conversions between variables and parameters

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11: Bibliography D

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Coulomb wave functions with complex values of the variable and the parameters. J. Math. Phys. 40 (12), pp. 6145–6166.

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External Links:

ISSN 0022-2488, Document, MathReview (Raj Wilson), ZentralBlatt

Referenced by:

§33.13.

Encodings:

BibTeX

12: 29.11 Lamé Wave Equation

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29.11.1 $$\frac{d^{2}w}{{dz}^2}+(h-\nu (\nu +1)k^2{\mathrm{sn}}^2(z,k)+k^2{\omega }^2{\mathrm{sn}}^4(z,k))w=0,$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $h$: real parameter, $k$: real parameter, $\nu$: real parameter, $\omega$: parameter and $w(z)$: solution

Referenced by:

§29.11, §29.11, §29.18(ii), §29.7(ii)

Permalink:

http://dlmf.nist.gov/29.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.11 and Ch.29

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13: 31.1 Special Notation

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$x$, $y$	real variables.
$z$, $\zeta$, $w$, $W$	complex variables.
…
$a$	complex parameter, $|a|\ge 1,a\ne 1$.
$q,\alpha ,\beta ,\gamma ,\delta ,ϵ,\nu$	complex parameters.

… ►Sometimes the parameters are suppressed.

14: 31.14 General Fuchsian Equation

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31.14.1 $$\frac{d^{2}w}{{dz}^2}+\left(\sum _{j=1}^N\frac{{\gamma }_j}{z-a_j}\right)\frac{dw}{dz}+\left(\sum _{j=1}^N\frac{q_j}{z-a_j}\right)w=0,$$ ${\sum }_{j=1}^N{q}_j=0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(i)

Permalink:

http://dlmf.nist.gov/31.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14 and Ch.31

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31.14.3 $$w(z)=\left(\prod _{j=1}^N{(z-a_j)}^{-{\gamma }_j/2}\right)W(z),$$

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $W(z)$: solution

Permalink:

http://dlmf.nist.gov/31.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

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31.14.4 $$\frac{d^{2}W}{{dz}^2}=\sum _{j=1}^N\left(\frac{{\tilde{\gamma }}_j}{{(z-a_j)}^2}+\frac{{\tilde{q}}_j}{z-a_j}\right)W,$$ ${\sum }_{j=1}^N{\tilde{q}}_j=0$,

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Defines:

$W(z)$: solution (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

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15: 32.1 Special Notation

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$m,n$	integers.
$x$	real variable.
$z$	complex variable.
$k$	real parameter.

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16: 15.7 Continued Fractions

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15.7.1 $$\frac{𝐅(a,b;c;z)}{𝐅(a,b+1;c+1;z)}=t_0-\frac{u_{1}z}{t_1-}\frac{u_{2}z}{t_2-}\frac{u_{3}z}{t_3-}\mathrm{\cdots },$$

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Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $t_n$: coefficient and $u_n$: coefficient

Permalink:

http://dlmf.nist.gov/15.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

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15.7.3 $$\frac{𝐅(a,b;c;z)}{𝐅(a,b+1;c+1;z)}=v_0-\frac{w_1}{v_1-}\frac{w_2}{v_2-}\frac{w_3}{v_3-}\mathrm{\cdots },$$

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Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $v_n$: coefficient and $w_n$: coefficient

Permalink:

http://dlmf.nist.gov/15.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

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15.7.5 $$\frac{𝐅(a,b;c;z)}{𝐅(a+1,b+1;c+1;z)}=x_0+\frac{y_1}{x_1+}\frac{y_2}{x_2+}\frac{y_3}{x_3+}\mathrm{\cdots },$$

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Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $y$: real variable, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

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17: 28.14 Fourier Series

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28.14.1 $${\mathrm{me}}_{\nu }(z,q)=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{2m}^{\nu }(q){\mathrm{e}}^{\mathrm{i}(\nu +2m)z},$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $q=h^2$: parameter, $z$: complex variable, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

A&S Ref:

20.3.8 (in slightly different form)

Referenced by:

§28.22(ii)

Permalink:

http://dlmf.nist.gov/28.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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28.14.2 $${\mathrm{ce}}_{\nu }(z,q)=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{2m}^{\nu }(q)\cos (\nu +2m)z,$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $z$: complex variable, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Permalink:

http://dlmf.nist.gov/28.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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28.14.3 $${\mathrm{se}}_{\nu }(z,q)=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{2m}^{\nu }(q)\sin (\nu +2m)z,$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $z$: complex variable, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Permalink:

http://dlmf.nist.gov/28.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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18: 28.32 Mathematical Applications

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28.32.2 $$\frac{{\partial }^{2}V}{{\partial x}^2}+\frac{{\partial }^{2}V}{{\partial y}^2}+k^{2}V=0$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $y$: real variable, $k$: parameter and $V(\xi ,\eta )$: solution

Referenced by:

§28.33(ii)

Permalink:

http://dlmf.nist.gov/28.32.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

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28.32.3 $$\frac{{\partial }^{2}V}{{\partial \xi }^2}+\frac{{\partial }^{2}V}{{\partial \eta }^2}+\frac{1}{2}{c}^2{k}^2(\cosh \left(2\xi \right)-\cos \left(2\eta \right))V=0.$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\eta$: variable, $\xi$: variable, $k$: parameter and $V(\xi ,\eta )$: solution

Referenced by:

§28.32(i), §28.33(ii)

Permalink:

http://dlmf.nist.gov/28.32.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

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28.32.4 $$\frac{{\partial }^{2}K}{{\partial z}^2}-\frac{{\partial }^{2}K}{{\partial \zeta }^2}=2q\left(\cos \left(2z\right)-\cos \left(2\zeta \right)\right)K.$$

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Symbols:

$\cos z$: cosine function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $q=h^2$: parameter, $z$: complex variable, $\zeta$: variable and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/28.32.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

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19: 18.33 Polynomials Orthogonal on the Unit Circle

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18.33.22 $$p^{\ast }(z)\equiv z^n\overline{p({\overline{z}}^{-1})}=\sum _{k=0}^n\overline{c_{n-k}}{z}^k.$$

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Symbols:

$\overline{z}$: complex conjugate, $\equiv$: equals by definition, $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer

Source:

Simon (2005a, (1.1.7))

Permalink:

http://dlmf.nist.gov/18.33.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

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20: 9.14 Incomplete Airy Functions

… ►Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …