of imaginary order

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21: 14.3 Definitions and Hypergeometric Representations

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14.3.7 $\mu +\nu \ne -1,-2,-3,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\mu$: general order, $\nu$: general degree and : real variable

A&S Ref:

8.1.3 (modified)

Permalink:

http://dlmf.nist.gov/14.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

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14.3.10

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mu$: general order, $\nu$: general degree and : real variable

Referenced by:

§14.20(i), §14.21(i), §14.21(iii), §14.6(i)

Permalink:

http://dlmf.nist.gov/14.3.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

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14.3.23

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\varphi }_{\lambda }^{(\alpha ,\beta )}\left(t\right)$: Jacobi function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\mu$: general order, $\nu$: general degree and : real variable

Referenced by:

§14.3(iv)

Permalink:

http://dlmf.nist.gov/14.3.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iv), §14.3 and Ch.14

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22: 25.9 Asymptotic Approximations

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25.9.1 $$\zeta \left(\sigma +\mathrm{i}t\right)=\sum _{1\le n\le x}\frac{1}{n^s}+\chi (s)\sum _{1\le n\le y}\frac{1}{n^{1-s}}+O\left(x^{-\sigma }\right)+O\left(y^{\sigma -1}{t}^{\frac{1}{2}-\sigma }\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable, $s$: complex variable and $\sigma$: fixed

Keywords:

asymptotic approximation

Source:

Titchmarsh (1986b, (4.12.4), p. 79)

Referenced by:

§25.9

Permalink:

http://dlmf.nist.gov/25.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.9 and Ch.25

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25.9.3 $$\zeta \left(\frac{1}{2}+\mathrm{i}t\right)=\sum _{n=1}^m\frac{1}{n^{\frac{1}{2}+\mathrm{i}t}}+\chi \left(\frac{1}{2}+\mathrm{i}t\right)\sum _{n=1}^m\frac{1}{n^{\frac{1}{2}-\mathrm{i}t}}+O\left(t^{-1/4}\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer and $n$: nonnegative integer

Keywords:

asymptotic approximation

Source:

Titchmarsh (1986b, (4.17.1), p. 88)

Referenced by:

§25.10(ii)

Permalink:

http://dlmf.nist.gov/25.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.9 and Ch.25

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23: 14.2 Differential Equations

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14.2.10 $$𝒲\left\{P_{\nu }^{\mu }\left(x\right),Q_{\nu }^{\mu }\left(x\right)\right\}=-{\mathrm{e}}^{\mu \pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\mathrm{\Gamma }\left(\nu -\mu +1\right)\left(x^2-1\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.1.8 (modified)

Permalink:

http://dlmf.nist.gov/14.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §14.2(iv), §14.2 and Ch.14

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14.2.11 $$P_{\nu +1}^{\mu }\left(x\right)Q_{\nu }^{\mu }\left(x\right)-P_{\nu }^{\mu }\left(x\right)Q_{\nu +1}^{\mu }\left(x\right)={\mathrm{e}}^{\mu \pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\mathrm{\Gamma }\left(\nu -\mu +2\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.2(iv), §14.2 and Ch.14

24: 2.5 Mellin Transform Methods

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2.5.14 $$|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|=\sqrt{2\pi }{\mathrm{e}}^{-\pi |y|/2}{|y|}^{x-(1/2)}\left(1+o\left(1\right)\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $o\left(x\right)$: order less than

Referenced by:

§2.5(iii)

Permalink:

http://dlmf.nist.gov/2.5.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

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2.5.36 $$E_{jk}(x)=\frac{1}{2\pi \mathrm{i}}{\int }_{q_{jk}-\mathrm{i}\mathrm{\infty }}^{q_{jk}+\mathrm{i}\mathrm{\infty }}{x}^{-z}{G}_{jk}(z)dz=o\left(x^{-q_{jk}}\right)$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $o\left(x\right)$: order less than, $G_{jk}(z)$: function, $q_{jk}>p_{jk}$: real number and $E_{jk}(x)$: function

Permalink:

http://dlmf.nist.gov/2.5.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §2.5(ii), §2.5 and Ch.2

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25: 11.6 Asymptotic Expansions

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11.6.7 $${𝐌}_{\nu }\left(\lambda \nu \right)\sim -\frac{{(\frac{1}{2}\lambda \nu )}^{\nu -1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{k!c_k(\mathrm{i}\lambda )}{{\nu }^k},$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$.

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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, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $c_k(\lambda )$: expansion function

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E7

Encodings:

pMML, png, TeX

See also:

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

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26: 28.1 Special Notation

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$m,n$	integers.
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$z=x+\mathrm{i}y$	complex variable.
$\nu$	order of the Mathieu function or modified Mathieu function. (When $\nu$ is an integer it is often replaced by $n$.)
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… ►Arscott (1964b) also uses $-\mathrm{i}\mu$ for $\nu$. …

27: 14.1 Special Notation

§14.1 Special Notation

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$x$, $y$, $\tau$	real variables.
$z=x+\mathrm{i}y$	complex variable.
$m$, $n$	unless stated otherwise, nonnegative integers, used for order and degree, respectively.
$\mu$, $\nu$	general order and degree, respectively.
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… ►The main functions treated in this chapter are the Legendre functions ${𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }\left(x\right)$, $P_{\nu }\left(z\right)$, $Q_{\nu }\left(z\right)$; Ferrers functions ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$; conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ (also known as Mehler functions). …

28: 11.2 Definitions

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11.2.2 $${𝐋}_{\nu }\left(z\right)=-\mathrm{i}{\mathrm{e}}^{-\frac{1}{2}\pi \mathrm{i}\nu }{𝐇}_{\nu }\left(\mathrm{i}z\right)={(\frac{1}{2}z)}^{\nu +1}\sum _{n=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2}z)}^{2n}}{\mathrm{\Gamma }\left(n+\frac{3}{2}\right)\mathrm{\Gamma }\left(n+\nu +\frac{3}{2}\right)}.$$

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Defines:

${𝐋}_{\nu }\left(z\right)$: modified Struve function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\nu$: real or complex order and $n$: integer order

A&S Ref:

12.2.1

Referenced by:

§11.13(ii), §11.2(i), §11.4(i), §11.4(iii), §11.5(ii), §11.5(i), §11.6(ii), §11.6(ii)

Permalink:

http://dlmf.nist.gov/11.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(i), §11.2 and Ch.11

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29: 2.10 Sums and Sequences

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2.10.16 $$S(\alpha ,\beta ,n)=\frac{{\mathrm{e}}^{\mathrm{i}\beta }}{{\mathrm{e}}^{\mathrm{i}\beta }-1}\left({\mathrm{e}}^{\mathrm{i}(n-1)\beta }{n}^{\alpha }-\alpha S(\alpha -1,\beta ,n)+O\left(n^{\alpha -1}\right)+O\left(1\right)\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $S(\alpha ,\beta ,n)$: sum

Referenced by:

§2.10(ii)

Permalink:

http://dlmf.nist.gov/2.10.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(ii), §2.10(ii), §2.10 and Ch.2

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2.10.18 $$S(\alpha ,\beta ,n)=\frac{{\mathrm{e}}^{\mathrm{i}n\beta }}{{\mathrm{e}}^{\mathrm{i}\beta }-1}{n}^{\alpha }+O\left(n^{\alpha -1}\right)+O\left(1\right),$$ $n\to \mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $S(\alpha ,\beta ,n)$: sum

Permalink:

http://dlmf.nist.gov/2.10.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(ii), §2.10(ii), §2.10 and Ch.2

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2.10.23 $${}_0{}^{}F_2^{}(-;1,1;x)={\int }_{-1/2}^{\mathrm{\infty }}\frac{x^t}{{(\mathrm{\Gamma }\left(t+1\right))}^3}dt+2\mathrm{\Re }{\int }_{-1/2}^{\mathrm{i}\mathrm{\infty }}\frac{x^t}{{(\mathrm{\Gamma }\left(t+1\right))}^3}\frac{dt}{{\mathrm{e}}^{-2\pi \mathrm{i}t}-1}={\int }_0^{\mathrm{\infty }}\frac{x^t}{{(\mathrm{\Gamma }\left(t+1\right))}^3}dt+O\left(1\right),$$ $x\to +\mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Permalink:

http://dlmf.nist.gov/2.10.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iii), §2.10(iii), §2.10 and Ch.2

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30: 30.11 Radial Spheroidal Wave Functions

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30.11.6

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\psi }_k^{(j)}(z)$: spherical function and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(iii), §30.11 and Ch.30

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