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31: 31.11 Expansions in Series of Hypergeometric Functions

… ►Taking $P_j=P_j^5$ or $P_j=P_j^6$ the coefficients $c_j$ satisfy the equations ►

31.11.4 $$L_0{c}_0+M_0{c}_1=0,$$

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Symbols:

$c_j$: coefficients, $L_j$: coefficients and $M_j$: coefficients

Permalink:

http://dlmf.nist.gov/31.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

►

31.11.5 $$K_j{c}_{j-1}+L_j{c}_j+M_j{c}_{j+1}=0,$$ $j=1,2,\mathrm{\dots }$,

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Symbols:

$j$: nonnegative integer, $c_j$: coefficients, $K_j$: coefficients, $L_j$: coefficients and $M_j$: coefficients

Referenced by:

§31.11(ii), Erratum (V1.1.7) for Subsection 31.11(ii)

Permalink:

http://dlmf.nist.gov/31.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

… ►

31.11.9 $$M_{-1}{P}_{-1}=0.$$

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Symbols:

$P_j$: General solution of generalized hypergeometric differential equation and $M_j$: coefficients

Referenced by:

§31.11(iii)

Permalink:

http://dlmf.nist.gov/31.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

… ►

31.11.13 $$\left(L_0/M_0\right)-\frac{K_1/M_1}{L_1/M_1-}\frac{K_2/M_2}{L_2/M_2-}\mathrm{\cdots }=0.$$

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Symbols:

$K_j$: coefficients, $L_j$: coefficients and $M_j$: coefficients

Referenced by:

§31.18

Permalink:

http://dlmf.nist.gov/31.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §31.11(iii), §31.11 and Ch.31

…

32: 10.20 Uniform Asymptotic Expansions for Large Order

… ►In the following formulas for the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, and $D_k(\zeta )$, $u_k$, $v_k$ are the constants defined in §9.7(i), and $U_k(p)$, $V_k(p)$ are the polynomials in $p$ of degree $3k$ defined in §10.41(ii). … ►Note: Another way of arranging the above formulas for the coefficients $A_k(\zeta ),B_k(\zeta ),C_k(\zeta )$, and $D_k(\zeta )$ would be by analogy with (12.10.42) and (12.10.46). … ►For (10.20.14) and further information on the coefficients see Temme (1997). … ►For resurgence properties of the coefficients (§2.7(ii)) see Howls and Olde Daalhuis (1999). … …

33: 5.10 Continued Fractions

… ►

5.10.1 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)+z-\left(z-\frac{1}{2}\right)\ln z-\frac{1}{2}\ln \left(2\pi \right)=\frac{a_0}{z+}\frac{a_1}{z+}\frac{a_2}{z+}\frac{a_3}{z+}\frac{a_4}{z+}\frac{a_5}{z+}\mathrm{\cdots },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $z$: complex variable and $a_k$: coefficient

Referenced by:

Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)

Permalink:

http://dlmf.nist.gov/5.10.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.10 and Ch.5

►where …

34: 25.19 Tables

… ►

Cloutman (1989) tabulates $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

►

Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta \left(s\right)$ for both real and complex $s$. §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of $\zeta (s,a)$, and §22.17 lists tables for some Dirichlet $L$-functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

35: 30.18 Software

… ►

SWF7: Coefficients ${\beta }_p$ in (30.9.1).

►

SWF8: Coefficients $c_p$ in (30.9.4).

►

SWF9: Coefficients ${\mathrm{\ell }}_p$ in (30.3.8).

…

36: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►

31.5.1

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Symbols:

$\gamma$: real or complex parameter, $n$: nonnegative integer, $a$: complex parameter, $P_j$: coefficient, $Q_j$: coefficient and $R_j$: coefficient

Permalink:

http://dlmf.nist.gov/31.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.5 and Ch.31

…

37: 33.9 Expansions in Series of Bessel Functions

… ►

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:

pMML, png, TeX

See also:

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

►where the function $𝗃$ is as in §10.47(ii), $a_{-1}=0$, $a_0=(2\mathrm{\ell }+1)!!C_{\mathrm{\ell }}\left(\eta \right)$, and … ►Here $b_{2\mathrm{\ell }}=b_{2\mathrm{\ell }+2}=0$, $b_{2\mathrm{\ell }+1}=1$, and ►

33.9.5 $$4{\eta }^2(k-2\mathrm{\ell })b_{k+1}+k{b}_{k-1}+b_{k-2}=0,$$ $k=2\mathrm{\ell }+2,2\mathrm{\ell }+3,\mathrm{\dots }$.

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Symbols:

$k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\eta$: real parameter and $b_k$: coefficient

A&S Ref:

14.4.3 ((Early printings contained an error. All editions miss $b_{2L}=0$.))

Permalink:

http://dlmf.nist.gov/33.9.E5

Encodings:

pMML, png, TeX

See also:

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

… ►

33.9.7 $${\lambda }_{\mathrm{\ell }}(\eta )\sim \sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{(-1)}^k(k-1)!b_k.$$

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Defines:

${\lambda }_{\mathrm{\ell }}(\eta )$: coefficient (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\eta$: real parameter and $b_k$: coefficient

A&S Ref:

14.4.4

Permalink:

http://dlmf.nist.gov/33.9.E7

Encodings:

pMML, png, TeX

See also:

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

…

38: 1.13 Differential Equations

… ►The equation … ►(More generally in (1.13.5) for $n$th-order differential equations, $f(z)$ is the coefficient multiplying the $(n-1)$th-order derivative of the solution divided by the coefficient multiplying the $n$th-order derivative of the solution, see Ince (1926, §5.2).) … ► $u$ and $z$ belong to domains $U$ and $D$ respectively, the coefficients $f(u,z)$ and $g(u,z)$ are continuous functions of both variables, and for each fixed $u$ (fixed $z$) the two functions are analytic in $z$ (in $u$). … ►The substitution $\xi =1/z$ in (1.13.1) gives … ►If $U(z)$ and $V(z)$ are respectively solutions of …

39: 4.47 Approximations

… ►Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arctan$, $\mathrm{arcsinh}$. Schonfelder (1980) gives 40D coefficients for $\sin$, $\cos$, $\tan$. …

40: 36.8 Convergent Series Expansions

… ►

36.8.3 $$\frac{3^{2/3}}{4{\pi }^2}{\mathrm{\Psi }}^{(\mathrm{H})}\left(3^{1/3}𝐱\right)=\mathrm{Ai}\left(x\right)\mathrm{Ai}\left(y\right)\sum _{n=0}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{c_n(x)c_n(y)}{n!}+\mathrm{Ai}\left(x\right){\mathrm{Ai}}^{\prime }\left(y\right)\sum _{n=2}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{c_n(x)d_n(y)}{n!}+{\mathrm{Ai}}^{\prime }\left(x\right)\mathrm{Ai}\left(y\right)\sum _{n=2}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{d_n(x)c_n(y)}{n!}+{\mathrm{Ai}}^{\prime }\left(x\right){\mathrm{Ai}}^{\prime }\left(y\right)\sum _{n=1}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{d_n(x)d_n(y)}{n!},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $y$: real parameter, $z$: real parameter, $m$: integer, $x$: real parameter, $c_n(t)$: coefficients and $d_n(t)$: coefficients

Referenced by:

§36.8

Permalink:

http://dlmf.nist.gov/36.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.8 and Ch.36

… ►

36.8.5 $$f_n(\zeta ,\overline{\zeta })=c_n(\zeta )c_n(\overline{\zeta })\mathrm{Ai}\left(\zeta \right)\mathrm{Bi}\left(\overline{\zeta }\right)+c_n(\zeta )d_n(\overline{\zeta })\mathrm{Ai}\left(\zeta \right){\mathrm{Bi}}^{\prime }\left(\overline{\zeta }\right)+d_n(\zeta )c_n(\overline{\zeta }){\mathrm{Ai}}^{\prime }\left(\zeta \right)\mathrm{Bi}\left(\overline{\zeta }\right)+d_n(\zeta )d_n(\overline{\zeta }){\mathrm{Ai}}^{\prime }\left(\zeta \right){\mathrm{Bi}}^{\prime }\left(\overline{\zeta }\right),$$

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Defines:

$f_n$: function (locally)

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\overline{z}$: complex conjugate, $m$: integer, $c_n(t)$: coefficients and $d_n(t)$: coefficients

Permalink:

http://dlmf.nist.gov/36.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.8 and Ch.36

…