Continued Fractions for Special Functions: …Handbook of Continued Fractions for Special Functions Special functions are pervasive in all fields of science and industry. The most

Continued Fractions for Special Functions:Handbook and Software

Team:

A. Cuyt,V.B. Petersen, J. Van Deun, H. Waadeland,

F. Backeljauw, S. Becuwe, M. Colman,W.B. Jones

Universiteit Antwerpen

Sør-Trøndelag University College

Norwegian University of Science and Technology

University of Colorado at Boulder

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Deliverables:

I Book: formulas with tables and graphsI Maple library: all fractionsI C++ library: all functionsI Web version to explore: all of the above

(www.cfsf.ua.ac.be)

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Book

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C++Related to larger-scale projects:

I dlmf.nist.govI functions.wolfram.com

Only 10% of the CF representations are also found there!

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special function: f(z)

continued fraction: b0(z) +K∞m=1

am(z)

bm(z)

f(z) = b0(z) +a1(z)

b1(z) +a2(z)

b2(z) +a3(z)

b3(z) + · · ·

= b0(z) +a1(z)

b1(z) +

a2(z)

b2(z) + · · ·

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I SF: limit-periodic representationI CF: larger convergence domain

Example:

Atan(z) =

∞∑m=0

(−1)m+1

2m+ 1z2m+1,

|z| < 1

=z

1 +

∞Km=2

(m− 1)2z2

2m− 1,

iz 6∈ (−∞,−1) ∪ (1,+∞)

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tail: tn(z) = K∞m=n+1

am(z)

bm(z)

Example:√1 + 4x− 1

2=

∞Km=1

x

1, x > −

14, tn(x) = f(x)

√2 − 1 =

11 +

21 +

11 +

21 +

. . . , t2n →√2 − 1

t2n+1 →√2

1 =∞Km=1

m(m+ 2)1

, tn →∞8 / 59

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approximant:

fn(z; 0) = b0(z) +n

Km=1

am(z)

bm(z)

modified approximant:

fn(z;wn) = b0(z) +n−1

Km=1

am(z)

bm(z) +

an(z)

bn(z) +wn

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. BookPart IPart IIPart IIIWeb

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1 3

Handbook of Continued

Fractions for Special Functions

Special functions are pervasive in all fields of science and

industry. The most well-known application areas are in physics,

engineering, chemistry, computer science and statistics.

Because of their importance, several books and websites and a

large collection of papers have been devoted to these functions.

Of the standard work on the subject, namely the Handbook

of Mathematical Functions with Formulas, Graphs and

Mathematical Tables edited by Milton Abramowitz and Irene

Stegun, the American National Institute of Standards claims to

have sold over 700 000 copies!

But so far no project has been devoted to the systematic study

of continued fraction representations for these functions.

This handbook is the result of such an endeavour. We emphasise

that only 10% of the continued fractions contained in this book,

can also be found in the Abramowitz and Stegun project or at

special functions websites!

ISBN 978-1-4020-6948-2

Handbook ofContinued Fractions for Special Functions

11

Cuyt | Petersen | VerdonkW

aadeland | Jones

A. CuytV. Brevik PetersenB. Verdonk H. Waadeland W. B. Jones

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Book. Part IPart IIPart IIIWeb

Maple

C++

Part I: Basic theory

I BasicsI Continued fraction representation of functionsI Convergence criteriaI Padé approximantsI Moment theory and orthogonal polynomials

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BookPart I. Part IIPart IIIWeb

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Part II: Numerics

I Continued fraction constructionI Truncation error boundsI Continued fraction evaluation

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BookPart IPart II. Part IIIWeb

Maple

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Part III: Special functions

I Elementary functionsI Incomplete gamma and relatedI Error function and relatedI Exponential integralsI Hypergeometric 2F1(a,n; c; z), n ∈ ZI Confluent hypergeometric 1F1(n;b; z), n ∈ ZI Parabolic cylinder functionsI Legendre functionsI Bessel functions of integer and fractional orderI Coulomb wave functionsI Zeta function and related

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BookPart IPart IIPart III. Web

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Formulas

Ln(1 + z) =z

1 +

∞

Km=2

(amz

1

), |Arg(1 + z)| < π, (11.2.2) – – –

– – –

– – – AS

a2k =k

2(2k − 1), a2k+1 =

k

2(2k + 1)

=2z

2 + z +

∞

Km=2

(−(m− 1)2z2

(2m− 1)(2 + z)

), (11.2.3) – – –

– – –

– – –

|Arg(1− z2/(2 + z)2

)| < π

Ln(

1 + z

1− z

)=

2z

1 +

∞

Km=1

(amz2

1

), |Arg(1− z2)| < π, (11.2.4) – – –

– – –

– – – AS

am =−m2

(2m− 1)(2m + 1).

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Tables

Table 11.2.3: Relative error of 20th partial sum and 20th approximants.

x Ln(1 + x) (11.2.1) (11.2.2) (11.2.3) (6.8.8)−0.9 −2.302585e+00 1.5e−02 2.8e−06 5.8e−12 1.5e−06

−0.4 −5.108256e−01 2.5e−10 1.8e−18 2.2e−36 2.4e−19

0.1 9.531018e−02 4.4e−23 5.3e−33 1.9e−65 1.3e−34

0.5 4.054651e−01 1.8e−08 1.9e−20 2.3e−40 2.0e−21

1.1 7.419373e−01 2.4e−01 2.8e−15 5.3e−30 5.5e−16

5 1.791759e+00 1.0e+13 4.2e−08 1.3e−15 2.0e−08

10 2.397895e+00 1.8e+19 5.3e−06 2.1e−11 3.3e−06

100 4.615121e+00 1.0e+40 1.8e−02 3.6e−04 2.5e−02

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Modification:

since in (11.2.2), limm→∞ amz = z/4 and

limm→∞ am+1 − 1

4

am − 14

= −1,

we find

w(z) =−1 +

√1 + z

2and {

w(1)2k (z) = w(z) + kz

2(2k+1) − z4

w(1)2k+1(z) = w(z) +

(k+1)z2(2k+1) − z

4

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Table 11.2.4: Relative error of 20th (modified) approximants.

x Ln(1 + x) (11.2.2) (11.2.2) (11.2.2)−0.9 −2.302585e+00 2.8e−06 1.9e−07 2.9e−10

−0.4 −5.108256e−01 1.8e−18 9.3e−20 4.3e−22

0.1 9.531018e−02 5.3e−33 2.7e−34 3.2e−37

0.5 4.054651e−01 1.9e−20 9.5e−22 5.5e−24

1.1 7.419373e−01 2.8e−15 1.5e−16 1.8e−18

5 1.791759e+00 4.2e−08 2.6e−09 1.1e−10

10 2.397895e+00 5.3e−06 3.7e−07 2.8e−08

100 4.615121e+00 1.8e−02 2.9e−03 9.2e−04

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Graphs

Figure 11.2.1: Complex region where f8(z; 0) of (11.2.2) guarantees ksignificant digits for Ln(1 + z) (from light to dark k = 6, 7, 8, 9).

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Figure 11.2.2: Number of significant digits guaranteed by the nth classicalapproximant of (11.2.2) (from light to dark n = 5, 6, 7) and the 5th modifiedapproximant evaluated with w

(1)5 (z) (darkest).

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Web interface

2F1(1/2, 1; 3/2; z) =1

2√zLn(1 +√z

1 −√z

)I series representationI continued fraction representations:

I C-fraction (15.3.7)I M-fraction (15.3.12)I Nörlund fraction (15.3.17)

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z 2F1 (1/2, 1; 3/2; z) =∞

Km=1

(cmz

1

), z ∈ C \ [1,+∞) (15.3.7a) – – –

– – –

– – –

with

c1 = 1, cm =−(m− 1)2

4(m− 1)2 − 1, m ≥ 2. (15.3.7b)

2F1(1/2, 1; 3/2; z) =1/2

1/2 + z/2 −z

3/2 + 3z/2 −4z

5/2 + 5z/2 − . . .(15.3.12) – – –

– – –

– – –

2F1(1/2, 1; 3/2; z) =1

1− z +z(1− z)

3/2 − 5/2z +

∞

Km=2

(m(m− 1/2)z(1− z)

(m + 1/2)− (2m + 1/2)z

),

<z < 1/2.(15.3.17) – – –

– – –

– – –

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Table 15.3.1: Relative error of the 5th partial sum and 5th (modified)approximants. More details can be found in the Examples 15.3.1, 15.3.2and 15.3.3.

x 2F1(1/2, 1; 3/2;x) (15.3.7) (15.3.12) (15.3.17)0.1 1.035488e+00 1.9e−08 1.4e−05 6.1e−06

0.2 1.076022e+00 7.9e−07 4.4e−04 3.4e−04

0.3 1.123054e+00 8.0e−06 3.1e−03 4.9e−03

0.4 1.178736e+00 4.7e−05 1.2e−02 4.3e−02

x 2F1(1/2, 1; 3/2;x) (15.3.7) (15.3.7) (15.3.7)0.1 1.035488e+00 1.9e−08 2.0e−10 1.6e−12

0.2 1.076022e+00 7.9e−07 8.7e−09 1.5e−10

0.3 1.123054e+00 8.0e−06 9.4e−08 2.7e−09

0.4 1.178736e+00 4.7e−05 5.9e−07 2.5e−08

. . .

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Table 15.3.2: Relative error of the 20th partial sum and 20th (modified)approximants. More details can be found in the Examples 15.3.1, 15.3.2and 15.3.3.

x 2F1(1/2, 1; 3/2;x) (15.3.7) (15.3.12) (15.3.17)0.1 1.035488e+00 4.0e−32 1.5e−20 1.5e−20

0.2 1.076022e+00 1.3e−25 1.5e−14 1.7e−13

0.3 1.123054e+00 1.4e−21 4.8e−11 7.6e−09

0.4 1.178736e+00 1.8e−18 1.4e−08 5.0e−05

x 2F1(1/2, 1; 3/2;x) (15.3.7) (15.3.7) (15.3.7)0.1 1.035488e+00 4.0e−32 2.6e−35 6.5e−38

0.2 1.076022e+00 1.3e−25 8.8e−29 4.8e−31

0.3 1.123054e+00 1.4e−21 1.1e−24 9.6e−27

0.4 1.178736e+00 1.8e−18 1.4e−21 1.9e−23

. . .

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Function categories

Elementary functions

Gamma function and related

functions

Error function and related

integrals

Exponential integrals and

related functions

Hypergeometric functions

2F1(a,b;c;z)

2F1(a,n;c;z)

2F1(a,1;c;z)

2F1(a,b;c;z) /

2F1(a,b+1;c+1;z)

2F1(a,b;c;z) /

2F1(a+1,b+1;c+1;z)

2F1(1/2,1;3/2;z)

2F1(2a,1;a+1;1/2)

Confluent hypergeometric

functions

Bessel functions

Hypergeometric functions » 2F1(1/2,1;3/2;z) » tabulate

Representation

Continue

Special functions :continued fraction and series representations

Handbook

A. CuytW. B. JonesV. PetersenB. VerdonkH. Waadeland

Software

F. BackeljauwS. BecuweM. ColmanA. CuytT. Docx

© Copyright 2008 Universiteit Antwerpen [email protected]

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Function categories



functions


integrals


related functions


2F1(a,b;c;z)

2F1(a,n;c;z)

2F1(a,1;c;z)

2F1(a,b;c;z) /

2F1(a,b+1;c+1;z)

2F1(a,b;c;z) /

2F1(a+1,b+1;c+1;z)

2F1(1/2,1;3/2;z)

2F1(2a,1;a+1;1/2)


functions

Bessel functions


Representation

Input

parameters none

base 1010

approximant (1 ≤ approximant ≤ 999)

z

output in tables relative error absolute error

Output

10th approximant (relative error)

z 2F1(1/2,1;3/2;z) (HY.1.4) (HY.3.7) (HY.3.7) (HY.3.7) (HY.3.12) (HY.3.12) (HY.3.12) (HY.3.17) (HY.3.17) (HY.3.17)

0.1 1.035488e+00 4.6e−13 2.4e−16 6.4e−19 3.0e−21 1.5e−10 4.5e−13 8.5e−15 7.5e−11 3.9e−12 5.4e−14

0.15 1.055046e+00 4.1e−11 1.8e−14 5.0e−17 3.6e−19 8.3e−09 2.8e−11 8.4e−13 7.6e−09 4.0e−10 8.6e−12

0.2 1.076022e+00 1.0e−09 4.3e−13 1.2e−15 1.2e−17 1.4e−07 5.4e−10 2.3e−11 2.4e−07 1.3e−08 4.0e−10

0.25 1.098612e+00 1.2e−08 5.4e−12 1.6e−14 2.1e−16 1.3e−06 5.5e−09 3.1e−10 4.2e−06 2.5e−07 9.8e−09

0.3 1.123054e+00 9.5e−08 4.6e−11 1.4e−13 2.3e−15 7.9e−06 3.8e−08 2.7e−09 5.1e−05 3.3e−06 1.7e−07

0.35 1.149640e+00 5.4e−07 3.0e−10 9.1e−13 1.9e−14 3.6e−05 1.9e−07 1.7e−08 4.9e−04 3.6e−05 2.4e−06

0.4 1.178736e+00 2.5e−06 1.6e−09 5.1e−12 1.3e−13 1.3e−04 8.3e−07 9.1e−08 4.0e−03 3.8e−04 3.4e−05

0.45 1.210806e+00 9.4e−06 7.5e−09 2.5e−11 7.4e−13 4.1e−04 3.0e−06 4.0e−07 3.1e−02 4.6e−03 6.7e−04

Continue


Handbook


Software



10

.1,.15,.2,.25,.3,.35,.4,.45

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Function categories



functions


integrals


related functions


2F1(a,b;c;z)

2F1(a,n;c;z)

2F1(a,1;c;z)

2F1(a,b;c;z) /

2F1(a,b+1;c+1;z)

2F1(a,b;c;z) /

2F1(a+1,b+1;c+1;z)

2F1(1/2,1;3/2;z)

2F1(2a,1;a+1;1/2)


functions

Bessel functions


Representation

Input

parameters none

base 1010


z

output in tables relative error absolute error

Output

z = 1/4 (relative error)

n 2F1(1/2,1;3/2;z) (HY.1.4) (HY.3.7) (HY.3.7) (HY.3.7) (HY.3.12) (HY.3.12) (HY.3.12) (HY.3.17) (HY.3.17) (HY.3.17)

5 1.098612e+00 2.2e−05 2.8e−06 3.1e−08 7.3e−10 1.3e−03 2.1e−05 2.0e−06 1.4e−03 1.7e−04 1.2e−05

10 1.098612e+00 1.2e−08 5.4e−12 1.6e−14 2.1e−16 1.3e−06 5.5e−09 3.1e−10 4.2e−06 2.5e−07 9.8e−09

15 1.098612e+00 8.4e−12 1.0e−17 1.3e−20 1.2e−22 1.3e−09 2.5e−12 9.7e−14 1.4e−08 5.5e−10 1.5e−11

20 1.098612e+00 6.3e−15 2.0e−23 1.4e−26 1.0e−28 1.3e−12 1.4e−15 4.2e−17 5.1e−11 1.5e−12 3.0e−14

25 1.098612e+00 5.0e−18 3.8e−29 1.8e−32 1.0e−34 1.2e−15 8.7e−19 2.1e−20 1.9e−13 4.3e−15 7.2e−17

Continue


Handbook


Software



5,10,15,20,25

.25

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C++

Maple library

symbolic:

f(z) = c0 + c1z+ c2z2 + . . .

f(z) = b0(z) +a1(z)

b1(z) +a2(z)

b2(z) +a3(z)

b3(z) + · · ·

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C++

Web interface

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions

Error function and related integrals » erfc » approximate

Representation

Continue


Handbook


Software



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CFHB live | Error function and related integrals » erfc http://www.cfhblive.ua.ac.be/evaluate/index.php

1 of 1 09/12/2008 03:11 PM

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions


Representation

Input

rhs J-fraction

parameters none

base 1010

digits (5 ≤ digits ≤ 999)


z

tail estimatenone standard improved

user defined (simregular form)

Continue


Handbook


Software



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13

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1 of 1 09/12/2008 03:12 PM

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions


Representation

Input

rhs J-fraction

parameters none

base 1010



z



Output

approximant 3.84214832712064746987580452585280852276469e-20

absolute error 1.792e-46

relative error 4.663e-27

tail estimate 0

Continue


Handbook


Software



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6.5

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1 of 1 09/12/2008 03:13 PM

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions


Representation

Input

rhs J-fraction

parameters none

base 1010



z



Continue


Handbook


Software



42

13

6.5

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1 of 1 09/12/2008 03:14 PM

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions


Representation

Input

rhs J-fraction

parameters none

base 1010



z



Output

approximant 3.84214832712064746987580500914386200233213e-20



tail estimate -6.67500000000000000000000000000000000000000e+01

Continue


Handbook


Software



42

13

6.5

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. C++Type S.IType S.IISeriesType F.IType F.IIFractionsCombineWeb

C++ library

numeric:

f(x) = ±d0.d1 . . .dt−1 × βe

u(t) =12β−t+1,

∣∣∣∣f(x) − f(x)

f(x)

∣∣∣∣ 6 2αu(t), f(x) > 0

⇒f(x)

1 + 2αu(t)6 f(x) 6

f(x)

1 − 2αu(t)

f(x) ≈ F(x) ≈ F(x) = f(x)

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truncation error:

f(x) ≈ F(x)

|f(x) − F(x)|

|f(x)|6 ?

round-off error:F(x) −→ F(x)

+,−,×,÷ ⊕,,⊗,�

|F(x) − F(x)||f(x)|

6 ?

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Example:

√π

2erf(x) =

∞∑m=0

(−1)m

m! (2m+ 1)x2m+1, 0 6 x 6 1

|c2m+1|↘ 0

= x

(1 + x2 c3

c1

(1 + x2 c5

c3(1 + . . .)

))c2m+1

c2m−1=

2m− 1m(2m+ 1)

, 1 6 m 6 231 − 1

β = 10, t = 40,α = 1, x = 0.8↙ ↘∣∣∣∣f(x) − F(x)

f(x)

∣∣∣∣ 6 1210−39

∣∣∣∣F(x) − F(x)f(x)

∣∣∣∣ 6 1210−39

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Example:

exp(x2)√π erfc(x) =

2x2x2+1

1 +

∞Km=1

am(x)

1, x > 0

=

2x2x2+1

1 +

∞Km=1

−2m(2m−1)(2x2+1+4m)(2x2+1+4(m−1))

1

0 > am(x)↘ −1/4

β = 10, t = 40,α = 1, x = 6.5↙ ↘∣∣∣∣f(x) − F(x)

f(x)

∣∣∣∣ 6 1210−39

∣∣∣∣F(x) − F(x)f(x)

∣∣∣∣ 6 1210−39

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C++. Type S.IType S.IISeriesType F.IType F.IIFractionsCombineWeb

Type S.I

monotonely decreasing am > 0

f(x) =

∞∑m=0

amxm

Example:

exp(x) =

∞∑m=0

xm

m!, x ∈ R

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C++Type S.I. Type S.IISeriesType F.IType F.IIFractionsCombineWeb

Type S.II

alternating am with |am| monotonely decreasing

Example:

erf(x) =2√π

∞∑m=0

(−1)mx2m+1

(2m+ 1)m!, x ∈ R

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C++Type S.IType S.II. SeriesType F.IType F.IIFractionsCombineWeb

F(x) =

n∑m=0

cmxm:

0 6 cmxm ↘ 0 :

∣∣∣∣∣∞∑

m=n+1

cmxm

∣∣∣∣∣ 6 cnxn

1 − R,

cmx

cm−16 R < 1

0 6 c0, |cmxm|↘ 0 :

∣∣∣∣∣∞∑

m=n+1

cmxm

∣∣∣∣∣ 6 |cn+1xn+1|, n odd

|f(x)| > |F(x)|⇒∣∣∣∣f(x) − F(x)

f(x)

∣∣∣∣ 6{

cn|xn|/(1−R)|F(x)|

|cn+1xn+1|

|F(x)|

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Example:

√π

2erf(x) =

∞∑m=0

(−1)m

m! (2m+ 1)x2m+1

|F(x)| > |x− x3/3| > 0.62

|c1x| = 0.80, |c3x3| = 0.17↘ |c61x61| 6 7.58× 10−41

n = 29 (2n+ 1 = 59),∣∣∣∣f(x) − F(x)

f(x)

∣∣∣∣ 6 1.27× 10−40

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F(x) = c0

(1 + c1

c0x(1 + c2

c1x(1 + . . . + cn

cn−1x). . .))

:

r0 = c0, r0 = r0(1 + ρ0), |ρ0| 6ε(r0)u(s)

1−ε(r0)u(s)

rm =cm

cm−1x, rm = rm(1 + ρm), |ρm| 6 ε(rm)u(s)

1−ε(rm)u(s)

∣∣∣∣F(x) − F(x)f(x)

∣∣∣∣ 6 ε(F)u(s)

1 − ε(F)u(s)

F+(|x|)

|F(x)|

F+(x) =

n∑m=0

|cm|xm,F+(|x|)

|F(x)|> 1

ε(F) = 1 + ε(r0) + n

(2 +

nmaxm=1

ε(rm)

)

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Example:

√π

2erf(x) =

∞∑m=0

c2m+1x2m+1

r0 = x, ε(r0) = 0, rm =x2(2m− 1)m(2m+ 1)

, ε(rm) = 4

n = 29, ε(F) = 175,F+(|x|)

|F(x)|6 2

ε(F)β−s+1

1 − ε(F)β−s+1/26

1210−39 ⇒ s > 43

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C++Type S.IType S.IISeries. Type F.IType F.IIFractionsCombineWeb

Type F.I

monotone behaviour of am(x)

f(x) =∞Km=1

am(x)

1

=a1(x)

1 +a2(x)

1 +a3(x)

1 + · · ·

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C++Type S.IType S.IISeries. Type F.IType F.IIFractionsCombineWeb

Example:

erfc(x) =∞Km=1

am(x)

1, x > 0

=

2x exp(−x2)√π(2x2+1)

1 +

∞Km=1

−2m(2m−1)(2x2+1+4m)(2x2+1+4(m−1))

1

am(x)↘ −1/4, tn(x)↘ −1/2

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C++Type S.IType S.IISeriesType F.I. Type F.IIFractionsCombineWeb

Type F.II

interval bound for am(x) from m > µ on

f(x) =∞Km=1

am(x)

1

crude tµ interval → finer tn interval

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C++Type S.IType S.IISeriesType F.I. Type F.IIFractionsCombineWeb

Example:

f(x) =2F1(a,k; c; x)

2F1(a,k+ 1; c+ 1; x)= 1 +

∞Km=1

amx

1,

x < 1

a2m+1 =−(a+m)(c− k+m)

(c+ 2m)(c+ 2m+ 1), m > 0

a2m =−(k+m)(c− a+m)

(c+ 2m− 1)(c+ 2m), m > 1

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C++Type S.IType S.IISeriesType F.IType F.II. FractionsCombineWeb

F(x) = fn(x;wn),bm(z) = 1:

Ln 6 tn ≈ wn 6 Rn

|f(x) − F(x)|

|f(x)|6Rn − Ln

1 + Ln

n−1∏m=1

Mm

Mm = max{∣∣∣∣ Lm

1 + Lm

∣∣∣∣ , ∣∣∣∣ Rm

1 + Rm

∣∣∣∣} , m = 1, . . . ,n− 1

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Example:


2x2x2 + 1 +

∞Km=2

am(x)

1

tn =∞K

m=n+1

am(x)

1, n > 1

Ln =∞K

m=n+1

−1/4

1= −

12

Rn =∞K

m=n+1

an+1(x)

1=

−1 +√

1 + 4an+1(x)

2

n = 13

M1 6 2.62× 10−4 ↗M12 6 3.43× 10−2

2(Rn − Ln) 6 2.64× 10−16,12∏m=1

Mm 6 1.27× 10−25

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F(x) = fn(z;wn),bm(z) = 1:

am = am(1 + δm), max(|δ1|, . . . , |δn|) 6 ∆ u(s)

|F(x) − F(x)||f(x)|

6 (4 + ∆)Mn − 1M− 1

u(s)

M = max{M1, . . . ,Mn}

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Example:


2x2x2 + 1 +

∞Km=1

am(x)

1

∆ =16

1 − 16u(s),

n = 13,

M = M13 6 3.84× 10−2

(4 +

161 − 8β−s+1

)Mn − 1M− 1

β−s+1

26

12β−39 ⇒ s > 42

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C++Type S.IType S.IISeriesType F.IType F.IIFractions. CombineWeb

Combinations

×,÷ of previous types:

relative error of the form∣∣∣∣∣∏hi=1(1 + δi)∏h+ki=h+1(1 + δi)

∣∣∣∣∣is bounded by 1 + ε if for 1 6 i 6 h+ k

|δi| 6diε

1 + ε,

h+k∑i=1

di = 1

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Example:

J5/2(x) = J1/2(x)J3/2(x)

J1/2(x)

J5/2(x)

J3/2(x)

J1/2(x) =

√2πx

sin(x)

Jν+1(x)

Jν(x)=x/(2ν+ 2)

1 +

∞Km=2

(ix)2/(4(ν+m− 1)(ν+m))

1,

x ∈ R, ν > 0

am(x)↗ 0

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special function series continued fractionγ(a, x) a > 0, x 6= 0Γ(a, x) a ∈ R, x > 0erf(x) |x| 6 1 identity via erfc(x)erfc(x) identity via erf(x) |x| > 1

dawson(x) |x| 6 1 |x| > 1Fresnel S(x) x ∈ RFresnel C(x) x ∈ REn(x), n > 0 n ∈ N, x > 02F1(a,n; c; x) a ∈ R,n ∈ Z,

c ∈ R \ Z−0 , x < 1

1F1(n; c; x) n ∈ Z,c ∈ R \ Z−

0 , x ∈ RIn(x) n = 0, x ∈ R n ∈ N, x ∈ RJn(x) n = 0, x ∈ R n ∈ N, x ∈ R

In+1/2(x) n = 0, x ∈ R n ∈ N, x ∈ RJn+1/2(x) n = 0, x ∈ R n ∈ N, x ∈ R

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C++Type S.IType S.IISeriesType F.IType F.IIFractionsCombine. Web

Web interface

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions

Error function and related integrals » erfc » evaluate

Representation

Input

parameters none

base 10^k10^k k= (base 2^k: 1 ≤ k ≤ 24; base 10^k: 1 ≤ k ≤ 7)


verbose yes no

x

Continue


Handbook


Software



1

40

6.5

54 / 59

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1 of 1 09/12/2008 04:38 PM

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions

Error function and related integrals » erfc » evaluate

Representation

Input

parameters none



verbose yes no

x

Output

value 3.842148327120647469875804543768776621449e-20

absolute error bound 6.655e-61 (CF)

relative error bound 1.732e-41 (CF)

tail estimate -3.691114343068670e-02

working precision 42 (CF)

approximant 13

Continue


Handbook


Software



1

40

6.5

55 / 59

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1 of 1 09/12/2008 03:16 PM

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions


Representation

Input

rhs J-fraction

parameters none

base 1010



z



Continue


Handbook


Software



42

13

6.5

-3.691114343068670e-02

56 / 59

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1 of 1 09/12/2008 03:17 PM

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions


Representation

Input

rhs J-fraction

parameters none

base 1010



z



Output

approximant 3.84214832712064746987580454376877662144924e-20



tail estimate -3.691114343068670e-02

Continue


Handbook


Software



42

13

6.5

-3.691114343068670e-02

57 / 59

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Function categories



functions


integrals


related functions



functions

Bessel functions

J(nu, z)

J(n, z)

J(nu+1, z) / J(nu, z)

j(n, z)

j(n+1, z) / j(n, z)

I(nu, z)

I(n, z)

I(nu+1, z) / I(nu, z)

i(n, z)

i(n+1, z) / i(n, z)

Bessel functions » I(n, z) » evaluate

Representation

Input

parameters n



verbose yes no

x

n

Continue


Handbook


Software



1

50

4.5

4

58 / 59

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59 / 59

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Continued Fractions for Special Functions: …Handbook of Continued Fractions for Special Functions Special functions are pervasive in all fields of science and industry. The most