Continued Fractions for Special Functions: Handbook and Software
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1 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 1 / 59
2 Project Project Deliverables: : formulas with tables and graphs library: all fractions C ++ library: all functions Web version to explore: all of the above ( 2 / 59
3 Project Related to larger-scale projects: dlmf.nist.gov functions.wolfram.com Only 10% of the CF representations are also found there! 3 / 59
4 Project 4 / 59
5 Project 5 / 59
6 Project special function: f(z) continued fraction: b 0 (z) + K m=1 a m (z) b m (z) f(z) = b 0 (z) + b 1 (z) + a 1 (z) a 2 (z) b 2 (z) + a 3(z) b 3 (z) + = b 0 (z) + a 1(z) a 2 (z) b 1 (z) + b 2 (z) + 6 / 59
7 Project SF: limit-periodic representation CF: larger convergence domain Example: Atan(z) = m=0 ( 1) m+1 2m + 1 z2m+1, z < 1 = z 1 + Km=2 (m 1) 2 z 2 2m 1, iz (, 1) (1, + ) 7 / 59
8 Project tail: t n (z) = K m=n+1 Example: 1 + 4x 1 2 = Km=1 a m (z) b m (z) x 1, x 1 4, t n(x) = f(x) = , t 2n 2 1 t 2n = Km=1 m(m + 2), t n 1 8 / 59
9 Project approximant: f n (z; 0) = b 0 (z) + n Km=1 a m (z) b m (z) modified approximant: f n (z; w n ) = b 0 (z) + n 1 Km=1 a m (z) a n (z) b m (z) + b n (z) + w n 9 / 59
10 Part I Part II Part III Web 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 Cuyt Petersen Verdonk Waadeland Jones A. Cuyt V. Brevik Petersen B. Verdonk H. Waadeland W. B. Jones 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 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 Handbook of Continued Fractions for Special Functions Handbook of Continued Fractions for Special Functions / 59
11 Part I: Basic theory Part I Part II Part III Web Basics Continued fraction representation of functions Convergence criteria Padé approximants Moment theory and orthogonal polynomials 11 / 59
12 Part II: Numerics Part I Part II Part III Web Continued fraction construction Truncation error bounds Continued fraction evaluation 12 / 59
13 Part I Part II Part III Web Part III: Special functions Elementary functions Incomplete gamma and related Error function and related Exponential integrals Hypergeometric 2 F 1 (a, n; c; z), n Z Confluent hypergeometric 1 F 1 (n; b; z), n Z Parabolic cylinder functions Legendre functions Bessel functions of integer and fractional order Coulomb wave functions Zeta function and related 13 / 59
14 Project Part I Part II Part III Web Ln Formulas Ln(1 + z) = z 1 + Km=2 = 2z 2 + z + ( ) 1 + z = 2z 1 z 1 + Km=1 ( am z ), Arg(1 + z) < π, (11.2.2) 1 Km=2 a 2k = k 2(2k 1), a k 2k+1 = 2(2k + 1) ( (m 1) 2 z 2 ( am z 2 1 a m = (2m 1)(2 + z) ), (11.2.3) Arg ( 1 z 2 /(2 + z) 2) < π ), Arg(1 z 2 ) < π, (11.2.4) m 2 (2m 1)(2m + 1). AS AS 14 / 59
15 Tables Part I Part II Part III Web Table : Relative error of 20 th partial sum and 20 th approximants. x Ln(1 + x) (11.2.1) (11.2.2) (11.2.3) (6.8.8) e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e / 59
16 Part I Part II Part III Web Modification: since in (11.2.2), lim m a m z = z/4 and a m lim m a m 1 4 = 1, we find w(z) = z 2 and { (1) w 2k (z) = w(z) + kz w (1) 2k+1 2(2k+1) z 4 (z) = w(z) + (k+1)z 2(2k+1) z 4 16 / 59
17 Part I Part II Part III Web Table : Relative error of 20 th (modified) approximants. x Ln(1 + x) (11.2.2) (11.2.2) (11.2.2) e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e / 59
18 Graphs Part I Part II Part III Web Figure : Complex region where f 8 (z; 0) of (11.2.2) guarantees k significant digits for Ln(1 + z) (from light to dark k = 6, 7, 8, 9). 18 / 59
19 Part I Part II Part III Web Figure : Number of significant digits guaranteed by the n th classical approximant of (11.2.2) (from light to dark n = 5, 6, 7) and the 5 th modified approximant evaluated with w (1) 5 (z) (darkest). 19 / 59
20 Web interface Part I Part II Part III Web 2F 1 ( 1 /2, 1; 3 /2; z) = 1 ( ) 1 + z 2 z Ln 1 z series representation continued fraction representations: C-fraction (15.3.7) M-fraction ( ) Nörlund fraction ( ) 20 / 59
21 Project Part I Part II Part III Web with z 2 F 1 ( 1 / 2, 1; 3 / 2; z) = c 1 = 1, c m = Km=1 ( cm z ), z C \ [1, + ) (15.3.7a) 1 (m 1)2 4(m 1) 2, 1 m 2. (15.3.7b) 1/2 z 2F 1 ( 1 / 2, 1; 3 / 2; z) = 1/2 + z/2 3/2 + 3z/2 4z 5/2 + 5z/2... ( ) 2F 1 ( 1 / 2, 1; 3 / 2; z) = 1 z(1 z) 1 z + 3/ 2 5 / 2z + Km=2 ( ) m(m 1/ 2)z(1 z), (m + 1 / 2) (2m + 1 / 2)z Rz < 1/2. ( ) 21 / 59
22 Part I Part II Part III Web Table : Relative error of the 5 th partial sum and 5 th (modified) approximants. More details can be found in the Examples , and x 2 F 1 ( 1 / 2, 1; 3 / 2; x) (15.3.7) ( ) ( ) e e e e e e e e e e e e e e e e 02 x 2 F 1 ( 1 / 2, 1; 3 / 2; x) (15.3.7) (15.3.7) (15.3.7) e e e e e e e e e e e e e e e e / 59
23 Part I Part II Part III Web Table : Relative error of the 20 th partial sum and 20 th (modified) approximants. More details can be found in the Examples , and x 2 F 1 ( 1 / 2, 1; 3 / 2; x) (15.3.7) ( ) ( ) e e e e e e e e e e e e e e e e 05 x 2 F 1 ( 1 / 2, 1; 3 / 2; x) (15.3.7) (15.3.7) (15.3.7) e e e e e e e e e e e e e e e e / 59
24 Part I Part II Part III Web Special functions : continued fraction and series representations Function categories Elementary functions Gamma function and related functions Error function and related integrals Exponential integrals and related functions Hypergeometric functions» 2F1(1/2,1;3/2;z)» tabulate Representation Handbook A. Cuyt W. B. Jones V. Petersen B. Verdonk H. Waadeland Software F. Backeljauw S. Becuwe M. Colman A. Cuyt T. Docx 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 Continue Copyright 2008 Universiteit Antwerpen contact@cfhblive.ua.ac.be 24 / 59
25 functions Bessel functions Project Part I Part II Part III Web Input parameters none base 10 approximant 10 (1 approximant 999) z.1,.15,.2,.25,.3,.35,.4,.45 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) e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e 04 Continue Copyright 2008 Universiteit Antwerpen contact@cfhblive.ua.ac.be 25 / 59
26 Confluent hypergeometric functions Bessel functions Project Part I Part II Part III Web Input parameters none base 10 approximant 5,10,15,20,25 (1 approximant 999) z.25 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) e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e 17 Continue Copyright 2008 Universiteit Antwerpen contact@cfhblive.ua.ac.be 26 / 59
27 library Library Web symbolic: f(z) = c 0 + c 1 z + c 2 z f(z) = b 0 (z) + b 1 (z) + a 1 (z) a 2 (z) b 2 (z) + a 3(z) b 3 (z) + 27 / 59
28 Web interface Library Web Special functions : continued fraction and series representations Handbook Software A. Cuyt F. Backeljauw W. B. Jones S. Becuwe V. Petersen M. Colman B. Verdonk A. Cuyt H. Waadeland T. Docx Function categories Error function and related integrals» erfc» approximate Elementary functions Gamma function and related functions Representation Error function and related integrals erf dawson erfc Fresnel C Fresnel S Exponential integrals and related functions Hypergeometric functions Confluent hypergeometric functions Bessel functions Continue Copyright 2008 Universiteit Antwerpen contact@cfhblive.ua.ac.be 28 / 59
29 CFHB live Error function and related integrals» erfc Project Library Web Special functions : continued fraction and series representations Function categories Elementary functions Gamma function and related functions Error function and related integrals» erfc» approximate Representation Handbook A. Cuyt W. B. Jones V. Petersen B. Verdonk H. Waadeland Software F. Backeljauw S. Becuwe M. Colman A. Cuyt T. Docx Error function and related integrals erf dawson erfc Fresnel C Fresnel S Exponential integrals and related functions Input rhs J-fraction Hypergeometric functions Confluent hypergeometric functions Bessel functions parameters none base 10 digits 42 (5 digits 999) approximant 13 (1 approximant 999) z 6.5 tail estimate none standard improved user defined (simregular form) Continue Copyright 2008 Universiteit Antwerpen contact@cfhblive.ua.ac.be 29 / 59
30 Special functions : continued fraction and series representations A. Cuyt F. Backeljauw W. B. Jones S. Becuwe V. Petersen M. Colman B. Verdonk A. Cuyt H. Waadeland T. Docx Project Library Web Function categories Elementary functions Gamma function and related functions Error function and related integrals erf dawson erfc Fresnel C Fresnel S Exponential integrals and related functions Hypergeometric functions Confluent hypergeometric functions Bessel functions Error function and related integrals» erfc» approximate Representation Input rhs parameters J-fraction none base 10 digits 42 (5 digits 999) approximant 13 (1 approximant 999) z 6.5 tail estimate none standard improved user defined (simregular form) Output approximant absolute error relative error e e e-27 tail estimate 0 Continue Copyright 2008 Universiteit Antwerpen contact@cfhblive.ua.ac.be 30 / 59
31 CFHB live Error function and related integrals» erfc Project Library Web Special functions : continued fraction and series representations Function categories Elementary functions Gamma function and related functions Error function and related integrals» erfc» approximate Representation Handbook A. Cuyt W. B. Jones V. Petersen B. Verdonk H. Waadeland Software F. Backeljauw S. Becuwe M. Colman A. Cuyt T. Docx Error function and related integrals erf dawson erfc Fresnel C Fresnel S Exponential integrals and related functions Input rhs J-fraction Hypergeometric functions Confluent hypergeometric functions Bessel functions parameters none base 10 digits 42 (5 digits 999) approximant 13 (1 approximant 999) z 6.5 tail estimate none standard improved user defined (simregular form) Continue Copyright 2008 Universiteit Antwerpen contact@cfhblive.ua.ac.be 31 / 59
32 Special functions : continued fraction and series representations A. Cuyt F. Backeljauw W. B. Jones S. Becuwe V. Petersen M. Colman B. Verdonk A. Cuyt H. Waadeland T. Docx Project Library Web Function categories Elementary functions Gamma function and related functions Error function and related integrals erf dawson erfc Fresnel C Fresnel S Exponential integrals and related functions Hypergeometric functions Confluent hypergeometric functions Bessel functions Error function and related integrals» erfc» approximate Representation Input rhs parameters J-fraction none base 10 digits 42 (5 digits 999) approximant 13 (1 approximant 999) z 6.5 tail estimate none standard improved user defined (simregular form) Output approximant absolute error relative error tail estimate e e e e+01 Continue Copyright 2008 Universiteit Antwerpen contact@cfhblive.ua.ac.be 32 / 59
33 Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web numeric: C ++ u(t) = 1 2 β t+1, library f(x) = ±d 0.d 1... d t 1 β e f(x) f(x) f(x) 2α u(t), f(x) > 0 f(x) 1 + 2αu(t) f(x) f(x) 1 2αu(t) f(x) F(x) F(x) = f(x) 33 / 59
34 Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web truncation error: f(x) F(x) round-off error: f(x) F(x) f(x) F(x) F(x) +,,,,,,? F(x) F(x) f(x)? 34 / 59
35 Example: Project Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web π 2 erf(x) = m=0 ( 1) m m! (2m + 1) x2m+1, 0 x 1 c 2m+1 0 ( = x 1 + x 2 c ( x 2 c )) 5 (1 +...) c 1 c 3 c 2m+1 = 2m 1 c 2m 1 m(2m + 1), 1 m f(x) F(x) f(x) β = 10, t = 40, α = 1, x = 0.8 F(x) F(x) f(x) / 59
36 Example: Project Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web exp(x 2 ) π erfc(x) = = 2x 2x x 2x Km=1 Km=1 a m (x), x > 0 1 2m(2m 1) (2x m)(2x (m 1)) 1 0 a m (x) 1/4 f(x) F(x) f(x) β = 10, t = 40, α = 1, x = 6.5 F(x) F(x) f(x) / 59
37 Type S.I Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web monotonely decreasing a m > 0 f(x) = a m x m m=0 Example: x m exp(x) = m!, m=0 x R 37 / 59
38 Type S.II Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web alternating a m with a m monotonely decreasing Example: erf(x) = 2 π m=0 ( 1) m x 2m+1 (2m + 1)m!, x R 38 / 59
39 Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web F(x) = n c m x m : m=0 0 c m x m 0 : 0 c 0, c m x m 0 : m=n+1 c m x m c nx n 1 R, c m x c m 1 R < 1 m=n+1 c m x m c n+1 x n+1, n odd f(x) F(x) f(x) F(x) f(x) { cn x n /(1 R) F(x) c n+1 x n+1 F(x) 39 / 59
40 Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web Example: π 2 erf(x) = m=0 ( 1) m m! (2m + 1) x2m+1 F(x) x x 3 / c 1 x = 0.80, c 3 x 3 = 0.17 c 61 x n = 29 (2n + 1 = 59), f(x) F(x) f(x) / 59
41 Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web ( ( ) )) F(x) = c 0 (1 + c 1 c 0 x 1 + c 2 c 1 x c n c n 1 x... : r 0 = c 0, r 0 = r 0 (1 + ρ 0 ), ρ 0 ɛ(r 0)u(s) 1 ɛ(r 0 )u(s) r m = c m x, r m = r m (1 + ρ m ), ρ m ɛ(r m)u(s) c 1 ɛ(r m )u(s) m 1 F(x) F(x) f(x) ɛ(f)u(s) F + ( x ) 1 ɛ(f)u(s) F(x) F + (x) = n c m x m, m=0 ( ɛ(f) = 1 + ɛ(r 0 ) + n F + ( x ) F(x) n max m=1 ɛ(r m) ) 41 / 59
42 Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web Example: π 2 erf(x) = m=0 c 2m+1 x 2m+1 r 0 = x, ɛ(r 0 ) = 0, r m = x2 (2m 1) m(2m + 1), ɛ(r m) = 4 n = 29, ɛ(f) = 175, F + ( x ) F(x) 2 ɛ(f)β s+1 1 ɛ(f)β s+1 / s / 59
43 Type F.I Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web monotone behaviour of a m (x) f(x) = = Km=1 a m (x) 1 a 1 (x) 1 + a 2(x) 1 + a 3(x) / 59
44 Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web Example: erfc(x) = = Km=1 a m (x), x > 0 1 2x exp( x 2 ) π(2x 2 +1) 1 + Km=1 2m(2m 1) (2x m)(2x (m 1)) 1 a m (x) 1/4, t n (x) 1/2 44 / 59
45 Type F.II Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web interval bound for a m (x) from m µ on f(x) = Km=1 a m (x) 1 crude t µ interval finer t n interval 45 / 59
46 Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web Example: f(x) = 2F 1 (a, k; c; x) 2F 1 (a, k + 1; c + 1; x) = 1 + x < 1 Km=1 a m x 1, (a + m)(c k + m) a 2m+1 = (c + 2m)(c + 2m + 1), m 0 a 2m = (k + m)(c a + m) (c + 2m 1)(c + 2m), m 1 46 / 59
47 Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web F(x) = f n (x; w n ), b m (z) = 1: f(x) F(x) f(x) L n t n w n R n R n L n 1 + L n n 1 m=1 M m { } L m M m = max 1 + L m, R m 1 + R m, m = 1,..., n 1 47 / 59
48 Example: Project Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web exp(x 2 ) π erfc(x) = t n = L n = R n = K m=n+1 1/4 K m=n+1 1 2x 2x a m (x), n 1 1 = 1 2 Km=2 a m (x) 1 a K n+1 (x) = a n+1 (x) m=n n = 13 M M (R n L n ) , 12 M m m=1 48 / 59
49 Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web F(x) = f n (z; w n ), b m (z) = 1: ã m = a m (1 + δ m ), F(x) F(x) f(x) max( δ 1,..., δ n ) u(s) (4 + ) Mn 1 M 1 u(s) M = max{m 1,..., M n } 49 / 59
50 Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web Example: exp(x 2 ) π erfc(x) = = 2x 2x u(s), n = 13, M = M Km=1 a m (x) 1 ( ) 16 M n β s+1 M 1 β s β 39 s / 59
51 Combinations Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web, of previous types: relative error of the form h i=1 (1 + δ i) h+k i=h+1 (1 + δ i) is bounded by 1 + ɛ if for 1 i h + k δ i d h+k iɛ 1 + ɛ, d i = 1 i=1 51 / 59
52 Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web Example: 2 J 1/2 (x) = πx sin(x) J ν+1 (x) x/(2ν + 2) = J ν (x) 1 + J 5/2 (x) = J 1/2 (x) J 3/2(x) J 5/2 (x) J 1/2 (x) J 3/2 (x) Km=2 (ix) 2 /(4(ν + m 1)(ν + m)), 1 x R, ν 0 a m (x) 0 52 / 59
53 Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web special function series continued fraction γ(a, x) a > 0, x 0 Γ(a, x) a R, x 0 erf(x) x 1 identity via erfc(x) erfc(x) identity via erf(x) x > 1 dawson(x) x 1 x > 1 Fresnel S(x) x R Fresnel C(x) x R E n (x), n > 0 n N, x > 0 2F 1 (a, n; c; x) a R, n Z, c R \ Z 0, x < 1 1F 1 (n; c; x) n Z, c R \ Z 0, x R I n (x) n = 0, x R n N, x R J n (x) n = 0, x R n N, x R I n+1/2 (x) n = 0, x R n N, x R J n+1/2 (x) n = 0, x R n N, x R 53 / 59
54 Web interface Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web Special functions : continued fraction and series representations Function categories Elementary functions Gamma function and related functions Error function and related integrals erf dawson erfc Fresnel C Fresnel S Exponential integrals and related functions Hypergeometric functions Error function and related integrals» erfc» evaluate Representation Handbook A. Cuyt W. B. Jones V. Petersen B. Verdonk H. Waadeland Software F. Backeljauw S. Becuwe M. Colman A. Cuyt T. Docx Confluent hypergeometric functions Bessel functions Input parameters none 10^k k= 1 (base 2^k: 1 k 24; base 10^k: 1 k 7) base 40 digits (5 digits 999) verbose yes no x 6.5 Continue Copyright 2008 Universiteit Antwerpen contact@cfhblive.ua.ac.be 54 / 59
55 continued fraction and series representations B. Verdonk A. Cuyt H. Waadeland T. Docx Project Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web Function categories Elementary functions Gamma function and related functions Error function and related integrals erf dawson erfc Fresnel C Fresnel S Exponential integrals and related functions Hypergeometric functions Confluent hypergeometric functions Bessel functions Error function and related integrals» erfc» evaluate Representation Input parameters none 10^k k= 1 (base 2^k: 1 k 24; base 10^k: 1 k 7) base 40 digits (5 digits 999) verbose yes no x 6.5 Output value absolute error bound relative error bound tail estimate working precision e e-61 (CF) 1.732e-41 (CF) e (CF) approximant 13 Continue Copyright 2008 Universiteit Antwerpen contact@cfhblive.ua.ac.be 55 / 59
56 CFHB live Error function and related integrals» erfc Project Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web Special functions : continued fraction and series representations Function categories Elementary functions Gamma function and related functions Error function and related integrals erf dawson erfc Fresnel C Fresnel S Exponential integrals and related functions Error function and related integrals» erfc» approximate Representation Input rhs J-fraction Handbook A. Cuyt W. B. Jones V. Petersen B. Verdonk H. Waadeland Software F. Backeljauw S. Becuwe M. Colman A. Cuyt T. Docx Hypergeometric functions Confluent hypergeometric functions Bessel functions parameters none base 10 digits 42 (5 digits 999) approximant 13 (1 approximant 999) z 6.5 tail estimate none standard improved user defined (simregular form) e-02 Continue Copyright 2008 Universiteit Antwerpen contact@cfhblive.ua.ac.be 56 / 59
57 Special functions : continued fraction and series representations A. Cuyt F. Backeljauw W. B. Jones S. Becuwe V. Petersen M. Colman B. Verdonk A. Cuyt H. Waadeland T. Docx Project Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web Function categories Elementary functions Gamma function and related functions Error function and related integrals erf dawson erfc Fresnel C Fresnel S Exponential integrals and related functions Hypergeometric functions Confluent hypergeometric functions Bessel functions Error function and related integrals» erfc» approximate Representation Input rhs parameters J-fraction none base 10 digits 42 (5 digits 999) approximant 13 (1 approximant 999) z 6.5 tail estimate none standard improved user defined (simregular form) e-02 Output approximant absolute error relative error tail estimate e e e e-02 Continue Copyright 2008 Universiteit Antwerpen contact@cfhblive.ua.ac.be 57 / 59
58 Special functions : continued fraction and series representations Handbook Software A. Cuyt F. Backeljauw W. B. Jones S. Becuwe V. Petersen M. Colman B. Verdonk A. Cuyt H. Waadeland T. Docx Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web Function categories Elementary functions Gamma function and related functions Error function and related integrals Exponential integrals and related functions Hypergeometric functions Confluent hypergeometric 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 10^k k= 1 (base 2^k: 1 k 24; base 10^k: 1 k 7) base 50 digits (5 digits 999) verbose yes no x 4.5 n 4 Continue Copyright 2008 Universiteit Antwerpen contact@cfhblive.ua.ac.be 58 / 59
59 H. Waadeland T. Docx Project Type S.I Type S.II Series Type F.I Type F.II Fractions Combine Web Function categories Elementary functions Gamma function and related functions Error function and related integrals Exponential integrals and related functions Hypergeometric functions Confluent hypergeometric 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 10^k k= 1 (base 2^k: 1 k 24; base 10^k: 1 k 7) base 50 digits (5 digits 999) verbose yes no x 4.5 n 4 Output value absolute error bound relative error bound tail estimate working precision approximant 1.106e-51, 1.017e-53, 2.128e-53, 6.851e-53 (CF), 1.036e-53 (SE) 1.256e-51, 1.471e-53, 3.818e-53, 1.486e-52 (CF), 5.931e-55 (SE) e-03, e-03, e-03, e-03 57, 59, 58, 57 (CF), 56 (SE) 25, 25, 24, 23 (CF), 33 (SE) Continue Copyright 2008 Universiteit Antwerpen contact@cfhblive.ua.ac.be 59 / 59
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