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21: 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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30.11.10 $$K_n^m(\gamma )=\frac{\sqrt{\pi }}{2}{\left(\frac{\gamma }{2}\right)}^m\frac{{(-1)}^m{a}_{n,\frac{1}{2}(m-n)}^{-m}({\gamma }^2)}{\mathrm{\Gamma }\left(\frac{3}{2}+m\right)A_n^{-m}({\gamma }^2){\mathrm{𝖯𝗌}}_n^m(0,{\gamma }^2)},$$ $n-m$ even,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $m$: nonnegative integer, $n\ge m$: integer degree, $A_n^m({\gamma }^2)$ , $K_n^m(\gamma )$: connection coefficient, ${\gamma }^2$: real parameter and $a_{n,k}^m({\gamma }^2)$: coefficients

Referenced by:

§30.16(iii)

Permalink:

http://dlmf.nist.gov/30.11.E10

Encodings:

pMML, png, TeX

See also:

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

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22: 5.4 Special Values and Extrema

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5.4.2

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $!!$: double factorial and $n$: nonnegative integer

Referenced by:

§5.4(i), §5.4(i)

Permalink:

http://dlmf.nist.gov/5.4.E2

Encodings:

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See also:

Annotations for §5.4(i), §5.4 and Ch.5

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5.4.16 $$\mathrm{\Im }\psi \left(\mathrm{i}y\right)=\frac{1}{2y}+\frac{\pi }{2}\coth \left(\pi y\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\coth z$: hyperbolic cotangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.11

Permalink:

http://dlmf.nist.gov/5.4.E16

Encodings:

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See also:

Annotations for §5.4(ii), §5.4 and Ch.5

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5.4.17 $$\mathrm{\Im }\psi \left(\frac{1}{2}+\mathrm{i}y\right)=\frac{\pi }{2}\tanh \left(\pi y\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\tanh z$: hyperbolic tangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.12

Permalink:

http://dlmf.nist.gov/5.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(ii), §5.4 and Ch.5

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5.4.18 $$\mathrm{\Im }\psi \left(1+\mathrm{i}y\right)=-\frac{1}{2y}+\frac{\pi }{2}\coth \left(\pi y\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\coth z$: hyperbolic cotangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.13

Referenced by:

§5.4(ii)

Permalink:

http://dlmf.nist.gov/5.4.E18

Encodings:

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See also:

Annotations for §5.4(ii), §5.4 and Ch.5

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23: 12.14 The Function $W(a,x)$

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12.14.8 $$W(a,x)=W(a,0)w_1(a,x)+W^{\prime }(a,0)w_2(a,x).$$

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Symbols:

$W(a,x)$: parabolic cylinder function, $x$: real variable, $a$: real or complex parameter, $w_1(a,x)$: even solution and $w_2(a,x)$: odd solution

A&S Ref:

19.17.1 (modification of)

Permalink:

http://dlmf.nist.gov/12.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(v), §12.14 and Ch.12

►Here $w_1(a,x)$ and $w_2(a,x)$ are the even and odd solutions of (12.2.3): ►

12.14.9 $$w_1(a,x)=\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n(a)\frac{x^{2n}}{(2n)!},$$

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Defines:

$w_1(a,x)$: even solution (locally)

Symbols:

$!$: factorial (as in $n!$), $x$: real variable, $n$: nonnegative integer, $a$: real or complex parameter and ${\alpha }_n(a)$: recursion

A&S Ref:

19.16.1 (modification of)

Permalink:

http://dlmf.nist.gov/12.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(v), §12.14 and Ch.12

… ►The even and odd solutions of (12.2.3) (see §12.14(v)) are given by … ►The coefficients $c_{2r}$ and $d_{2r}$ are obtainable by equating real and imaginary parts in …