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11: 36.8 Convergent Series Expansions

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${\mathrm{\Psi }}_K\left(𝐱\right)={\displaystyle \frac{2}{K+2}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\mathrm{i}}^n\cos \left({\displaystyle \frac{\pi (n(K+1)-1)}{2(K+2)}}\right)\mathrm{\Gamma }\left({\displaystyle \frac{n+1}{K+2}}\right)a_n(𝐱),$ $K$ odd,

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36.8.4 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)=2{\pi }^2{\left(\frac{2}{3}\right)}^{2/3}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(-\mathrm{i}{(2/3)}^{2/3}z\right)}^n}{n!}\mathrm{\Re }\left(f_n(\frac{x+\mathrm{i}y}{{12}^{1/3}},\frac{x-\mathrm{i}y}{{12}^{1/3}})\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter, $m$: integer, $x$: real parameter and $f_n$: function

Referenced by:

§36.8

Permalink:

http://dlmf.nist.gov/36.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.8 and Ch.36

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12: 19.2 Definitions

… ►Let $s^2(t)$ be a cubic or quartic polynomial in $t$ with simple zeros, and let $r(s,t)$ be a rational function of $s$ and $t$ containing at least one odd power of $s$. …Thus the elliptic part of (19.2.1) is …

13: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►We integrate by parts twice giving: … ►Other applications follow from the fact that $ℒ$ is suitable for describing vibrations, especially standing waves, which arise in many parts of engineering and the physical sciences, see Birkhoff and Rota (1989, §§10.3 and 10.16). … ►For $f(x)$ even in $x$ this yields the Fourier cosine transform pair (1.14.9) & (1.14.11), and for $f(x)$ odd the Fourier sine transform pair (1.14.10) & (1.14.12). … ►Note that the notations of (1.18.32) and (1.18.47) are used to distinguish the contributions from the discrete and continuous parts of the spectrum. … ►Then $dim{N}_z$ is constant for $\mathrm{\Im }z>0$ and also constant for . …

14: 36.2 Catastrophes and Canonical Integrals

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36.2.15

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit and $K$: codimension

Referenced by:

§36.2(ii)

Permalink:

http://dlmf.nist.gov/36.2.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(ii), §36.2 and Ch.36

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$$\frac{{\partial }^p}{{\partial x_1}^p}{\mathrm{\Psi }}_K\left(\mathrm{𝟎}\right)=\frac{2}{K+2}\mathrm{\Gamma }\left(\frac{p+1}{K+2}\right)\cos \left(\frac{\pi }{2}\left(\frac{p+1}{K+2}+p\right)\right),$$ $K$ odd,

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36.2.20 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,0\right)=2{\pi }^2{(\frac{2}{3})}^{2/3}\mathrm{\Re }\left(\mathrm{Ai}\left(\frac{x+\mathrm{i}y}{{12}^{1/3}}\right)\mathrm{Bi}\left(\frac{x-\mathrm{i}y}{{12}^{1/3}}\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $y$: real parameter and $x$: real parameter

Referenced by:

§36.2(ii), §36.7(iii), §36.8

Permalink:

http://dlmf.nist.gov/36.2.E20

Encodings:

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See also:

Annotations for §36.2(ii), §36.2 and Ch.36

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15: 28.29 Definitions and Basic Properties

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28.29.6

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Symbols:

$\mathrm{\Re }$: real part and $\nu$: complex parameter

Referenced by:

§28.29(ii)

Permalink:

http://dlmf.nist.gov/28.29.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(ii), §28.29 and Ch.28

… ►In the symmetric case $Q(z)=Q(-z)$, $w_{\text{I}}(z,\lambda )$ is an even solution and $w_{\text{II}}(z,\lambda )$ is an odd solution; compare §28.2(ii). …

16: 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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17: 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:

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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): … ►

12.14.10 $$w_2(a,x)=\sum _{n=0}^{\mathrm{\infty }}{\beta }_n(a)\frac{x^{2n+1}}{(2n+1)!},$$

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Defines:

$w_2(a,x)$: odd solution (locally)

Symbols:

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

A&S Ref:

19.16.2 (modification of)

Permalink:

http://dlmf.nist.gov/12.14.E10

Encodings:

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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 …

18: 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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30.11.11 $$K_n^m(\gamma )=\frac{\sqrt{\pi }}{2}{\left(\frac{\gamma }{2}\right)}^{m+1}\frac{{(-1)}^m{a}_{n,\frac{1}{2}(m-n+1)}^{-m}({\gamma }^2)}{\mathrm{\Gamma }\left(\frac{5}{2}+m\right)A_n^{-m}({\gamma }^2)({d{\mathrm{𝖯𝗌}}_n^m(z,{\gamma }^2)/dz|}_{z=0})},$$ $n-m$ odd.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $z$: complex variable, $m$: 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.E11

Encodings:

pMML, png, TeX

See also:

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

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