of imaginary argument

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21: 20.1 Special Notation

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$m$, $n$	integers.
$z$ $(\in ℂ)$	the argument.
$\tau$ $(\in ℂ)$	the lattice parameter, $\mathrm{\Im }\tau >0$.
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►The main functions treated in this chapter are the theta functions ${\theta }_j\left(z\right|\tau )={\theta }_j(z,q)$ where $j=1,2,3,4$ and $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$. When $\tau$ is fixed the notation is often abbreviated in the literature as ${\theta }_j(z)$, or even as simply ${\theta }_j$, it being then understood that the argument is the primary variable. … ►Primes on the $\theta$ symbols indicate derivatives with respect to the argument of the $\theta$ function. …

22: 30.7 Graphics

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§30.7(iv) Functions of Complex Argument

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Figure 30.7.16: $|{\mathrm{𝑃𝑠}}_0^0(x+\mathrm{i}y,4)|$, $-2\le x\le 2$, $-2\le y\le 2$. Magnify 3D Help

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Figure 30.7.17: $|{\mathrm{𝑃𝑠}}_0^0(x+\mathrm{i}y,-4)|$, $-2\le x\le 2$, $-2\le y\le 2$. Magnify 3D Help

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Figure 30.7.18: $|{\mathrm{𝑃𝑠}}_1^1(x+\mathrm{i}y,4)|$, $-2\le x\le 2$, $-2\le y\le 2$. Magnify 3D Help

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Figure 30.7.19: $|{\mathrm{𝑃𝑠}}_1^1(x+\mathrm{i}y,-4)|$, $-2\le x\le 2$, $-2\le y\le 2$. Magnify 3D Help

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23: 10.17 Asymptotic Expansions for Large Argument

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10.17.12 $$H_{\nu }^{(2)}{}_{}{}^{\prime }\left(z\right)\sim -\mathrm{i}{\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-\mathrm{i}\omega }\sum _{k=0}^{\mathrm{\infty }}{(-\mathrm{i})}^k\frac{b_k(\nu )}{z^k},$$ $-2\pi +\delta \le \mathrm{ph}z\le \pi -\delta$.

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Symbols:

$H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $\omega$ and $b_k(\nu )$: polynomial coefficient

A&S Ref:

9.2.14

Referenced by:

§10.18(iii)

Permalink:

http://dlmf.nist.gov/10.17.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(ii), §10.17 and Ch.10

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10.17.14 $$\left|R_{\mathrm{\ell }}^{\pm }(\nu ,z)\right|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)\right),$$

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Defines:

$R_{\mathrm{\ell }}^{\pm }(\nu ,z)$: remainder (locally)

Symbols:

$\exp z$: exponential function, $\mathrm{i}$: imaginary unit, ${𝒱}_{a,b}\left(f\right)$: total variation, $z$: complex variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

Referenced by:

§10.17(iv), Erratum (V1.0.10) for Equation (10.17.14)

Permalink:

http://dlmf.nist.gov/10.17.E14

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

Originally the factor ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)$ in the argument to the exponential was written incorrectly as ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$.

Reported 2014-09-27 by Gergő Nemes

See also:

Annotations for §10.17(iv), §10.17 and Ch.10

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24: 30.6 Functions of Complex Argument

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30.6.4 $${\mathrm{𝑃𝑠}}_n^m(x\pm \mathrm{i}0,{\gamma }^2)={(\mp \mathrm{i})}^m{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),$$

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Symbols:

$\mathrm{i}$: imaginary unit, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

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30.6.5 $${\mathrm{𝑄𝑠}}_n^m(x\pm \mathrm{i}0,{\gamma }^2)={(\mp \mathrm{i})}^m\left({\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)\mp \frac{1}{2}\mathrm{i}\pi {\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

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25: 35.2 Laplace Transform

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

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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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26: 9.1 Special Notation

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$k$	nonnegative integer, except in §9.9(iii).
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$z(=x+\mathrm{i}y)$	complex variable.
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primes	derivatives with respect to argument.

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27: 4.15 Graphics

… ►The corresponding surfaces for $\arccos \left(x+\mathrm{i}y\right)$, $\mathrm{arccot}\left(x+\mathrm{i}y\right)$, $\mathrm{arcsec}\left(x+\mathrm{i}y\right)$ can be visualized from Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equations (4.23.16)–(4.23.18).

28: 8.3 Graphics

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§8.3(ii) Complex Argument

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Figure 8.3.8: $\mathrm{\Gamma }(0.25,x+\mathrm{i}y)$, $-3\le x\le 3$, $-3\le y\le 3$. … Magnify 3D Help

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Figure 8.3.9: $\gamma (0.25,x+\mathrm{i}y)$, $-3\le x\le 3$, $-3\le y\le 3$. … Magnify 3D Help

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Figure 8.3.11: $\mathrm{\Gamma }(1,x+\mathrm{i}y)$, $-3\le x\le 3$, $-3\le y\le 3$. Magnify 3D Help

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Figure 8.3.16: ${\gamma }^{\ast }(2.5,x+\mathrm{i}y)$, $-3\le x\le 3$, $-3\le y\le 3$. Magnify 3D Help

29: Errata

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10.17.14 $$\left|R_{\mathrm{\ell }}^{\pm }(\nu ,z)\right|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)\right)$$

Originally the factor ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)$ in the argument to the exponential was written incorrectly as ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$.

Reported 2014-09-27 by Gergő Nemes.

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30: 10.67 Asymptotic Expansions for Large Argument

§10.67 Asymptotic Expansions for Large Argument

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§10.67(i) ${\mathrm{ber}}_{\nu }x,{\mathrm{bei}}_{\nu }x,{\ker }_{\nu }x,{\mathrm{kei}}_{\nu }x$, and Derivatives

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§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu =0$

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