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31: 19.12 Asymptotic Approximations

… ►Asymptotic approximations for $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$, with different variables, are given in Karp et al. (2007). They are useful primarily when $(1-k)/(1-\sin \varphi )$ is either small or large compared with 1. …

32: 10.69 Uniform Asymptotic Expansions for Large Order

§10.69 Uniform Asymptotic Expansions for Large Order

… ►

10.69.1 $$\xi ={(1+\mathrm{i}{x}^2)}^{1/2}.$$

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Defines:

$\xi$ (locally)

Symbols:

$\mathrm{i}$: imaginary unit and $x$: real variable

A&S Ref:

9.10.41

Permalink:

http://dlmf.nist.gov/10.69.E1

Encodings:

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See also:

Annotations for §10.69 and Ch.10

… ►

10.69.2 $${\mathrm{ber}}_{\nu }\left(\nu x\right)+\mathrm{i}{\mathrm{bei}}_{\nu }\left(\nu x\right)\sim \frac{{\mathrm{e}}^{\nu \xi }}{{(2\pi \nu \xi )}^{1/2}}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{\nu }\sum _{k=0}^{\mathrm{\infty }}\frac{U_k({\xi }^{-1})}{{\nu }^k},$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter, $U_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.37

Referenced by:

§10.69

Permalink:

http://dlmf.nist.gov/10.69.E2

Encodings:

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See also:

Annotations for §10.69 and Ch.10

… ► … ►

33: 33.9 Expansions in Series of Bessel Functions

… ►

§33.9(i) Spherical Bessel Functions

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33.9.1 $$F_{\mathrm{\ell }}(\eta ,\rho )=\rho \sum _{k=0}^{\mathrm{\infty }}{a}_k{𝗃}_{\mathrm{\ell }+k}\left(\rho \right),$$

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Symbols:

$F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $a_k$: coefficients

A&S Ref:

14.4.5

Referenced by:

§33.9(i), §33.9(i)

Permalink:

http://dlmf.nist.gov/33.9.E1

Encodings:

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See also:

Annotations for §33.9(i), §33.9 and Ch.33

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§33.9(ii) Bessel Functions and Modified Bessel Functions

… ►

33.9.3 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\frac{(2\mathrm{\ell }+1)!}{{(2\eta )}^{2\mathrm{\ell }+1}}{\rho }^{-\mathrm{\ell }}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{b}_k{t}^{k/2}{I}_k\left(2\sqrt{t}\right),$$ $\eta >0$,

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $t$: parameter and $b_k$: coefficient

A&S Ref:

14.4.1

Referenced by:

§33.9(ii), §33.9(ii)

Permalink:

http://dlmf.nist.gov/33.9.E3

Encodings:

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See also:

Annotations for §33.9(ii), §33.9 and Ch.33

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33.9.4 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\frac{(2\mathrm{\ell }+1)!}{{(2\left|\eta \right|)}^{2\mathrm{\ell }+1}}{\rho }^{-\mathrm{\ell }}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{b}_k{t}^{k/2}{J}_k\left(2\sqrt{t}\right),$$ .

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!$: factorial (as in $n!$), $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $t$: parameter and $b_k$: coefficient

Referenced by:

§33.9(ii), §33.9(ii)

Permalink:

http://dlmf.nist.gov/33.9.E4

Encodings:

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See also:

Annotations for §33.9(ii), §33.9 and Ch.33

…

34: 27.14 Unrestricted Partitions

… ►For large $n$ ►

27.14.8 $$p\left(n\right)\sim \frac{{\mathrm{e}}^{K\sqrt{n}}}{4n\sqrt{3}},$$

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Symbols:

$\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $p\left(n\right)$: total number of partitions of $n$ , $n$: positive integer and $K$: change of variable

A&S Ref:

24.2.1 III

Permalink:

http://dlmf.nist.gov/27.14.E8

Encodings:

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See also:

Annotations for §27.14(iii), §27.14 and Ch.27

… ►

27.14.9 $$p\left(n\right)=\frac{1}{\pi \sqrt{2}}\sum _{k=1}^{\mathrm{\infty }}\sqrt{k}{A}_k(n){\left[\frac{d}{dt}\frac{\sinh \left(K\sqrt{t}/k\right)}{\sqrt{t}}\right]}_{t=n-(1/24)},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sinh z$: hyperbolic sine function, $p\left(n\right)$: total number of partitions of $n$ , $k$: positive integer, $n$: positive integer, $K$: change of variable and $A_k(n)$: coefficients

A&S Ref:

24.2.1 I.C

Referenced by:

§27.20

Permalink:

http://dlmf.nist.gov/27.14.E9

Encodings:

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See also:

Annotations for §27.14(iii), §27.14 and Ch.27

…

35: 10.40 Asymptotic Expansions for Large Argument

… ►

10.40.2 $$K_{\nu }\left(z\right)\sim {\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.2

Referenced by:

§10.40(i), §10.40(i), §10.40(ii), §10.40(iii), §10.41(iv), §10.45

Permalink:

http://dlmf.nist.gov/10.40.E2

Encodings:

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See also:

Annotations for §10.40(i), §10.40 and Ch.10

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36: 13.20 Uniform Asymptotic Approximations for Large $\mu$

§13.20 Uniform Asymptotic Approximations for Large $\mu$

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§13.20(i) Large $\mu$, Fixed $\kappa$

… ► … ►

§13.20(v) Large $\mu$, Other Expansions

… ►

37: Bibliography T

… ►

Atlas for Computing Mathematical Functions: An Illustrated Guide for Practitioners. John Wiley & Sons Inc., New York.

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Notes:

With CD-ROM containing a large collection of mathematical function software written in Fortran 90 and Mathematica (an edition with the same software in C and Mathematica exists also). The functions are computed for real variables only. Maximum accuracy 12D.

External Links:

ISBN 0-471-18171-4, MathReview, ZentralBlatt

Referenced by:

§19.36(ii), § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

…

38: 11.13 Methods of Computation

… ►Although the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. For large $|z|$ and/or $|\nu |$ the asymptotic expansions given in §11.6 should be used instead. … ►For complex variables the methods described in §§3.5(viii) and 3.5(ix) are available. … ►Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. The solution ${𝐊}_{\nu }\left(x\right)$ needs to be integrated backwards for small $x$, and either forwards or backwards for large $x$ depending whether or not $\nu$ exceeds $\frac{1}{2}$. …

39: 10.70 Zeros

§10.70 Zeros

►Asymptotic approximations for large zeros are as follows. …If $m$ is a large positive integer, then …

40: 15.19 Methods of Computation

… ►Large values of $|a|$ or $|b|$, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. … ►Initial values for moderate values of $|a|$ and $|b|$ can be obtained by the methods of §15.19(i), and for large values of $|a|$, $|b|$, or $|c|$ via the asymptotic expansions of §§15.12(ii) and 15.12(iii). ►For example, in the half-plane $\mathrm{\Re }z\le \frac{1}{2}$ we can use (15.12.2) or (15.12.3) to compute $F(a,b;c+N+1;z)$ and $F(a,b;c+N;z)$, where $N$ is a large positive integer, and then apply (15.5.18) in the backward direction. … … ►In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of the Gauss hypergeometric function, when the variable and parameters are real and one of the numerator parameters is a positive integer. …