complex variables and parameters

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11: 10.33 Continued Fractions

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10.33.1 $$\frac{I_{\nu }\left(z\right)}{I_{\nu -1}\left(z\right)}=\frac{1}{2\nu z^{-1}+}\frac{1}{2(\nu +1)z^{-1}+}\frac{1}{2(\nu +2)z^{-1}+}\mathrm{\cdots },$$ $z\ne 0$,

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.74(v)

Permalink:

http://dlmf.nist.gov/10.33.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.33 and Ch.10

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10.33.2 $$\frac{I_{\nu }\left(z\right)}{I_{\nu -1}\left(z\right)}=\frac{\frac{1}{2}z/\nu }{1+}\frac{\frac{1}{4}{z}^2/(\nu (\nu +1))}{1+}\frac{\frac{1}{4}{z}^2/((\nu +1)(\nu +2))}{1+}\mathrm{\cdots },$$ $\nu \ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.74(v)

Permalink:

http://dlmf.nist.gov/10.33.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.33 and Ch.10

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12: 10.36 Other Differential Equations

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10.36.1 $$z^2(z^2+{\nu }^2)w^{\prime \prime }+z(z^2+3{\nu }^2)w^{\prime }-\left({(z^2+{\nu }^2)}^2+z^2-{\nu }^2\right)w=0,$$ $w={𝒵}_{\nu }^{\prime }\left(z\right)$,

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Symbols:

${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.36.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.36 and Ch.10

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10.36.2 $$z^2{w}^{\prime \prime }+z(1\pm 2z)w^{\prime }+(\pm z-{\nu }^2)w=0,$$ $w={\mathrm{e}}^{\mp z}{𝒵}_{\nu }\left(z\right)$.

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.41

Permalink:

http://dlmf.nist.gov/10.36.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.36 and Ch.10

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

ⓘ

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:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

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

ⓘ

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

ⓘ

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:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

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

ⓘ

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

14: 15.7 Continued Fractions

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15.7.1 $$\frac{𝐅(a,b;c;z)}{𝐅(a,b+1;c+1;z)}=t_0-\frac{u_{1}z}{t_1-}\frac{u_{2}z}{t_2-}\frac{u_{3}z}{t_3-}\mathrm{\cdots },$$

ⓘ

Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $t_n$: coefficient and $u_n$: coefficient

Permalink:

http://dlmf.nist.gov/15.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

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15.7.3 $$\frac{𝐅(a,b;c;z)}{𝐅(a,b+1;c+1;z)}=v_0-\frac{w_1}{v_1-}\frac{w_2}{v_2-}\frac{w_3}{v_3-}\mathrm{\cdots },$$

ⓘ

Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $v_n$: coefficient and $w_n$: coefficient

Permalink:

http://dlmf.nist.gov/15.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

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15.7.5 $$\frac{𝐅(a,b;c;z)}{𝐅(a+1,b+1;c+1;z)}=x_0+\frac{y_1}{x_1+}\frac{y_2}{x_2+}\frac{y_3}{x_3+}\mathrm{\cdots },$$

ⓘ

Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $y$: real variable, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

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

ⓘ

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

ⓘ

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

ⓘ

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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16: 15.5 Derivatives and Contiguous Functions

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15.5.1 $$\frac{d}{dz}F(a,b;c;z)=\frac{ab}{c}F(a+1,b+1;c+1;z),$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.5(i), §15.5 and Ch.15

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15.5.2 $$\frac{d^n}{{dz}^n}F(a,b;c;z)=\frac{{\left(a\right)}_n{\left(b\right)}_n}{{\left(c\right)}_n}F(a+n,b+n;c+n;z).$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.5(i), §15.5 and Ch.15

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15.5.3 $${\left(z\frac{d}{dz}z\right)}^n\left(z^{a-1}F(a,b;c;z)\right)={\left(a\right)}_n{z}^{a+n-1}F(a+n,b;c;z).$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§18.17(iv)

Permalink:

http://dlmf.nist.gov/15.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.5(i), §15.5 and Ch.15

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15.5.4 $$\frac{d^n}{{dz}^n}\left(z^{c-1}F(a,b;c;z)\right)={\left(c-n\right)}_n{z}^{c-n-1}F(a,b;c-n;z).$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§18.17(iv)

Permalink:

http://dlmf.nist.gov/15.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.5(i), §15.5 and Ch.15

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15.5.12 $$(b-a)F(a,b;c;z)+aF(a+1,b;c;z)-bF(a,b+1;c;z)=0,$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §15.5(ii), §15.5 and Ch.15

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17: 15.1 Special Notation

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15.1.1 $${}_2{}^{}F_1^{}(a,b;c;z)=F(a,b;c;z)=F(\genfrac{}{}{0pt}{}{a,b}{c};z),$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.1.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.1 and Ch.15

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15.1.2 $$\frac{F(a,b;c;z)}{\mathrm{\Gamma }\left(c\right)}=𝐅(a,b;c;z)=𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)={}_2{}^{}𝐅_1^{}(a,b;c;z),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.1.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.1 and Ch.15

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18: 28.11 Expansions in Series of Mathieu Functions

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28.11.1 $$f(z)={\alpha }_0{\mathrm{ce}}_0(z,q)+\sum _{n=1}^{\mathrm{\infty }}\left({\alpha }_n{\mathrm{ce}}_n(z,q)+{\beta }_n{\mathrm{se}}_n(z,q)\right),$$

ⓘ

Defines:

$f(z)$: function (locally)

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{se}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, ${\alpha }_n$ and ${\beta }_n$

Referenced by:

§28.11

Permalink:

http://dlmf.nist.gov/28.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11 and Ch.28

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28.11.3 $$1=2\sum _{n=0}^{\mathrm{\infty }}{A}_0^{2n}(q){\mathrm{ce}}_{2n}(z,q),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

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28.11.4 $$\cos 2mz=\sum _{n=0}^{\mathrm{\infty }}{A}_{2m}^{2n}(q){\mathrm{ce}}_{2n}(z,q),$$ $m\ne 0$,

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

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28.11.5 $$\cos (2m+1)z=\sum _{n=0}^{\mathrm{\infty }}{A}_{2m+1}^{2n+1}(q){\mathrm{ce}}_{2n+1}(z,q),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

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28.11.6 $$\sin (2m+1)z=\sum _{n=0}^{\mathrm{\infty }}{B}_{2m+1}^{2n+1}(q){\mathrm{se}}_{2n+1}(z,q),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

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19: 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

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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

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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

►

25.14.6 $$\mathrm{\Phi }(z,s,a)=\frac{1}{2}{a}^{-s}+{\int }_0^{\mathrm{\infty }}\frac{z^x}{{(a+x)}^s}dx-2{\int }_0^{\mathrm{\infty }}\frac{\sin \left(x\ln z-s\arctan \left(x/a\right)\right)}{{(a^2+x^2)}^{s/2}({\mathrm{e}}^{2\pi x}-1)}dx,$$ $\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$.

ⓘ

Symbols:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Erdélyi et al. (1953a, (1.11.4), p. 28)

Notes:

For the case see Erdélyi et al. (1953a, (1.11.4), p. 28). In the case $\left|z\right|=1$ one checks that the first integral converges absolutely iff $\mathrm{\Re }s>1$.

Referenced by:

Erratum (V1.1.4) for Equation (25.14.6)

Permalink:

http://dlmf.nist.gov/25.14.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The constraint which originally read “$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $|z|=1,\mathrm{\Re }a>0$” has been improved to be “$\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$”.

Suggested 2021-08-23 by Gergő Nemes

See also:

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

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20: 10.30 Limiting Forms

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10.30.1 $$I_{\nu }\left(z\right)\sim {(\frac{1}{2}z)}^{\nu }/\mathrm{\Gamma }\left(\nu +1\right),$$ $\nu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.7

Referenced by:

§10.30(i)

Permalink:

http://dlmf.nist.gov/10.30.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.30(i), §10.30 and Ch.10

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10.30.2 $$K_{\nu }\left(z\right)\sim \frac{1}{2}\mathrm{\Gamma }\left(\nu \right){(\frac{1}{2}z)}^{-\nu },$$ $\mathrm{\Re }\nu >0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.9

Referenced by:

§10.30(i)

Permalink:

http://dlmf.nist.gov/10.30.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.30(i), §10.30 and Ch.10

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10.30.4 $$I_{\nu }\left(z\right)\sim {\mathrm{e}}^z/\sqrt{2\pi z},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: complex parameter and $\delta$: small positive constant

Permalink:

http://dlmf.nist.gov/10.30.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.30(ii), §10.30 and Ch.10

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10.30.5 $$I_{\nu }\left(z\right)\sim {\mathrm{e}}^{\pm (\nu +\frac{1}{2})\pi \mathrm{i}}{\mathrm{e}}^{-z}/\sqrt{2\pi z},$$ $\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$.

ⓘ

Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: complex parameter and $\delta$: small positive constant

Permalink:

http://dlmf.nist.gov/10.30.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.30(ii), §10.30 and Ch.10

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