Mathematical Hyperbolic Functions

8/18/2019 Mathematical Hyperbolic Functions

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10/4/2015 Hyperbolic function - Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Hyperbolic_function 3/20

sinh, cosh and tanh

9 Identities10 Relationship to the exponential function11 Hyperbolic functions for complex numbers12 See also13 References

14 External links

Standard analytic expressions

The hyperbolic functions are:

Hyperbolic sine:

.

Hyperbolic cosine:

.

Hyperbolic tangent:

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-https://en.wikipedia.org/wiki/File:Sinh_cosh_tanh.svg


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csch, sech and coth

.

Hyperbolic cotangent:

.

Hyperbolic secant:

.

Hyperbolic cosecant:

.

Hyperbolic functions can be introduced via imaginary circular angles:

Hyperbolic sine:

https://en.wikipedia.org/wiki/Hyperbolic_angle#Imaginary_circular_anglehttps://en.wikipedia.org/wiki/File:Csch_sech_coth.svg


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(a) cosh( x) is the average of

e x and e−x

.

Hyperbolic cosine:

.

Hyperbolic tangent:

.

Hyperbolic cotangent:

.Hyperbolic secant:

.

Hyperbolic cosecant:

.

where i is the imaginary unit with the property that i2 = −1.

The complex forms in the definitions above derive from Euler's formula.

Useful relations

https://en.wikipedia.org/wiki/Euler%27s_formulahttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Imaginary_unithttps://en.wikipedia.org/wiki/Arithmetic_meanhttps://en.wikipedia.org/wiki/File:Hyperbolic_and_exponential;_cosh.svg


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(b) sinh( x) is half the

difference of e x and e−x

Odd and even functions:

Hence:

It can be seen that cosh x and sech x are even functions; the others areodd functions.

Hyperbolic sine and cosine satisfy the identity

which is similar to the Pythagorean trigonometric identity. One also has

https://en.wikipedia.org/wiki/Pythagorean_trigonometric_identityhttps://en.wikipedia.org/wiki/Odd_functionshttps://en.wikipedia.org/wiki/Even_functionhttps://en.wikipedia.org/wiki/Subtractionhttps://en.wikipedia.org/wiki/File:Hyperbolic_and_exponential;_sinh.svg


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for the other functions.

The hyperbolic tangent is the solution to the differential equation with f(0)=0 and thenonlinear boundary value problem:[12]

It can be shown that the area under the curve of cosh ( x) over a finite interval is always equal to the arc

length corresponding to that interval:[13]

Sums of arguments:

particularly

Sum and difference of cosh and sinh:

http://-/?-http://-/?-https://en.wikipedia.org/wiki/Boundary_value_problemhttps://en.wikipedia.org/wiki/Nonlinearhttps://en.wikipedia.org/wiki/Differential_equation


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Inverse functions as logarithms

Main article: Inverse hyperbolic function

Derivatives

https://en.wikipedia.org/wiki/Inverse_hyperbolic_function


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Second derivatives

The second derivatives of sinh and cosh are the same function:

All functions with this property are linear combinations of sinh and cosh, notably the exponentialfunctions and , and the zero function .

Standard integrals

For a full list, see list of integrals of hyperbolic functions.

https://en.wikipedia.org/wiki/List_of_integrals_of_hyperbolic_functionshttps://en.wikipedia.org/wiki/Zero_functionhttps://en.wikipedia.org/wiki/Exponential_functionhttps://en.wikipedia.org/wiki/Linear_combinationhttps://en.wikipedia.org/wiki/Second_derivative


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The following integrals can be proved using hyperbolic substitution.

https://en.wikipedia.org/wiki/Hyperbolic_substitution


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where C is the constant of integration.

Taylor series expressions

It is possible to express the above functions as Taylor series:

The function sinh x has a Taylor series expression with only odd exponents for x. Thus it is an oddfunction, that is, −sinh x = sinh(− x), and sinh 0 = 0.

The function cosh x has a Taylor series expression with only even exponents for x. Thus it is an evenfunction, that is, symmetric with respect to the y-axis. The sum of the sinh and cosh series is the infiniteseries expression of the exponential function.

https://en.wikipedia.org/wiki/Exponential_functionhttps://en.wikipedia.org/wiki/Infinite_serieshttps://en.wikipedia.org/wiki/Even_functionhttps://en.wikipedia.org/wiki/Odd_functionhttps://en.wikipedia.org/wiki/Taylor_serieshttps://en.wikipedia.org/wiki/Constant_of_integration


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where

is the nth Bernoulli number is the nth Euler number

Comparison with circular functions

The hyperbolic functions represent an expansion of trigonometry beyond the circular functions. Bothtypes depend on an argument, either circular angle or hyperbolic angle.

Since the area of a circular sector is it will be equal to u when r = square root of 2. In the diagram

such a circle is tangent to the hyperbola x y = 1 at (1,1). The yellow sector depicts an area and anglemagnitude. Similarly, the red augmentation depicts an area and magnitude as hyperbolic angle.

https://en.wikipedia.org/wiki/Right_trianglehttps://en.wikipedia.org/wiki/Square_root_of_2https://en.wikipedia.org/wiki/Circular_sector#Areahttps://en.wikipedia.org/wiki/Hyperbolic_anglehttps://en.wikipedia.org/wiki/Anglehttps://en.wikipedia.org/wiki/Argument_of_a_functionhttps://en.wikipedia.org/wiki/Circular_functionhttps://en.wikipedia.org/wiki/Trigonometryhttps://en.wikipedia.org/wiki/Euler_numberhttps://en.wikipedia.org/wiki/Bernoulli_number


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Circle and hyperbola tangent at(1,1) display geometry of circular functions in terms of circular sector area u andhyperbolic functionsdepending on hyperbolic sector

area u.

The legs of the two right triangles with hypotenuse on the ray defining the angles are of length 2 timesthe circular and hyperbolic functions.

Mellon Haskell of University of California, Berkeley described the basis of hyperbolic functions inareas of hyperbolic sectors in an 1895 article in Bulletin of the American Mathematical Society (see

External links). He refers to the hyperbolic angle as an invariant measure with respect to the squeezemapping just as circular angle is invariant under rotation.

Identities

The hyperbolic functions satisfy many identities, all of them similar in

form to the trigonometric identities. In fact, Osborn's rule[14] statesthat one can convert any trigonometric identity into a hyperbolicidentity by expanding it completely in terms of integral powers of sines and cosines, changing sine to sinh and cosine to cosh, andswitching the sign of every term which contains a product of 2, 6, 10,14, ... sinhs. This yields for example the addition theorems

the "double argument formulas"

http://-/?-https://en.wikipedia.org/wiki/Trigonometric_identityhttps://en.wikipedia.org/wiki/Squeeze_mappinghttps://en.wikipedia.org/wiki/Invariant_measurehttps://en.wikipedia.org/wiki/Bulletin_of_the_American_Mathematical_Societyhttps://en.wikipedia.org/wiki/Hyperbolic_sectorhttps://en.wikipedia.org/wiki/University_of_California,_Berkeleyhttps://en.wikipedia.org/wiki/Right_trianglehttps://en.wikipedia.org/wiki/Hyperbolic_sectorhttps://en.wikipedia.org/wiki/Sector_of_a_circlehttps://en.wikipedia.org/wiki/File:Circular_and_hyperbolic_angle.svg


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and the "half-argument formulas"[15]

Note: This is equivalent to its circular counterpart multiplied by −1. Note: This corresponds to its circular counterpart.

The derivative of sinh x is cosh x and the derivative of cosh x is sinh x; this is similar to trigonometricfunctions, albeit the sign is different (i.e., the derivative of cos x is −sin x).

The Gudermannian function gives a direct relationship between the circular functions and thehyperbolic ones that does not involve complex numbers.

The graph of the function a cosh( x/a) is the catenary, the curve formed by a uniform flexible chainhanging freely between two fixed points under uniform gravity.

https://en.wikipedia.org/wiki/Catenaryhttps://en.wikipedia.org/wiki/Gudermannian_functionhttps://en.wikipedia.org/wiki/Derivativehttp://-/?-


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Relationship to the exponential function

From the definitions of the hyperbolic sine and cosine, we can derive the following identities:

and

These expressions are analogous to the expressions for sine and cosine, based on Euler's formula, assums of complex exponentials.

Hyperbolic functions for complex numbers

Since the exponential function can be defined for any complex argument, we can extend the definitionsof the hyperbolic functions also to complex arguments. The functions sinh z and cosh z are thenholomorphic.

Relationships to ordinary trigonometric functions are given by Euler's formula for complex numbers:

so:

https://en.wikipedia.org/wiki/Euler%27s_formulahttps://en.wikipedia.org/wiki/Holomorphic_functionhttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Exponential_functionhttps://en.wikipedia.org/wiki/Euler%27s_formula


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Wikimedia

Thus, hyperbolic functions are periodic with respect to the imaginary component, with period (

for hyperbolic tangent and cotangent).

Hyperbolic functions in the complex plane

See also

e (mathematical constant)
https://en.wikipedia.org/wiki/E_(mathematical_constant)https://en.wikipedia.org/wiki/File:Complex_Csch.jpghttps://en.wikipedia.org/wiki/File:Complex_Sech.jpghttps://en.wikipedia.org/wiki/File:Complex_Coth.jpghttps://en.wikipedia.org/wiki/File:Complex_Tanh.jpghttps://en.wikipedia.org/wiki/File:Complex_Cosh.jpghttps://en.wikipedia.org/wiki/File:Complex_Sinh.jpghttps://en.wikipedia.org/wiki/Periodic_function

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Mathematical Hyperbolic Functions