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21: 8.8 Recurrence Relations and Derivatives

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8.8.1 $$\gamma (a+1,z)=a\gamma (a,z)-z^a{\mathrm{e}}^{-z},$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.22

Permalink:

http://dlmf.nist.gov/8.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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8.8.2 $$\mathrm{\Gamma }(a+1,z)=a\mathrm{\Gamma }(a,z)+z^a{\mathrm{e}}^{-z}.$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.19(v)

Permalink:

http://dlmf.nist.gov/8.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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8.8.5 $$P(a+1,z)=P(a,z)-\frac{z^a{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a+1\right)},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.21

Referenced by:

§8.25(v)

Permalink:

http://dlmf.nist.gov/8.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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8.8.6 $$Q(a+1,z)=Q(a,z)+\frac{z^a{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a+1\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.25(v), §8.4

Permalink:

http://dlmf.nist.gov/8.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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8.8.12 $$Q(a+n,z)=Q(a,z)+z^a{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{z^k}{\mathrm{\Gamma }\left(a+k+1\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $a$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/8.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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22: 21.8 Abelian Functions

… ►An Abelian function is a $2g$-fold periodic, meromorphic function of $g$ complex variables. In consequence, Abelian functions are generalizations of elliptic functions (§23.2(iii)) to more than one complex variable. …

23: 22.8 Addition Theorems

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22.8.1 $$\mathrm{sn}(u+v)=\frac{\mathrm{sn}u\mathrm{cn}v\mathrm{dn}v+\mathrm{sn}v\mathrm{cn}u\mathrm{dn}u}{1-k^2{\mathrm{sn}}^{2}u{\mathrm{sn}}^{2}v},$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

A&S Ref:

16.17.1

Referenced by:

§22.18(iv)

Permalink:

http://dlmf.nist.gov/22.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(i), §22.8 and Ch.22

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22.8.2 $$\mathrm{cn}(u+v)=\frac{\mathrm{cn}u\mathrm{cn}v-\mathrm{sn}u\mathrm{dn}u\mathrm{sn}v\mathrm{dn}v}{1-k^2{\mathrm{sn}}^{2}u{\mathrm{sn}}^{2}v},$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

A&S Ref:

16.17.2

Referenced by:

§22.6(ii)

Permalink:

http://dlmf.nist.gov/22.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(i), §22.8 and Ch.22

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22.8.3 $$\mathrm{dn}(u+v)=\frac{\mathrm{dn}u\mathrm{dn}v-k^2\mathrm{sn}u\mathrm{cn}u\mathrm{sn}v\mathrm{cn}v}{1-k^2{\mathrm{sn}}^{2}u{\mathrm{sn}}^{2}v}.$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

A&S Ref:

16.17.3

Permalink:

http://dlmf.nist.gov/22.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(i), §22.8 and Ch.22

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22.8.5 $$\mathrm{sd}(u+v)=\frac{\mathrm{sd}u\mathrm{cd}v\mathrm{nd}v+\mathrm{sd}v\mathrm{cd}u\mathrm{nd}u}{1+k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^{2}u{\mathrm{sd}}^{2}v},$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(i), §22.8 and Ch.22

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22.8.27 $$\mathrm{dn}{z}_1\mathrm{dn}{z}_3=\mathrm{dn}{z}_2\mathrm{dn}{z}_4=k^{\prime }.$$

ⓘ

Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.8.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(iii), §22.8 and Ch.22

…

24: 22.10 Maclaurin Series

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22.10.4 $$\mathrm{sn}(z,k)=\sin z-\frac{k^2}{4}(z-\sin z\cos z)\cos z+O\left(k^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.1

Referenced by:

§22.10(ii), §22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.5 $$\mathrm{cn}(z,k)=\cos z+\frac{k^2}{4}(z-\sin z\cos z)\sin z+O\left(k^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.2

Referenced by:

§22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.6 $$\mathrm{dn}(z,k)=1-\frac{k^2}{2}{\sin }^{2}z+O\left(k^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.3

Referenced by:

§22.10(ii), §22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.7 $$\mathrm{sn}(z,k)=\tanh z-\frac{k_{}^{\prime }{}_{}{}^2}{4}(z-\sinh z\cosh z){\mathrm{sech}}^{2}z+O\left(k_{}^{\prime }{}_{}{}^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.1

Referenced by:

§22.10(ii)

Permalink:

http://dlmf.nist.gov/22.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.8 $$\mathrm{cn}(z,k)=\mathrm{sech}z+\frac{k_{}^{\prime }{}_{}{}^2}{4}(z-\sinh z\cosh z)\tanh z\mathrm{sech}z+O\left(k_{}^{\prime }{}_{}{}^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.2

Permalink:

http://dlmf.nist.gov/22.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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25: 8.4 Special Values

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8.4.7 $$\gamma (n+1,z)=n!(1-{\mathrm{e}}^{-z}{e}_n(z)),$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\gamma (a,z)$: incomplete gamma function, $z$: complex variable, $n$: nonnegative integer and $e_n(z)$: functions

Referenced by:

§8.11(v)

Permalink:

http://dlmf.nist.gov/8.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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8.4.8 $$\mathrm{\Gamma }(n+1,z)=n!{\mathrm{e}}^{-z}{e}_n(z),$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $n$: nonnegative integer and $e_n(z)$: functions

Permalink:

http://dlmf.nist.gov/8.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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8.4.9 $$P(n+1,z)=1-{\mathrm{e}}^{-z}{e}_n(z),$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $z$: complex variable, $n$: nonnegative integer and $e_n(z)$: functions

A&S Ref:

6.5.13

Permalink:

http://dlmf.nist.gov/8.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

►

8.4.10 $$Q(n+1,z)={\mathrm{e}}^{-z}{e}_n(z),$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $n$: nonnegative integer and $e_n(z)$: functions

Permalink:

http://dlmf.nist.gov/8.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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8.4.12 $${\gamma }^{\ast }(-n,z)=z^n,$$

ⓘ

Symbols:

${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

6.5.14

Referenced by:

§8.13(i)

Permalink:

http://dlmf.nist.gov/8.4.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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26: 7.18 Repeated Integrals of the Complementary Error Function

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7.18.6 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{z}^k}{2^{n-k}k!\mathrm{\Gamma }\left(1+\frac{1}{2}(n-k)\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.4

Permalink:

http://dlmf.nist.gov/7.18.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iii), §7.18 and Ch.7

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7.18.7 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)=-\frac{z}{n}{\mathrm{i}}^{n-1}\mathrm{erfc}\left(z\right)+\frac{1}{2n}{\mathrm{i}}^{n-2}\mathrm{erfc}\left(z\right),$$ $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.5

Referenced by:

§7.22(iii)

Permalink:

http://dlmf.nist.gov/7.18.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iii), §7.18 and Ch.7

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7.18.10 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)=\frac{{\mathrm{e}}^{-z^2}}{2^n\sqrt{\pi }}U(\frac{1}{2}n+\frac{1}{2},\frac{1}{2},z^2).$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§7.18(iv), §7.18(iv), Erratum (V1.0.21) for Paragraph Confluent Hypergeometric Functions (in §7.18(iv))

Permalink:

http://dlmf.nist.gov/7.18.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

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7.18.11 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)=\frac{{\mathrm{e}}^{-z^2/2}}{\sqrt{2^{n-1}\pi }}U(n+\frac{1}{2},z\sqrt{2}).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.13

Referenced by:

§12.7(ii)

Permalink:

http://dlmf.nist.gov/7.18.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

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7.18.12 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)=\frac{1}{\sqrt{2^{n-1}\pi }}{\mathrm{𝐻ℎ}}_n\left(\sqrt{2}z\right).$$

ⓘ

Defines:

${\mathrm{𝐻ℎ}}_n\left(z\right)$: probability function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.10

Referenced by:

§12.7(ii)

Permalink:

http://dlmf.nist.gov/7.18.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

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27: 14.34 Software

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§14.34(iii) Legendre Functions: Complex Argument and/or Parameters

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28: Publications

… ►

Dissemination of 3D Visualizations of Complex Function Data for the NIST Digital Library of Mathematical Functions, CODATA Data Science Journal 6 (2007), pp. S146–S154.

…

29: 35.2 Laplace Transform

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35.2.1 $$g(𝐙)={\int }_{𝛀}\mathrm{etr}\left(-𝐙𝐗\right)f(𝐗)d𝐗,$$

ⓘ

Defines:

$g$: Laplace transformation of $f$ (locally)

Symbols:

$\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, $𝐗$: real symmetric $m\times m$ matrix, $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $𝐙$: complex symmetric $m\times m$ matrix and $f(𝐗)$: complex-valued function of matrix argument

Keywords:

Laplace transform

Referenced by:

§35.2

Permalink:

http://dlmf.nist.gov/35.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.2, §35.2 and Ch.35

… ►Then (35.2.1) converges absolutely on the region $\mathrm{\Re }\left(𝐙\right)>{𝐗}_0$, and $g(𝐙)$ is a complex analytic function of all elements $z_{j,k}$ of $𝐙$. … ►

35.2.2 $$f(𝐗)=\frac{1}{{(2\pi \mathrm{i})}^{m(m+1)/2}}\int \mathrm{etr}\left(𝐙𝐗\right)g(𝐙)d𝐙,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\mathrm{i}$: imaginary unit, $\int$: integral, $𝐗$: real symmetric $m\times m$ matrix, $d$: differential of matrix, $𝐙$: complex symmetric $m\times m$ matrix, $f(𝐗)$: complex-valued function of matrix argument, $m$: positive integer and $g$: Laplace transformation of $f$

Permalink:

http://dlmf.nist.gov/35.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.2, §35.2 and Ch.35

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35.2.3

ⓘ

Defines:

$\ast$: convolution operator (locally)

Symbols:

$\int$: integral, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $d$: differential of matrix, : less-than for matrices, $f(𝐗)$: complex-valued function of matrix argument and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.6(ii)

Permalink:

http://dlmf.nist.gov/35.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.2, §35.2 and Ch.35

30: 6.6 Power Series

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6.6.2 $$E_1\left(z\right)=-\gamma -\ln z-\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^n}{n!n}.$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.1.11

Referenced by:

§6.4, §6.5

Permalink:

http://dlmf.nist.gov/6.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

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6.6.3 $$E_1\left(z\right)=-\ln z+{\mathrm{e}}^{-z}\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!}\psi \left(n+1\right),$$

ⓘ

Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§2.5(iii), §6.6, §8.19(iv)

Permalink:

http://dlmf.nist.gov/6.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

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6.6.6 $$\mathrm{Ci}\left(z\right)=\gamma +\ln z+\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^{2n}}{(2n)!(2n)}.$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ci}\left(z\right)$: cosine integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.2.16

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

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