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11: 8 Incomplete Gamma and Related
Functions

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12: 28 Mathieu Functions and Hill’s Equation

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13: 27.15 Chinese Remainder Theorem

… ►For example, suppose a lengthy calculation involves many 10-digit integers. Most of the calculation can be done with five-digit integers as follows. …Their product $m$ has 20 digits, twice the number of digits in the data. By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod $m_1$), (mod $m_2$), (mod $m_3$), and (mod $m_4$), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

14: 15.7 Continued Fractions

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15: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

16: 23 Weierstrass Elliptic and Modular
Functions

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17: 26.15 Permutations: Matrix Notation

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26.15.2 $$inv(\sigma )=\sum a_{gh}{a}_{k\mathrm{\ell }},$$

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Symbols:

$h$: nonnegative integer, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer and $\sigma$: permutation

Permalink:

http://dlmf.nist.gov/26.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.15 and Ch.26

►where the sum is over and $n\ge h>\mathrm{\ell }\ge 1$. … ►A permutation with restricted position specifies a subset $B\subseteq \{1,2,\mathrm{\dots },n\}\times \{1,2,\mathrm{\dots },n\}$. … ►

26.15.3 $$R(x,B)=\sum _{j=0}^n{r}_j(B)x^j.$$

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Symbols:

$x$: real variable, $j$: nonnegative integer, $n$: nonnegative integer, $B$: subset, $r_j(B)$: number of rook positions and $R(x,B)$: rook polynomial

Permalink:

http://dlmf.nist.gov/26.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.15 and Ch.26

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26.15.6 $$N(x,B)=\sum _{k=0}^n{N}_k(B)x^k,$$

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Symbols:

$x$: real variable, $k$: nonnegative integer, $n$: nonnegative integer, $B$: subset and $N(x,B)$: generating function

Permalink:

http://dlmf.nist.gov/26.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.15 and Ch.26

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18: 11.6 Asymptotic Expansions

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11.6.1 $${𝐊}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.1.29

Referenced by:

§11.6(i), §11.6(i), §11.6(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

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11.6.2 $${𝐌}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^{k+1}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐌}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.2.6

Referenced by:

§11.6(i), §11.6(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

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11.6.3 $${\int }_0^z{𝐊}_0\left(t\right)dt-\frac{2}{\pi }(\ln \left(2z\right)+\gamma )\sim \frac{2}{\pi }\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k+1}\frac{(2k)!(2k-1)!}{{(k!)}^2{(2z)}^{2k}},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\gamma$: Euler’s constant, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.1.32

Referenced by:

§11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

►

11.6.4 $${\int }_0^z{𝐌}_0\left(t\right)dt+\frac{2}{\pi }(\ln \left(2z\right)+\gamma )\sim \frac{2}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{(2k)!(2k-1)!}{{(k!)}^2{(2z)}^{2k}},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

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Symbols:

$\gamma$: Euler’s constant, ${𝐌}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.2.8

Referenced by:

§11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

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$c_3(\lambda )=20{\lambda }^{-6}-4{\lambda }^{-4},$

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19: 10.44 Sums

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10.44.1 $${𝒵}_{\nu }\left(\lambda z\right)={\lambda }^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}\frac{{({\lambda }^2-1)}^k{(\frac{1}{2}z)}^k}{k!}{𝒵}_{\nu \pm k}\left(z\right),$$ .

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Symbols:

$!$: factorial (as in $n!$), ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.51

Referenced by:

§10.44(i), §10.66

Permalink:

http://dlmf.nist.gov/10.44.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.44(i), §10.44 and Ch.10

►If $𝒵=I$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary. … ►

10.44.3 $${𝒵}_{\nu }\left(u\pm v\right)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{(\pm 1)}^k{𝒵}_{\nu +k}\left(u\right)I_k\left(v\right),$$ .

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $k$: nonnegative integer and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.44.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.44(ii), §10.44(ii), §10.44 and Ch.10

►The restriction is unnecessary when $𝒵=I$ and $\nu$ is an integer. … ►

10.44.6 $$K_n\left(z\right)=\frac{n!{(\frac{1}{2}z)}^{-n}}{2}\sum _{k=0}^{n-1}{(-1)}^k\frac{{(\frac{1}{2}z)}^k{I}_k\left(z\right)}{k!(n-k)}+{(-1)}^{n-1}\left(\ln \left(\frac{1}{2}z\right)-\psi \left(n+1\right)\right)I_n\left(z\right)+{(-1)}^n\sum _{k=1}^{\mathrm{\infty }}\frac{(n+2k)I_{n+2k}\left(z\right)}{k(n+k)},$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.53

Permalink:

http://dlmf.nist.gov/10.44.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.44(iii), §10.44 and Ch.10

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20: 5.11 Asymptotic Expansions

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5.11.2 $$\psi \left(z\right)\sim \ln z-\frac{1}{2z}-\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{2k{z}^{2k}}.$$

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Symbols:

$B_n$: Bernoulli numbers, $\sim$: Poincaré asymptotic expansion, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.3.18

Referenced by:

§5.11(i), §5.11(ii), §5.4(iii)

Permalink:

http://dlmf.nist.gov/5.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(i), §5.11 and Ch.5

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5.11.5 $$g_k=\sqrt{2}{\left(\frac{1}{2}\right)}_k{a}_{2k},$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $k$: nonnegative integer, $g_k$: coefficients and $a_k$: coefficient

Referenced by:

§5.11(i)

Permalink:

http://dlmf.nist.gov/5.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(i), §5.11 and Ch.5

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5.11.6 $$a_0{a}_k+\frac{1}{2}{a}_1{a}_{k-1}+\frac{1}{3}{a}_2{a}_{k-2}+\mathrm{\cdots }+\frac{1}{k+1}{a}_k{a}_0=\frac{1}{k}{a}_{k-1},$$ $k\ge 1$.

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Symbols:

$k$: nonnegative integer and $a_k$: coefficient

Referenced by:

§5.11(i)

Permalink:

http://dlmf.nist.gov/5.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(i), §5.11 and Ch.5

… ►Wrench (1968) gives exact values of $g_k$ up to $g_{20}$. … ►

5.11.8 $$\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)\sim \left(z+h-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+\sum _{k=2}^{\mathrm{\infty }}\frac{{(-1)}^k{B}_k\left(h\right)}{k(k-1)z^{k-1}},$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

Referenced by:

§12.10(ii), (5.11.8), §5.11(i), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8), Erratum (V1.0.11) for Equation (5.11.8)

Permalink:

http://dlmf.nist.gov/5.11.E8

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.11):

Originally $h$ in (5.11.8) was unnecessarily restricted to lie in the interval $[0,1]$. In fact, $h$ may lie anywhere in the complex plane.

Suggested 2015-02-28 by Nico Temme

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 $\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z+h\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.11(i), §5.11(i), §5.11 and Ch.5

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