53)

on119863119901 cap 119861119901 From here

119888119899 =119886119899

119894119899minus1= 1198943119899+1

sdot 119886119899 (54)



119888119896119901+1 = (119894)3(119896119901+1)+1

120572119896 = (minus1)119896120572119896 (55)


119901119911 and

(46)







sinh119901119911 =infin

sum

119896=1

119888119896119911119896 119911 isin 119863119901 (56)





sh119901 (119909) =

sinh119901119909 119909 isin [0120587119901

2)

sin119901119909 119909 isin (minus120587119901

2 0)

(57)


119901119909 ge 0 on [0 120574119901)



minus (10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901minus2

1199061015840)1015840


119906 = 0

119906 (0) = 0

1199061015840(0) = 1

(58)



(1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(59)



sin119901119909 =infin

sum

119896=1

119888119896119909119896= sh119901 (119909)


2 0)

(60)


119896=0120572119896 sdot 119909 sdot |119909|


infin

sum

119896=0

120572119896 sdot 119909 sdot |119909|119896119901=

infin

sum

119896=0

120572119896 sdot (minus1)119896119901sdot 119909119896119901+1

= 119866 (119909) (61)


infin

sum

119896=1

119888119896119909119896= 119866 (119909) on (minus120574119901 0) (62)








s119901 (119909) =

sin119901119909 119909 isin [0120587119901

2)

sinh119901119909 119909 isin (minus120587119901

2 0)

(63)



harrs119901(119909)

andharr



def= (dd119909)

harrs119901(119909)

andharr

ch119901(119909)def= (dd119909)

harr

sh119901(119909)


harrs119901119909 =


sin119901119909 119909 isin [0120587119901

2)

harr

sh119901119909 =

sin119901119909 119909 isin (minus120587119901

2 0)

sinh119901119909 119909 isin [0 +infin)

(64)


harrs119901119909 = sin119901119909 119909 isin R

harr

sh119901119909 = sinh119901119909 119909 isin R

(65)







harrs119901(119909) and



(harrs119901 (119909))

119901

+ (harrc119901 (119909))

119901

= 1 (66)


In[1] = pip[p ] =




Out[1] =120587csc(120587119901)

119901

In[2] = s3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == - 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -5 pip[3]][[1]])


In[3] = sh3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -pip[3] 5][[1]])


In[4] = sin3[x ] = (u[x]


v[x] == -2 Abs[u[x]]u[x]

u[0] == 0 v[0] == 1

uv x -4 pip[3] 4 pip[3]][[1]])



1199043(119909) for




sh119901(119909) andharr


(harr

ch119901 (119909))119901

minus (harr

sh119901 (119909))119901

= 1 (67)

7 Visualizations


harrs119901(119909) and

harr







328 (119909) = 119909 minus1199094

12minus1199097

252minus83 11990910

90 720minus1 817 119909

13

7 076 160

minus199 691 119909

16

2 377 589 760minus12 324 719 119909

19

406 567 848 960

minus22 008 573 061 119909

22

1 878 343 462 195 200

minus107 355 387 043 119909

25

22 540 121 546 342 400

minus89 152 153 354 993 119909

28

44 304 862 911 490 621 440

(68)




harr



harr


computingharr



12058734 12058732minus12058732 minus12058734

minus15

minus10

minus05

05

10

15

(a)

12058734 12058732minus12058732 minus12058734

01

02

minus02

minus01

(b)


harrs3(119909)

12058734 12058732minus12058732 minus12058734

05

10

minus10

minus05

(a)

0 12058734minus12058734 12058732

10minus15

10minus10

10minus5

10minus1

(b)


harrs3(119909) minus 1198723(119909)|


sh4(119909)harr

sh30(119909)harr


harrs31(119909)


argsinh119901119911

def= int

119911

0

1

(1 + 119904119901)1119901

d119904

= 11991121198651 (

1

1199011

119901 1 +

1

119901 minus 119911119901) 119911 isin D ⊊ C

(69)




minus2

minus1

1

2


(a)


1

2

minus1

(b)


1

2

minus2

minus1

(c)


119901)) for








In[4] = Plot[



( plots the graph )




1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)


1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)




sh3(119909)harr

sh31(119909)harr

ch3(119909) andharr



harr



harr

sh119901(119909)harr



8 Conclusion



1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

1

2

3

minus3

minus2

minus1

(b)





andharr


sh119901(119909) andharr


2 4minus4 minus2

minus3

minus2

minus1

1

2

3

(a)

42minus4 minus2

minus3

minus2

minus1

1

2

3

(b)


harr

sh3(119909)


int120587119901

0

10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

d119909

int120587119901

0|119906|119901 d119909

(70)

in 1198821119901




p = 2

u

minus4

minus2

0

2

4

p = 3

p = 4 p = 5

p = 10 p = 11

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

u

minus2

minus4

0

2

4

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

40 2minus2minus4

u998400

minus2 0 2 4minus4

u998400





p = 2

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 3

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 5

u

minus2

minus4

0

2

4

minus2 0 2 4minus4

u998400

p = 11

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

u

p = 10

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400


sh119901(119909) on 1199061015840=harr















Appendix










119879119901 (119909) = sin119901119909 (A1)


arcsin119901119909 = 119909 sdot 21198651 (1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896 sdot 119901 + 1) sdot 119896 sdot Γ (1119901)119909119896119901+1

(A2)





119887119895 =

Γ (119897 + 1119901)


0 otherwise(B1)




sum

119895=1

119887119895 sdot 119909119895 (B2)


119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1



119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983


1198871)

119898119894


1198871)

119898119899minus1

(B3)


0 such that





1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)





(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)


Competing Interests


Acknowledgments



References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













(1199081015840(119911))119901minus2

11990810158401015840(119911) + 119908 (119911)

119901minus1

= (dd119911119908 (119911))

119901minus2 d2

d1199112119908 (119911) + 119908 (119911)

119901minus1


119901minus2 d2

d1199112(minus119894 sdot 119906 (119894 sdot 119911))


= 119894 sdot [d

d (119894 sdot 119911)119906 (119894 sdot 119911)]

119901minus2 d2

d (119894 sdot 119911)2119906 (119894 sdot 119911)


= 119894 sdot [dd119904119906 (119904)]

119901minus2 d2


119901minus2sdot 119906 (119904)119901minus1

= 119894 sdot ([dd119904119906 (119904)]

119901minus2 d2

d1199042119906 (119904) minus (minus119894)

119901minus2sdot 119906 (119904)119901minus1)

= 119894 sdot ([dd119904119906 (119904)]

119901minus2 d2

d1199042119906 (119904) minus 119906 (119904)

119901minus1) = 0

(52)



infin

119899=1119888119899 sdot


sum

119899=1


infin

sum

119899=1

119888119899 sdot 119894119899minus1

sdot 119911119899 (53)

on119863119901 cap 119861119901 From here

119888119899 =119886119899

119894119899minus1= 1198943119899+1

sdot 119886119899 (54)



119888119896119901+1 = (119894)3(119896119901+1)+1

120572119896 = (minus1)119896120572119896 (55)


119901119911 and

(46)







sinh119901119911 =infin

sum

119896=1

119888119896119911119896 119911 isin 119863119901 (56)





sh119901 (119909) =

sinh119901119909 119909 isin [0120587119901

2)

sin119901119909 119909 isin (minus120587119901

2 0)

(57)


119901119909 ge 0 on [0 120574119901)



minus (10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901minus2

1199061015840)1015840


119906 = 0

119906 (0) = 0

1199061015840(0) = 1

(58)



(1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(59)



sin119901119909 =infin

sum

119896=1

119888119896119909119896= sh119901 (119909)


2 0)

(60)


119896=0120572119896 sdot 119909 sdot |119909|


infin

sum

119896=0

120572119896 sdot 119909 sdot |119909|119896119901=

infin

sum

119896=0

120572119896 sdot (minus1)119896119901sdot 119909119896119901+1

= 119866 (119909) (61)


infin

sum

119896=1

119888119896119909119896= 119866 (119909) on (minus120574119901 0) (62)








s119901 (119909) =

sin119901119909 119909 isin [0120587119901

2)

sinh119901119909 119909 isin (minus120587119901

2 0)

(63)



harrs119901(119909)

andharr



def= (dd119909)

harrs119901(119909)

andharr

ch119901(119909)def= (dd119909)

harr

sh119901(119909)


harrs119901119909 =


sin119901119909 119909 isin [0120587119901

2)

harr

sh119901119909 =

sin119901119909 119909 isin (minus120587119901

2 0)

sinh119901119909 119909 isin [0 +infin)

(64)


harrs119901119909 = sin119901119909 119909 isin R

harr

sh119901119909 = sinh119901119909 119909 isin R

(65)







harrs119901(119909) and



(harrs119901 (119909))

119901

+ (harrc119901 (119909))

119901

= 1 (66)


In[1] = pip[p ] =




Out[1] =120587csc(120587119901)

119901

In[2] = s3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == - 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -5 pip[3]][[1]])


In[3] = sh3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -pip[3] 5][[1]])


In[4] = sin3[x ] = (u[x]


v[x] == -2 Abs[u[x]]u[x]

u[0] == 0 v[0] == 1

uv x -4 pip[3] 4 pip[3]][[1]])



1199043(119909) for




sh119901(119909) andharr


(harr

ch119901 (119909))119901

minus (harr

sh119901 (119909))119901

= 1 (67)

7 Visualizations


harrs119901(119909) and

harr







328 (119909) = 119909 minus1199094

12minus1199097

252minus83 11990910

90 720minus1 817 119909

13

7 076 160

minus199 691 119909

16

2 377 589 760minus12 324 719 119909

19

406 567 848 960

minus22 008 573 061 119909

22

1 878 343 462 195 200

minus107 355 387 043 119909

25

22 540 121 546 342 400

minus89 152 153 354 993 119909

28

44 304 862 911 490 621 440

(68)




harr



harr


computingharr



12058734 12058732minus12058732 minus12058734

minus15

minus10

minus05

05

10

15

(a)

12058734 12058732minus12058732 minus12058734

01

02

minus02

minus01

(b)


harrs3(119909)

12058734 12058732minus12058732 minus12058734

05

10

minus10

minus05

(a)

0 12058734minus12058734 12058732

10minus15

10minus10

10minus5

10minus1

(b)


harrs3(119909) minus 1198723(119909)|


sh4(119909)harr

sh30(119909)harr


harrs31(119909)


argsinh119901119911

def= int

119911

0

1

(1 + 119904119901)1119901

d119904

= 11991121198651 (

1

1199011

119901 1 +

1

119901 minus 119911119901) 119911 isin D ⊊ C

(69)




minus2

minus1

1

2


(a)


1

2

minus1

(b)


1

2

minus2

minus1

(c)


119901)) for








In[4] = Plot[



( plots the graph )




1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)


1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)




sh3(119909)harr

sh31(119909)harr

ch3(119909) andharr



harr



harr

sh119901(119909)harr



8 Conclusion



1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

1

2

3

minus3

minus2

minus1

(b)





andharr


sh119901(119909) andharr


2 4minus4 minus2

minus3

minus2

minus1

1

2

3

(a)

42minus4 minus2

minus3

minus2

minus1

1

2

3

(b)


harr

sh3(119909)


int120587119901

0

10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

d119909

int120587119901

0|119906|119901 d119909

(70)

in 1198821119901




p = 2

u

minus4

minus2

0

2

4

p = 3

p = 4 p = 5

p = 10 p = 11

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

u

minus2

minus4

0

2

4

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

40 2minus2minus4

u998400

minus2 0 2 4minus4

u998400





p = 2

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 3

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 5

u

minus2

minus4

0

2

4

minus2 0 2 4minus4

u998400

p = 11

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

u

p = 10

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400


sh119901(119909) on 1199061015840=harr















Appendix










119879119901 (119909) = sin119901119909 (A1)


arcsin119901119909 = 119909 sdot 21198651 (1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896 sdot 119901 + 1) sdot 119896 sdot Γ (1119901)119909119896119901+1

(A2)





119887119895 =

Γ (119897 + 1119901)


0 otherwise(B1)




sum

119895=1

119887119895 sdot 119909119895 (B2)


119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1



119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983


1198871)

119898119894


1198871)

119898119899minus1

(B3)


0 such that





1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)





(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)


Competing Interests


Acknowledgments



References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus (10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901minus2

1199061015840)1015840


119906 = 0

119906 (0) = 0

1199061015840(0) = 1

(58)



(1199061015840)119901minus2

11990610158401015840minus 119906119901minus1

= 0

119906 (0) = 0

1199061015840(0) = 1

(59)



sin119901119909 =infin

sum

119896=1

119888119896119909119896= sh119901 (119909)


2 0)

(60)


119896=0120572119896 sdot 119909 sdot |119909|


infin

sum

119896=0

120572119896 sdot 119909 sdot |119909|119896119901=

infin

sum

119896=0

120572119896 sdot (minus1)119896119901sdot 119909119896119901+1

= 119866 (119909) (61)


infin

sum

119896=1

119888119896119909119896= 119866 (119909) on (minus120574119901 0) (62)








s119901 (119909) =

sin119901119909 119909 isin [0120587119901

2)

sinh119901119909 119909 isin (minus120587119901

2 0)

(63)



harrs119901(119909)

andharr



def= (dd119909)

harrs119901(119909)

andharr

ch119901(119909)def= (dd119909)

harr

sh119901(119909)


harrs119901119909 =


sin119901119909 119909 isin [0120587119901

2)

harr

sh119901119909 =

sin119901119909 119909 isin (minus120587119901

2 0)

sinh119901119909 119909 isin [0 +infin)

(64)


harrs119901119909 = sin119901119909 119909 isin R

harr

sh119901119909 = sinh119901119909 119909 isin R

(65)







harrs119901(119909) and



(harrs119901 (119909))

119901

+ (harrc119901 (119909))

119901

= 1 (66)


In[1] = pip[p ] =




Out[1] =120587csc(120587119901)

119901

In[2] = s3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == - 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -5 pip[3]][[1]])


In[3] = sh3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -pip[3] 5][[1]])


In[4] = sin3[x ] = (u[x]


v[x] == -2 Abs[u[x]]u[x]

u[0] == 0 v[0] == 1

uv x -4 pip[3] 4 pip[3]][[1]])



1199043(119909) for




sh119901(119909) andharr


(harr

ch119901 (119909))119901

minus (harr

sh119901 (119909))119901

= 1 (67)

7 Visualizations


harrs119901(119909) and

harr







328 (119909) = 119909 minus1199094

12minus1199097

252minus83 11990910

90 720minus1 817 119909

13

7 076 160

minus199 691 119909

16

2 377 589 760minus12 324 719 119909

19

406 567 848 960

minus22 008 573 061 119909

22

1 878 343 462 195 200

minus107 355 387 043 119909

25

22 540 121 546 342 400

minus89 152 153 354 993 119909

28

44 304 862 911 490 621 440

(68)




harr



harr


computingharr



12058734 12058732minus12058732 minus12058734

minus15

minus10

minus05

05

10

15

(a)

12058734 12058732minus12058732 minus12058734

01

02

minus02

minus01

(b)


harrs3(119909)

12058734 12058732minus12058732 minus12058734

05

10

minus10

minus05

(a)

0 12058734minus12058734 12058732

10minus15

10minus10

10minus5

10minus1

(b)


harrs3(119909) minus 1198723(119909)|


sh4(119909)harr

sh30(119909)harr


harrs31(119909)


argsinh119901119911

def= int

119911

0

1

(1 + 119904119901)1119901

d119904

= 11991121198651 (

1

1199011

119901 1 +

1

119901 minus 119911119901) 119911 isin D ⊊ C

(69)




minus2

minus1

1

2


(a)


1

2

minus1

(b)


1

2

minus2

minus1

(c)


119901)) for








In[4] = Plot[



( plots the graph )




1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)


1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)




sh3(119909)harr

sh31(119909)harr

ch3(119909) andharr



harr



harr

sh119901(119909)harr



8 Conclusion



1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

1

2

3

minus3

minus2

minus1

(b)





andharr


sh119901(119909) andharr


2 4minus4 minus2

minus3

minus2

minus1

1

2

3

(a)

42minus4 minus2

minus3

minus2

minus1

1

2

3

(b)


harr

sh3(119909)


int120587119901

0

10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

d119909

int120587119901

0|119906|119901 d119909

(70)

in 1198821119901




p = 2

u

minus4

minus2

0

2

4

p = 3

p = 4 p = 5

p = 10 p = 11

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

u

minus2

minus4

0

2

4

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

40 2minus2minus4

u998400

minus2 0 2 4minus4

u998400





p = 2

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 3

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 5

u

minus2

minus4

0

2

4

minus2 0 2 4minus4

u998400

p = 11

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

u

p = 10

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400


sh119901(119909) on 1199061015840=harr















Appendix










119879119901 (119909) = sin119901119909 (A1)


arcsin119901119909 = 119909 sdot 21198651 (1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896 sdot 119901 + 1) sdot 119896 sdot Γ (1119901)119909119896119901+1

(A2)





119887119895 =

Γ (119897 + 1119901)


0 otherwise(B1)




sum

119895=1

119887119895 sdot 119909119895 (B2)


119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1



119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983


1198871)

119898119894


1198871)

119898119899minus1

(B3)


0 such that





1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)





(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)


Competing Interests


Acknowledgments



References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










In[1] = pip[p ] =




Out[1] =120587csc(120587119901)

119901

In[2] = s3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == - 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -5 pip[3]][[1]])


In[3] = sh3[x ] = (u[x]

NDSolve[

u[x] == (v[x])and(12) v[x] == 2 u[x]and2

u[0] == 0 v[0] == 1

u v x -pip[3] 5][[1]])


In[4] = sin3[x ] = (u[x]


v[x] == -2 Abs[u[x]]u[x]

u[0] == 0 v[0] == 1

uv x -4 pip[3] 4 pip[3]][[1]])



1199043(119909) for




sh119901(119909) andharr


(harr

ch119901 (119909))119901

minus (harr

sh119901 (119909))119901

= 1 (67)

7 Visualizations


harrs119901(119909) and

harr







328 (119909) = 119909 minus1199094

12minus1199097

252minus83 11990910

90 720minus1 817 119909

13

7 076 160

minus199 691 119909

16

2 377 589 760minus12 324 719 119909

19

406 567 848 960

minus22 008 573 061 119909

22

1 878 343 462 195 200

minus107 355 387 043 119909

25

22 540 121 546 342 400

minus89 152 153 354 993 119909

28

44 304 862 911 490 621 440

(68)




harr



harr


computingharr



12058734 12058732minus12058732 minus12058734

minus15

minus10

minus05

05

10

15

(a)

12058734 12058732minus12058732 minus12058734

01

02

minus02

minus01

(b)


harrs3(119909)

12058734 12058732minus12058732 minus12058734

05

10

minus10

minus05

(a)

0 12058734minus12058734 12058732

10minus15

10minus10

10minus5

10minus1

(b)


harrs3(119909) minus 1198723(119909)|


sh4(119909)harr

sh30(119909)harr


harrs31(119909)


argsinh119901119911

def= int

119911

0

1

(1 + 119904119901)1119901

d119904

= 11991121198651 (

1

1199011

119901 1 +

1

119901 minus 119911119901) 119911 isin D ⊊ C

(69)




minus2

minus1

1

2


(a)


1

2

minus1

(b)


1

2

minus2

minus1

(c)


119901)) for








In[4] = Plot[



( plots the graph )




1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)


1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)




sh3(119909)harr

sh31(119909)harr

ch3(119909) andharr



harr



harr

sh119901(119909)harr



8 Conclusion



1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

1

2

3

minus3

minus2

minus1

(b)





andharr


sh119901(119909) andharr


2 4minus4 minus2

minus3

minus2

minus1

1

2

3

(a)

42minus4 minus2

minus3

minus2

minus1

1

2

3

(b)


harr

sh3(119909)


int120587119901

0

10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

d119909

int120587119901

0|119906|119901 d119909

(70)

in 1198821119901




p = 2

u

minus4

minus2

0

2

4

p = 3

p = 4 p = 5

p = 10 p = 11

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

u

minus2

minus4

0

2

4

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

40 2minus2minus4

u998400

minus2 0 2 4minus4

u998400





p = 2

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 3

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 5

u

minus2

minus4

0

2

4

minus2 0 2 4minus4

u998400

p = 11

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

u

p = 10

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400


sh119901(119909) on 1199061015840=harr















Appendix










119879119901 (119909) = sin119901119909 (A1)


arcsin119901119909 = 119909 sdot 21198651 (1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896 sdot 119901 + 1) sdot 119896 sdot Γ (1119901)119909119896119901+1

(A2)





119887119895 =

Γ (119897 + 1119901)


0 otherwise(B1)




sum

119895=1

119887119895 sdot 119909119895 (B2)


119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1



119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983


1198871)

119898119894


1198871)

119898119899minus1

(B3)


0 such that





1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)





(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)


Competing Interests


Acknowledgments



References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










12058734 12058732minus12058732 minus12058734

minus15

minus10

minus05

05

10

15

(a)

12058734 12058732minus12058732 minus12058734

01

02

minus02

minus01

(b)


harrs3(119909)

12058734 12058732minus12058732 minus12058734

05

10

minus10

minus05

(a)

0 12058734minus12058734 12058732

10minus15

10minus10

10minus5

10minus1

(b)


harrs3(119909) minus 1198723(119909)|


sh4(119909)harr

sh30(119909)harr


harrs31(119909)


argsinh119901119911

def= int

119911

0

1

(1 + 119904119901)1119901

d119904

= 11991121198651 (

1

1199011

119901 1 +

1

119901 minus 119911119901) 119911 isin D ⊊ C

(69)




minus2

minus1

1

2


(a)


1

2

minus1

(b)


1

2

minus2

minus1

(c)


119901)) for








In[4] = Plot[



( plots the graph )




1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)


1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)




sh3(119909)harr

sh31(119909)harr

ch3(119909) andharr



harr



harr

sh119901(119909)harr



8 Conclusion



1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

1

2

3

minus3

minus2

minus1

(b)





andharr


sh119901(119909) andharr


2 4minus4 minus2

minus3

minus2

minus1

1

2

3

(a)

42minus4 minus2

minus3

minus2

minus1

1

2

3

(b)


harr

sh3(119909)


int120587119901

0

10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

d119909

int120587119901

0|119906|119901 d119909

(70)

in 1198821119901




p = 2

u

minus4

minus2

0

2

4

p = 3

p = 4 p = 5

p = 10 p = 11

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

u

minus2

minus4

0

2

4

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

40 2minus2minus4

u998400

minus2 0 2 4minus4

u998400





p = 2

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 3

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 5

u

minus2

minus4

0

2

4

minus2 0 2 4minus4

u998400

p = 11

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

u

p = 10

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400


sh119901(119909) on 1199061015840=harr















Appendix










119879119901 (119909) = sin119901119909 (A1)


arcsin119901119909 = 119909 sdot 21198651 (1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896 sdot 119901 + 1) sdot 119896 sdot Γ (1119901)119909119896119901+1

(A2)





119887119895 =

Γ (119897 + 1119901)


0 otherwise(B1)




sum

119895=1

119887119895 sdot 119909119895 (B2)


119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1



119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983


1198871)

119898119894


1198871)

119898119899minus1

(B3)


0 such that





1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)





(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)


Competing Interests


Acknowledgments



References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus2

minus1

1

2


(a)


1

2

minus1

(b)


1

2

minus2

minus1

(c)


119901)) for








In[4] = Plot[



( plots the graph )




1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)


1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)




sh3(119909)harr

sh31(119909)harr

ch3(119909) andharr



harr



harr

sh119901(119909)harr



8 Conclusion



1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

1

2

3

minus3

minus2

minus1

(b)





andharr


sh119901(119909) andharr


2 4minus4 minus2

minus3

minus2

minus1

1

2

3

(a)

42minus4 minus2

minus3

minus2

minus1

1

2

3

(b)


harr

sh3(119909)


int120587119901

0

10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

d119909

int120587119901

0|119906|119901 d119909

(70)

in 1198821119901




p = 2

u

minus4

minus2

0

2

4

p = 3

p = 4 p = 5

p = 10 p = 11

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

u

minus2

minus4

0

2

4

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

40 2minus2minus4

u998400

minus2 0 2 4minus4

u998400





p = 2

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 3

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 5

u

minus2

minus4

0

2

4

minus2 0 2 4minus4

u998400

p = 11

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

u

p = 10

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400


sh119901(119909) on 1199061015840=harr















Appendix










119879119901 (119909) = sin119901119909 (A1)


arcsin119901119909 = 119909 sdot 21198651 (1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896 sdot 119901 + 1) sdot 119896 sdot Γ (1119901)119909119896119901+1

(A2)





119887119895 =

Γ (119897 + 1119901)


0 otherwise(B1)




sum

119895=1

119887119895 sdot 119909119895 (B2)


119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1



119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983


1198871)

119898119894


1198871)

119898119899minus1

(B3)


0 such that





1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)





(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)


Competing Interests


Acknowledgments



References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)


1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(b)




sh3(119909)harr

sh31(119909)harr

ch3(119909) andharr



harr



harr

sh119901(119909)harr



8 Conclusion



1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

1

2

3

minus3

minus2

minus1

(b)





andharr


sh119901(119909) andharr


2 4minus4 minus2

minus3

minus2

minus1

1

2

3

(a)

42minus4 minus2

minus3

minus2

minus1

1

2

3

(b)


harr

sh3(119909)


int120587119901

0

10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

d119909

int120587119901

0|119906|119901 d119909

(70)

in 1198821119901




p = 2

u

minus4

minus2

0

2

4

p = 3

p = 4 p = 5

p = 10 p = 11

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

u

minus2

minus4

0

2

4

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

40 2minus2minus4

u998400

minus2 0 2 4minus4

u998400





p = 2

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 3

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 5

u

minus2

minus4

0

2

4

minus2 0 2 4minus4

u998400

p = 11

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

u

p = 10

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400


sh119901(119909) on 1199061015840=harr















Appendix










119879119901 (119909) = sin119901119909 (A1)


arcsin119901119909 = 119909 sdot 21198651 (1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896 sdot 119901 + 1) sdot 119896 sdot Γ (1119901)119909119896119901+1

(A2)





119887119895 =

Γ (119897 + 1119901)


0 otherwise(B1)




sum

119895=1

119887119895 sdot 119909119895 (B2)


119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1



119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983


1198871)

119898119894


1198871)

119898119899minus1

(B3)


0 such that





1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)





(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)


Competing Interests


Acknowledgments



References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2minus2 minus1

minus3

minus2

minus1

1

2

3

(a)

1 2minus2 minus1

1

2

3

minus3

minus2

minus1

(b)





andharr


sh119901(119909) andharr


2 4minus4 minus2

minus3

minus2

minus1

1

2

3

(a)

42minus4 minus2

minus3

minus2

minus1

1

2

3

(b)


harr

sh3(119909)


int120587119901

0

10038161003816100381610038161003816119906101584010038161003816100381610038161003816

119901

d119909

int120587119901

0|119906|119901 d119909

(70)

in 1198821119901




p = 2

u

minus4

minus2

0

2

4

p = 3

p = 4 p = 5

p = 10 p = 11

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

u

minus2

minus4

0

2

4

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

40 2minus2minus4

u998400

minus2 0 2 4minus4

u998400





p = 2

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 3

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 5

u

minus2

minus4

0

2

4

minus2 0 2 4minus4

u998400

p = 11

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

u

p = 10

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400


sh119901(119909) on 1199061015840=harr















Appendix










119879119901 (119909) = sin119901119909 (A1)


arcsin119901119909 = 119909 sdot 21198651 (1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896 sdot 119901 + 1) sdot 119896 sdot Γ (1119901)119909119896119901+1

(A2)





119887119895 =

Γ (119897 + 1119901)


0 otherwise(B1)




sum

119895=1

119887119895 sdot 119909119895 (B2)


119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1



119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983


1198871)

119898119894


1198871)

119898119899minus1

(B3)


0 such that





1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)





(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)


Competing Interests


Acknowledgments



References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










p = 2

u

minus4

minus2

0

2

4

p = 3

p = 4 p = 5

p = 10 p = 11

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

u

minus2

minus4

0

2

4

u

minus4

minus2

0

2

4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

minus2 0 2 4minus4

u998400

40 2minus2minus4

u998400

minus2 0 2 4minus4

u998400





p = 2

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 3

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 5

u

minus2

minus4

0

2

4

minus2 0 2 4minus4

u998400

p = 11

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

u

p = 10

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400


sh119901(119909) on 1199061015840=harr















Appendix










119879119901 (119909) = sin119901119909 (A1)


arcsin119901119909 = 119909 sdot 21198651 (1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896 sdot 119901 + 1) sdot 119896 sdot Γ (1119901)119909119896119901+1

(A2)





119887119895 =

Γ (119897 + 1119901)


0 otherwise(B1)




sum

119895=1

119887119895 sdot 119909119895 (B2)


119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1



119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983


1198871)

119898119894


1198871)

119898119899minus1

(B3)


0 such that





1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)





(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)


Competing Interests


Acknowledgments



References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










p = 2

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 3

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 4

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

p = 5

u

minus2

minus4

0

2

4

minus2 0 2 4minus4

u998400

p = 11

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400

u

p = 10

u

minus4

minus2

0

2

4

minus2 0 2 4minus4

u998400


sh119901(119909) on 1199061015840=harr















Appendix










119879119901 (119909) = sin119901119909 (A1)


arcsin119901119909 = 119909 sdot 21198651 (1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896 sdot 119901 + 1) sdot 119896 sdot Γ (1119901)119909119896119901+1

(A2)





119887119895 =

Γ (119897 + 1119901)


0 otherwise(B1)




sum

119895=1

119887119895 sdot 119909119895 (B2)


119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1



119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983


1198871)

119898119894


1198871)

119898119899minus1

(B3)


0 such that





1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)





(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)


Competing Interests


Acknowledgments



References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Appendix










119879119901 (119909) = sin119901119909 (A1)


arcsin119901119909 = 119909 sdot 21198651 (1

1199011

119901 1 +

1

119901 119909119901)

=

infin

sum

119896=0

Γ (119896 + 1119901)

(119896 sdot 119901 + 1) sdot 119896 sdot Γ (1119901)119909119896119901+1

(A2)





119887119895 =

Γ (119897 + 1119901)


0 otherwise(B1)




sum

119895=1

119887119895 sdot 119909119895 (B2)


119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1



119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983


1198871)

119898119894


1198871)

119898119899minus1

(B3)


0 such that





1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)





(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)


Competing Interests


Acknowledgments



References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












sum

119895=1

119887119895 sdot 119909119895 (B2)


119886119899 =1

119899 sdot 1198871198991

sum1198981 1198982119898119899minus1



119898111989821198983 sdot (1198872

1198871)

1198981

sdot (1198873

1198871)

1198982

(1198874

1198871)

1198983


1198871)

119898119894


1198871)

119898119899minus1

(B3)


0 such that





1198871 =Γ (1119901)

1 sdot 1 sdot Γ (1119901)= 1 (B5)





(B6)

where

119896 = lfloor119899 minus 1

119901rfloor (B7)


Competing Interests


Acknowledgments



References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra







