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41: 4.25 Continued Fractions

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4.25.1 $$\tan z=\frac{z}{1-}\frac{z^2}{3-}\frac{z^2}{5-}\frac{z^2}{7-}\mathrm{\cdots },$$ $z\ne \pm \frac{1}{2}\pi$, $\pm \frac{3}{2}\pi$, $\mathrm{\dots }$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.94

Permalink:

http://dlmf.nist.gov/4.25.E1

Encodings:

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See also:

Annotations for §4.25 and Ch.4

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4.25.2 $$\tan \left(az\right)=\frac{a\tan z}{1+}\frac{(1-a^2){\tan }^{2}z}{3+}\frac{(4-a^2){\tan }^{2}z}{5+}\frac{(9-a^2){\tan }^{2}z}{7+}\mathrm{\cdots },$$ , $az\ne \pm \frac{1}{2}\pi ,\pm \frac{3}{2}\pi ,\mathrm{\dots }$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $\tan z$: tangent function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.3.95

Permalink:

http://dlmf.nist.gov/4.25.E2

Encodings:

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See also:

Annotations for §4.25 and Ch.4

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4.25.3 $$\frac{\arcsin z}{\sqrt{1-z^2}}=\frac{z}{1-}\frac{1\cdot 2z^2}{3-}\frac{1\cdot 2z^2}{5-}\frac{3\cdot 4z^2}{7-}\frac{3\cdot 4z^2}{9-}\mathrm{\cdots },$$

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Symbols:

$\arcsin z$: arcsine function and $z$: complex variable

A&S Ref:

4.4.44

Permalink:

http://dlmf.nist.gov/4.25.E3

Encodings:

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See also:

Annotations for §4.25 and Ch.4

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4.25.4 $$\arctan z=\frac{z}{1+}\frac{z^2}{3+}\frac{4z^2}{5+}\frac{9z^2}{7+}\frac{16z^2}{9+}\mathrm{\cdots },$$

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Symbols:

$\arctan z$: arctangent function and $z$: complex variable

A&S Ref:

4.4.43

Referenced by:

§3.10(ii)

Permalink:

http://dlmf.nist.gov/4.25.E4

Encodings:

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Annotations for §4.25 and Ch.4

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4.25.5 $${\mathrm{e}}^{2a\arctan \left(1/z\right)}=1+\frac{2a}{z-a+}\frac{a^2+1}{3z+}\frac{a^2+4}{5z+}\frac{a^2+9}{7z+}\mathrm{\cdots },$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\arctan z$: arctangent function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.2.42

Permalink:

http://dlmf.nist.gov/4.25.E5

Encodings:

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See also:

Annotations for §4.25 and Ch.4

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42: 4.19 Maclaurin Series and Laurent Series

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4.19.1 $$\sin z=z-\frac{z^3}{3!}+\frac{z^5}{5!}-\frac{z^7}{7!}+\mathrm{\cdots },$$

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Symbols:

$!$: factorial (as in $n!$), $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.65

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.19.E1

Encodings:

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See also:

Annotations for §4.19 and Ch.4

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4.19.2 $$\cos z=1-\frac{z^2}{2!}+\frac{z^4}{4!}-\frac{z^6}{6!}+\mathrm{\cdots }.$$

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Symbols:

$\cos z$: cosine function, $!$: factorial (as in $n!$) and $z$: complex variable

A&S Ref:

4.3.66

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

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4.19.3 $$\tan z=z+\frac{z^3}{3}+\frac{2}{15}{z}^5+\frac{17}{315}{z}^7+\mathrm{\cdots }+\frac{{(-1)}^{n-1}{2}^{2n}(2^{2n}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$ ,

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\tan z$: tangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.67

Referenced by:

§4.19

Permalink:

http://dlmf.nist.gov/4.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

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4.19.4 $$\csc z=\frac{1}{z}+\frac{z}{6}+\frac{7}{360}{z}^3+\frac{31}{15120}{z}^5+\mathrm{\cdots }+\frac{{(-1)}^{n-1}2(2^{2n-1}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$ ,

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $!$: factorial (as in $n!$), $n$: integer and $z$: complex variable

A&S Ref:

4.3.68

Referenced by:

§4.33

Permalink:

http://dlmf.nist.gov/4.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

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4.19.5 $$\sec z=1+\frac{z^2}{2}+\frac{5}{24}{z}^4+\frac{61}{720}{z}^6+\mathrm{\cdots }+\frac{{(-1)}^n{E}_{2n}}{(2n)!}{z}^{2n}+\mathrm{\cdots },$$ ,

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Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\sec z$: secant function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.69

Referenced by:

§24.1, §24.19(i), §24.2(ii)

Permalink:

http://dlmf.nist.gov/4.19.E5

Encodings:

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See also:

Annotations for §4.19 and Ch.4

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43: 6.6 Power Series

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

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

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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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6.6.4 $$\mathrm{Ein}\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}{z}^n}{n!n},$$

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

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6.6.5 $$\mathrm{Si}\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^{2n+1}}{(2n+1)!(2n+1)},$$

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Symbols:

$!$: factorial (as in $n!$), $\mathrm{Si}\left(z\right)$: sine integral, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.2.14

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

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

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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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44: 8.5 Confluent Hypergeometric Representations

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8.5.1 $$\gamma (a,z)=a^{-1}{z}^a{\mathrm{e}}^{-z}M(1,1+a,z)=a^{-1}{z}^{a}M(a,1+a,-z),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.12

Referenced by:

§8.11(ii), §8.14

Permalink:

http://dlmf.nist.gov/8.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

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8.5.2 $${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-z}𝐌(1,1+a,z)=𝐌(a,1+a,-z).$$

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Symbols:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.7

Permalink:

http://dlmf.nist.gov/8.5.E2

Encodings:

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See also:

Annotations for §8.5 and Ch.8

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8.5.3 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-z}U(1-a,1-a,z)=z^a{\mathrm{e}}^{-z}U(1,1+a,z).$$

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Symbols:

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

Referenced by:

§8.14, §8.19(vi)

Permalink:

http://dlmf.nist.gov/8.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

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8.5.4 $$\gamma (a,z)=a^{-1}{z}^{\frac{1}{2}a-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}z}{M}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.5

Permalink:

http://dlmf.nist.gov/8.5.E4

Encodings:

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See also:

Annotations for §8.5 and Ch.8

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8.5.5 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}a-\frac{1}{2}}{W}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

45: 10.56 Generating Functions

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10.56.1 $$\frac{\cos \sqrt{z^2-2zt}}{z}=\frac{\cos z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{t^n}{n!}{𝗃}_{n-1}\left(z\right),$$

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Symbols:

$\cos z$: cosine function, $!$: factorial (as in $n!$), ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

Referenced by:

§10.56, Ch.10

Permalink:

http://dlmf.nist.gov/10.56.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

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10.56.2 $$\frac{\sin \sqrt{z^2-2zt}}{z}=\frac{\sin z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{t^n}{n!}{𝗒}_{n-1}\left(z\right).$$

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Symbols:

$!$: factorial (as in $n!$), $\sin z$: sine function, ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind, $n$: integer and $z$: complex variable

Referenced by:

§10.56

Permalink:

http://dlmf.nist.gov/10.56.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

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10.56.3 $$\frac{\cosh \sqrt{z^2+2\mathrm{i}zt}}{z}=\frac{\cosh z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{{(\mathrm{i}t)}^n}{n!}{𝗂}_{n-1}^{(1)}\left(z\right),$$

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Symbols:

$!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56

Permalink:

http://dlmf.nist.gov/10.56.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

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10.56.4 $$\frac{\sinh \sqrt{z^2+2\mathrm{i}zt}}{z}=\frac{\sinh z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{{(\mathrm{i}t)}^n}{n!}{𝗂}_{n-1}^{(2)}\left(z\right),$$

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Symbols:

$!$: factorial (as in $n!$), $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56

Permalink:

http://dlmf.nist.gov/10.56.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

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10.56.5 $$\frac{\exp \left(-\sqrt{z^2+2\mathrm{i}zt}\right)}{z}=\frac{{\mathrm{e}}^{-z}}{z}+\frac{2}{\pi }\sum _{n=1}^{\mathrm{\infty }}\frac{{(-\mathrm{i}t)}^n}{n!}{𝗄}_{n-1}\left(z\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, ${𝗄}_n\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56, Ch.10

Permalink:

http://dlmf.nist.gov/10.56.E5

Encodings:

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See also:

Annotations for §10.56 and Ch.10

46: 4.37 Inverse Hyperbolic Functions

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4.37.10 $$\mathrm{arcsinh}\left(-z\right)=-\mathrm{arcsinh}z.$$

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Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function and $z$: complex variable

A&S Ref:

4.6.11

Permalink:

http://dlmf.nist.gov/4.37.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iii), §4.37 and Ch.4

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4.37.13 $$\mathrm{arccsch}\left(-z\right)=-\mathrm{arccsch}z.$$

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Symbols:

$\mathrm{arccsch}z$: inverse hyperbolic cosecant function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iii), §4.37 and Ch.4

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4.37.26 $$z=\sinh w,$$

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Symbols:

$\sinh z$: hyperbolic sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(v), §4.37 and Ch.4

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4.37.27 $$z=\cosh w,$$

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Symbols:

$\cosh z$: hyperbolic cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(v), §4.37 and Ch.4

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4.37.28 $$z=\tanh w,$$

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Symbols:

$\tanh z$: hyperbolic tangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(v), §4.37 and Ch.4

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47: 8.8 Recurrence Relations and Derivatives

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

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

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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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8.8.3 $$w(a+2,z)-(a+1+z)w(a+1,z)+azw(a,z)=0.$$

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Symbols:

$z$: complex variable, $a$: parameter and $w(a,z)$: function

Permalink:

http://dlmf.nist.gov/8.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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8.8.4 $$z{\gamma }^{\ast }(a+1,z)={\gamma }^{\ast }(a,z)-\frac{{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a+1\right)}.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/8.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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

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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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48: 5.6 Inequalities

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5.6.4 .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $s$: real or complex variable and $x$: real variable

Referenced by:

§5.6(i)

Permalink:

http://dlmf.nist.gov/5.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(i), §5.6(i), §5.6 and Ch.5

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5.6.5 $$\exp \left((1-s)\psi \left(x+s^{1/2}\right)\right)\le \frac{\mathrm{\Gamma }\left(x+1\right)}{\mathrm{\Gamma }\left(x+s\right)}\le \exp \left((1-s)\psi \left(x+\frac{1}{2}(s+1)\right)\right),$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $\exp z$: exponential function, $s$: real or complex variable and $x$: real variable

Referenced by:

§5.6(i)

Permalink:

http://dlmf.nist.gov/5.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(i), §5.6(i), §5.6 and Ch.5

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§5.6(ii) Complex Variables

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5.6.8 $$\left|\frac{\mathrm{\Gamma }\left(z+a\right)}{\mathrm{\Gamma }\left(z+b\right)}\right|\le \frac{1}{{|z|}^{b-a}}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(ii), §5.6 and Ch.5

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5.6.9 $$|\mathrm{\Gamma }\left(z\right)|\le {(2\pi )}^{1/2}{|z|}^{x-(1/2)}{\mathrm{e}}^{-\pi |y|/2}\exp \left(\frac{1}{6}{|z|}^{-1}\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $x$: real variable, $y$: real variable and $z$: complex variable

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(ii), §5.6 and Ch.5

49: 25.14 Lerch’s Transcendent

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25.14.1 $$\mathrm{\Phi }(z,s,a)\equiv \sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{{(a+n)}^s},$$ ; $\mathrm{\Re }s>1,|z|=1$.

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Defines:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent

Symbols:

$\equiv$: equals by definition, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Erdélyi et al. (1953a, (1.11.1), p. 27)

Referenced by:

(25.14.2), (25.14.3), §25.14(ii), §25.14, Erratum (V1.0.21) for Equation (25.14.1)

Permalink:

http://dlmf.nist.gov/25.14.E1

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.21):

The previous constraint $a\ne 0,-1,-2,\mathrm{\dots },$ was removed. A clarification regarding the correct constraints for Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$ has been added in the text immediately below.

See also:

Annotations for §25.14(i), §25.14 and Ch.25

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25.14.2 $$\zeta (s,a)=\mathrm{\Phi }(1,s,a),$$ $\mathrm{\Re }s>1$, $a\ne 0,-1,-2,\mathrm{\dots }$,

ⓘ

Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $\mathrm{\Re }$: real part, $a$: real or complex parameter and $s$: complex variable

Keywords:

specialization

Proof sketch:

Derivable from (25.11.1), (25.14.1).

Permalink:

http://dlmf.nist.gov/25.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.14(i), §25.14 and Ch.25

►

25.14.3 $${\mathrm{Li}}_s\left(z\right)=z\mathrm{\Phi }(z,s,1),$$ $\mathrm{\Re }s>1$, $|z|\le 1$.

ⓘ

Symbols:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $\mathrm{\Re }$: real part, $s$: complex variable and $z$: complex variable

Keywords:

specialization

Proof sketch:

Derivable from (25.12.10) and (25.14.1).

Referenced by:

(25.12.12)

Permalink:

http://dlmf.nist.gov/25.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.14(i), §25.14 and Ch.25

… ►

25.14.4 $$\mathrm{\Phi }(z,s,a)=z^m\mathrm{\Phi }(z,s,a+m)+\sum _{n=0}^{m-1}\frac{z^n}{{(a+n)}^s}.$$

ⓘ

Symbols:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $m$: nonnegative integer, $n$: nonnegative integer, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

recurrence

Source:

Erdélyi et al. (1953a, (1.11.2), p. 27)

Permalink:

http://dlmf.nist.gov/25.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.14(ii), §25.14 and Ch.25

►

25.14.5 $$\mathrm{\Phi }(z,s,a)=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}{\mathrm{e}}^{-ax}}{1-z{\mathrm{e}}^{-x}}dx,$$ $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $z=1$; $\mathrm{\Re }s>0$, $\mathrm{\Re }a>0$ if $z\in ℂ\setminus [1,\mathrm{\infty })$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $[a,b)$: half-closed interval, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $\setminus$: set subtraction, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Erdélyi et al. (1953a, (1.11.3), p. 27)

Referenced by:

Erratum (V1.1.4) for Equation (25.14.5)

Permalink:

http://dlmf.nist.gov/25.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.14(ii), §25.14 and Ch.25

…

50: 6.11 Relations to Other Functions

… ►

6.11.1 $$E_1\left(z\right)=\mathrm{\Gamma }(0,z).$$

ⓘ

Symbols:

$E_1\left(z\right)$: exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

A&S Ref:

5.1.45 (special case of)

Referenced by:

§6.11

Permalink:

http://dlmf.nist.gov/6.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

… ►

6.11.2 $$E_1\left(z\right)={\mathrm{e}}^{-z}U(1,1,z),$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $z$: complex variable

Referenced by:

§6.11, 5th item, §6.6

Permalink:

http://dlmf.nist.gov/6.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

►

6.11.3 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)=U(1,1,-\mathrm{i}z).$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.11, 5th item

Permalink:

http://dlmf.nist.gov/6.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6