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41: 12.17 Physical Applications

… ►Setting $w=U(\xi )V(\eta )W(\zeta )$ and separating variables, we obtain … ►Buchholz (1969) collects many results on boundary-value problems involving PCFs. Miller (1974) treats separation of variables by group theoretic methods. Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator. ►Problems on high-frequency scattering in homogeneous media by parabolic cylinders lead to asymptotic methods for integrals involving PCFs. …

42: 30.5 Functions of the Second Kind

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30.5.1 $${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2),$$ $n=m,m+1,m+2,\mathrm{\dots }$.

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Symbols:

${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: 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/30.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

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30.5.2 $${\mathrm{𝖰𝗌}}_n^m(-x,{\gamma }^2)={(-1)}^{n-m+1}{\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2),$$

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Symbols:

${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: 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/30.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

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30.5.3 $${\mathrm{𝖰𝗌}}_n^m(x,0)={𝖰}_n^m\left(x\right);$$

ⓘ

Symbols:

: Ferrers function of the second kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer and $n\ge m$: integer degree

Permalink:

http://dlmf.nist.gov/30.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

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30.5.4 $$𝒲\left\{{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),{\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)\right\}=\frac{(n+m)!}{(1-x^2)(n-m)!}{A}_n^m({\gamma }^2)A_n^{-m}({\gamma }^2)(\ne 0),$$

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Symbols:

$𝒲$: Wronskian, $!$: factorial (as in $n!$), ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: 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, $A_n^m({\gamma }^2)$ and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

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43: 10.40 Asymptotic Expansions for Large Argument

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10.40.11 $$|R_{\mathrm{\ell }}(\nu ,z)|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\mathrm{\infty }}\left(t^{-1}\right)\right),$$

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Defines:

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

Symbols:

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

Permalink:

http://dlmf.nist.gov/10.40.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(iii), §10.40 and Ch.10

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10.40.12

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, ${𝒱}_{a,b}\left(f\right)$: total variation and $z$: complex variable

Referenced by:

§10.40(i), §10.40(iii), Erratum (V1.0.11) for Equation (10.40.12)

Permalink:

http://dlmf.nist.gov/10.40.E12

Encodings:

pMML, png, TeX

Correction (effective with 1.0.11):

Originally the third constraint was written incorrectly as $\pi \le |\mathrm{ph}z|\le \frac{3}{2}\pi$.

Suggested 2014-11-05 by Gergő Nemes

See also:

Annotations for §10.40(iii), §10.40 and Ch.10

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44: 23.11 Integral Representations

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23.11.2 $$\mathrm{\wp }\left(z\right)=\frac{1}{z^2}+8{\int }_0^{\mathrm{\infty }}s\left({\mathrm{e}}^{-s}{\sinh }^2\left(\frac{1}{2}zs\right)f_1(s,\tau )+{\mathrm{e}}^{\mathrm{i}\tau s}{\sin }^2\left(\frac{1}{2}zs\right)f_2(s,\tau )\right)ds,$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $𝕃$: lattice, $z$: complex, $\tau$: complex variable and $f_j(s,\tau )$: functions

Permalink:

http://dlmf.nist.gov/23.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.11 and Ch.23

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23.11.3 $$\zeta \left(z\right)=\frac{1}{z}+{\int }_0^{\mathrm{\infty }}\left({\mathrm{e}}^{-s}\left(zs-\sinh \left(zs\right)\right)f_1(s,\tau )-{\mathrm{e}}^{\mathrm{i}\tau s}\left(zs-\sin \left(zs\right)\right)f_2(s,\tau )\right)ds,$$

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $𝕃$: lattice, $z$: complex, $\tau$: complex variable and $f_j(s,\tau )$: functions

Permalink:

http://dlmf.nist.gov/23.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.11 and Ch.23

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45: 30.7 Graphics

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►►

Figure 30.7.5: ${\mathrm{𝖯𝗌}}_n^0(x,4)$, $n=0,1,2,3$, $-1\le x\le 1$. Magnify

►

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Figure 30.7.6: ${\mathrm{𝖯𝗌}}_n^0(x,-4)$, $n=0,1,2,3$, $-1\le x\le 1$. Magnify

►

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Figure 30.7.7: ${\mathrm{𝖯𝗌}}_n^1(x,30)$, $n=1,2,3,4$, $-1\le x\le 1$. Magnify

►

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Figure 30.7.8: ${\mathrm{𝖯𝗌}}_n^1(x,-30)$, $n=1,2,3,4$, $-1\le x\le 1$. Magnify

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►►

Figure 30.7.11: ${\mathrm{𝖰𝗌}}_n^0(x,4)$, $n=0,1,2,3$, . Magnify

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46: 10.17 Asymptotic Expansions for Large Argument

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

ⓘ

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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10.17.15

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, ${𝒱}_{a,b}\left(f\right)$: total variation, $z$: complex variable and $\chi (\mathrm{\ell })$

Referenced by:

§10.17(iv)

Permalink:

http://dlmf.nist.gov/10.17.E15

Encodings:

pMML, png, TeX

See also:

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

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47: 30.4 Functions of the First Kind

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30.4.1 $${\int }_{-1}^1{\left({\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)\right)}^{2}dx=\frac{2}{2n+1}\frac{(n+m)!}{(n-m)!},$$

ⓘ

Symbols:

$dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

A&S Ref:

21.7.11

Referenced by:

§30.4(iii)

Permalink:

http://dlmf.nist.gov/30.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(i), §30.4 and Ch.30

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30.4.2 $${\mathrm{𝖯𝗌}}_n^m(x,0)={𝖯}_n^m\left(x\right);$$

ⓘ

Symbols:

: Ferrers function of the first kind, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer and $n\ge m$: integer degree

Referenced by:

§30.4(iv)

Permalink:

http://dlmf.nist.gov/30.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(i), §30.4 and Ch.30

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30.4.3 $${\mathrm{𝖯𝗌}}_n^m(-x,{\gamma }^2)={(-1)}^{n-m}{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2).$$

ⓘ

Symbols:

${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: 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/30.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(ii), §30.4 and Ch.30

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30.4.4 $${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)={(1-x^2)}^{\frac{1}{2}m}\sum _{k=0}^{\mathrm{\infty }}{g}_k{x}^k,$$ $-1\le x\le 1$,

ⓘ

Symbols:

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

A&S Ref:

21.7.18

Permalink:

http://dlmf.nist.gov/30.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(iii), §30.4 and Ch.30

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30.4.7 $$f(x)=\sum _{n=m}^{\mathrm{\infty }}{c}_n{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),$$

ⓘ

Symbols:

${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $f(x)$: function and $c_n$: coefficient

Referenced by:

§30.4(iv)

Permalink:

http://dlmf.nist.gov/30.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(iv), §30.4 and Ch.30

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48: Bibliography G

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Algorithm 259: Legendre functions for arguments larger than one. Comm. ACM 8 (8), pp. 488–492.

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Notes:

Variable-precision Algol. For remark see ACM Trans. Math. Software 3(2) (1977), p. 204–205

External Links:

ISSN 0001-0782, Document

Referenced by:

§14.34(ii).

Encodings:

BibTeX

…

49: 20.3 Graphics

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§20.3(i) $\theta$-Functions: Real Variable and Real Nome

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►

►

Figure 20.3.13: ${\theta }_4(\pi x,q)$, $0\le x\le 2$, $0\le q\le 0.99$. Magnify 3D Help

►

§20.3(ii) $\theta$-Functions: Complex Variable and Real Nome

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§20.3(iii) $\theta$-Functions: Real Variable and Complex Lattice Parameter

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►

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Figure 20.3.21: ${\theta }_4\left(0\right|u+\mathrm{i}v)$, $-1\le u\le 1$, $0.005\le v\le 0.1$. Magnify 3D Help

50: 14.19 Toroidal (or Ring) Functions

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14.19.2 $$P_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(\frac{1}{2}-\mu \right)}{{\pi }^{1/2}{\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}𝐅(\frac{1}{2}-\mu ,\frac{1}{2}+\nu -\mu ;1-2\mu ;1-{\mathrm{e}}^{-2\xi }),$$ $\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine 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, $\mu$: general order, $\nu$: general degree and $\xi >0$: variable

Referenced by:

§14.19(ii), Erratum (V1.0.1) for Equation (14.19.2)

Permalink:

http://dlmf.nist.gov/14.19.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.1):

Originally the argument to $𝐅$ was incorrect and the condition on $\mu$ too weak:

$$P_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(1-2\mu \right)2^{2\mu }}{\mathrm{\Gamma }\left(1-\mu \right){\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}𝐅(\frac{1}{2}-\mu ,\frac{1}{2}+\nu -\mu ;1-2\mu ;{\mathrm{e}}^{-2\xi }),$$ $\mu \ne \frac{1}{2}$.

Also, the factor multiplying $𝐅$ was rewritten to clarify the poles, which determine the correct condition on $\mu$.

Reported 2010-11-02 by Alvaro Valenzuela

See also:

Annotations for §14.19(ii), §14.19 and Ch.14

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