values on the cut

SearchAdvancedHelp

(0.001 seconds)

31—40 of 45 matching pages

31: 30.6 Functions of Complex Argument

… ►of (30.2.1) with $\mu =m$ and $\lambda ={\lambda }_n^m\left({\gamma }^2\right)$ are real when $z\in (1,\mathrm{\infty })$, and their principal values (§4.2(i)) are obtained by analytic continuation to $ℂ\setminus (-\mathrm{\infty },1]$. … ►

Values on $(-1,1)$

►

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

ⓘ

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

►

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

ⓘ

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

…

32: 4.13 Lambert $W$-Function

… ► $W_0\left(z\right)$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },-{\mathrm{e}}^{-1}]$, real-valued when $z>-{\mathrm{e}}^{-1}$, and has a square root branch point at $z=-{\mathrm{e}}^{-1}$. …The other branches $W_k\left(z\right)$ are single-valued analytic functions on $ℂ\setminus (-\mathrm{\infty },0]$, have a logarithmic branch point at $z=0$, and, in the case $k=\pm 1$, have a square root branch point at $z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}$ respectively. … ►

►►

Figure 4.13.2: The $W\left(z\right)$ function on the first 5 Riemann sheets. … Magnify

►Alternative notations are $\mathrm{Wp}\left(x\right)$ for $W_0\left(x\right)$, $\mathrm{Wm}\left(x\right)$ for $W_{-1}\left(x+0\mathrm{i}\right)$, both previously used in this section, the Wright $\omega$-function $\omega \left(z\right)=W\left({\mathrm{e}}^z\right)$, which is single-valued, satisfies … ►where $t\ge 0$ for $W_0$, $t\le 0$ for $W_{\pm 1}$ on the relevant branch cuts, …

33: 14.1 Special Notation

… ►Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise. ►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). …

34: 14.24 Analytic Continuation

… ►the limiting value being taken in (14.24.1) when $2\nu$ is an odd integer. … ►the limiting value being taken in (14.24.4) when $\mu \in \mathbb{Z}$. … ►The behavior of $P_{\nu }^{-\mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ as $z\to -1$ from the left on the upper or lower side of the cut from $-\mathrm{\infty }$ to $1$ can be deduced from (14.8.7)–(14.8.11), combined with (14.24.1) and (14.24.2) with $s=\pm 1$.

35: 14.22 Graphics

… ►In the graphics shown in this section, height corresponds to the absolute value of the function and color to the phase. … ►

►

►

Figure 14.22.1: $P_{1/2}^0\left(x+\mathrm{i}y\right)$, $-5\le x\le 5$, $-5\le y\le 5$. There is a cut along the real axis from $-\mathrm{\infty }$ to $-1$. Magnify 3D Help

►

►

►

Figure 14.22.2: $P_{1/2}^{-1/2}\left(x+\mathrm{i}y\right)$, $-5\le x\le 5$, $-5\le y\le 5$. There is a cut along the real axis from $-\mathrm{\infty }$ to $1$. Magnify 3D Help

►

►

►

Figure 14.22.3: $P_{1/2}^{-1}\left(x+\mathrm{i}y\right)$, $-5\le x\le 5$, $-5\le y\le 5$. There is a cut along the real axis from $-\mathrm{\infty }$ to $1$. Magnify 3D Help

►

►

►

Figure 14.22.4: ${𝑸}_0^0\left(x+\mathrm{i}y\right)$, $-5\le x\le 5$, $-5\le y\le 5$. There is a cut along the real axis from $-1$ to $1$. Magnify 3D Help

36: 30.9 Asymptotic Approximations and Expansions

… ►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … ►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). … ►The asymptotic behavior of ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ and ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$ as $x\to \pm 1$ is given in Erdélyi et al. (1955, p. 151). …

37: 5.12 Beta Function

… ►In this section all fractional powers have their principal values, except where noted otherwise. … ►In (5.12.8) the fractional powers have their principal values when $w>0$ and $z>0$, and are continued via continuity. … ►In (5.12.11) and (5.12.12) the fractional powers are continuous on the integration paths and take their principal values at the beginning. …when $\mathrm{\Re }b>0$, $a$ is not an integer and the contour cuts the real axis between $-1$ and the origin. …

38: 15.3 Graphics

… ►In Figures 15.3.5 and 15.3.6, height corresponds to the absolute value of the function and color to the phase. … ►

►

►

Figure 15.3.5: $F(\frac{4}{3},\frac{9}{16};\frac{14}{5};x+\mathrm{i}y),0\le x\le 2,-0.5\le y\le 0.5$. (There is a cut along the real axis from $1$ to $\mathrm{\infty }$.) Magnify 3D Help

…

39: 30.13 Wave Equation in Prolate Spheroidal Coordinates

… ►In most applications the solution $w$ has to be a single-valued function of $(x,y,z)$, which requires $\mu =m$ (a nonnegative integer) and … ►

30.13.13 $$w_2(\eta )=a_2{\mathrm{𝖯𝗌}}_n^m(\eta ,{\gamma }^2)+b_2{\mathrm{𝖰𝗌}}_n^m(\eta ,{\gamma }^2).$$

ⓘ

Symbols:

${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $m$: nonnegative integer, $n\ge m$: integer degree, $b_j$: coefficients, $\eta$: prolate spheroidal coordinate, $w_j$: solution to DE, ${\gamma }^2={\kappa }^2{c}^2$: parameter and $a_j$: coefficients

Referenced by:

§30.13(v), §30.14(iv), §30.14(v)

Permalink:

http://dlmf.nist.gov/30.13.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(iv), §30.13 and Ch.30

… ►For the Dirichlet boundary-value problem of the region ${\xi }_1\le \xi \le {\xi }_2$ between two ellipsoids, the eigenvalues are determined from …

40: 22.18 Mathematical Applications

… ►The half-open rectangle $(-K,K)\times [-K^{\prime },K^{\prime }]$ maps onto $ℂ$ cut along the intervals $(-\mathrm{\infty },-1]$ and $[1,\mathrm{\infty })$. … ►in which $a,b,c,d,e,f$ are real constants, can be achieved in terms of single-valued functions. …