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11: 30.2 Differential Equations

… ►The Liouville normal form of equation (30.2.1) is …

12: 31.14 General Fuchsian Equation

… ►

Normal Form

…

13: William P. Reinhardt

… ►Reinhardt firmly believes that the Mandelbrot set is a special function, and notes with interest that the natural boundaries of analyticity of many “more normal” special functions are also fractals. …

14: 7.20 Mathematical Applications

… ►The normal distribution function with mean $m$ and standard deviation $\sigma$ is given by …For applications in statistics and probability theory, also for the role of the normal distribution functions (the error functions and probability integrals) in the asymptotics of arbitrary probability density functions, see Johnson et al. (1994, Chapter 13) and Patel and Read (1982, Chapters 2 and 3).

15: 28.12 Definitions and Basic Properties

… ►

28.12.2 $${\lambda }_{\nu }\left(-q\right)={\lambda }_{\nu }\left(q\right)={\lambda }_{-\nu }\left(q\right).$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(i), §28.12 and Ch.28

►As in §28.7 values of $q$ for which (28.2.16) has simple roots $\lambda$ are called normal values with respect to $\nu$. For real values of $\nu$ and $q$ all the ${\lambda }_{\nu }\left(q\right)$ are real, and $q$ is normal. … ►If $q$ is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of $z$ and $q$ by the normalization ►

28.12.5 $${\int }_0^{\pi }{\mathrm{me}}_{\nu }(x,q){\mathrm{me}}_{\nu }(-x,q)dx=\pi .$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $q=h^2$: parameter, $x$: real variable and $\nu$: complex parameter

Referenced by:

§28.14

Permalink:

http://dlmf.nist.gov/28.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(ii), §28.12 and Ch.28

…

16: 31.9 Orthogonality

… ►

31.9.2 $${\int }_{\zeta }^{(1+,0+,1-,0-)}{t}^{\gamma -1}{(1-t)}^{\delta -1}{(t-a)}^{ϵ-1}{w}_m(t)w_k(t)dt={\delta }_{m,k}{\theta }_m.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $\zeta$: point and ${\theta }_m$: normalization constant

Permalink:

http://dlmf.nist.gov/31.9.E2

Encodings:

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See also:

Annotations for §31.9(i), §31.9 and Ch.31

… ►The normalization constant ${\theta }_m$ is given by ►

31.9.3 $${\theta }_m={(1-{\mathrm{e}}^{2\pi \mathrm{i}\gamma })(1-{\mathrm{e}}^{2\pi \mathrm{i}\delta }){\zeta }^{\gamma }{(1-\zeta )}^{\delta }{(\zeta -a)}^{ϵ}\frac{f_0(q,\zeta )}{f_1(q,\zeta )}\frac{\partial }{\partial q}𝒲\left\{f_0(q,\zeta ),f_1(q,\zeta )\right\}|}_{q=q_m},$$

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Defines:

${\theta }_m$: normalization constant (locally)

Symbols:

$𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\zeta$: point and $f_0(q_m,z)$, $f_1(q_m,z)$: coefficients

Permalink:

http://dlmf.nist.gov/31.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.9(i), §31.9 and Ch.31

… ►For further information, including normalization constants, see Sleeman (1966a). …

17: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

… ►

33.5.6 $$C_{\mathrm{\ell }}\left(0\right)=\frac{2^{\mathrm{\ell }}\mathrm{\ell }!}{(2\mathrm{\ell }+1)!}=\frac{1}{(2\mathrm{\ell }+1)!!}.$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!!$: double factorial, $!$: factorial (as in $n!$) and $\mathrm{\ell }$: nonnegative integer

Permalink:

http://dlmf.nist.gov/33.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.5(ii), §33.5 and Ch.33

… ►

33.5.9 $$C_{\mathrm{\ell }}\left(\eta \right)\sim \frac{{\mathrm{e}}^{-\pi \eta /2}}{(2\mathrm{\ell }+1)!!}\sim {\mathrm{e}}^{-\pi \eta /2}\frac{{\mathrm{e}}^{\mathrm{\ell }}}{\sqrt{2}{(2\mathrm{\ell })}^{\mathrm{\ell }+1}}.$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!!$: double factorial, $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

Referenced by:

§33.23(iv), §33.5(iv)

Permalink:

http://dlmf.nist.gov/33.5.E9

Encodings:

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See also:

Annotations for §33.5(iv), §33.5 and Ch.33

18: 30.1 Special Notation

… ►where $d_{mn}(\gamma )$ is a normalization constant determined by …

19: 33.7 Integral Representations

… ►

33.7.2 $$H_{\mathrm{\ell }}^{-}(\eta ,\rho )=\frac{{\mathrm{e}}^{-\mathrm{i}\rho }{\rho }^{-\mathrm{\ell }}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{\mathrm{\ell }-\mathrm{i}\eta }{(t+2\mathrm{i}\rho )}^{\mathrm{\ell }+\mathrm{i}\eta }dt,$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.3.1 (modified)

Referenced by:

§33.7

Permalink:

http://dlmf.nist.gov/33.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.7 and Ch.33

►

33.7.3 $$H_{\mathrm{\ell }}^{-}(\eta ,\rho )=\frac{-\mathrm{i}{\mathrm{e}}^{-\pi \eta }{\rho }^{\mathrm{\ell }+1}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{\int }_0^{\mathrm{\infty }}\left(\frac{\exp \left(-\mathrm{i}(\rho \tanh t-2\eta t)\right)}{{(\cosh t)}^{2\mathrm{\ell }+2}}+\mathrm{i}{(1+t^2)}^{\mathrm{\ell }}\exp \left(-\rho t+2\eta \arctan t\right)\right)dt,$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\arctan z$: arctangent function, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.3.3

Referenced by:

§33.7

Permalink:

http://dlmf.nist.gov/33.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.7 and Ch.33

►

33.7.4 $$H_{\mathrm{\ell }}^{+}(\eta ,\rho )=\frac{\mathrm{i}{\mathrm{e}}^{-\pi \eta }{\rho }^{\mathrm{\ell }+1}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{\int }_{-1}^{-\mathrm{i}\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}\rho t}{(1-t)}^{\mathrm{\ell }-\mathrm{i}\eta }{(1+t)}^{\mathrm{\ell }+\mathrm{i}\eta }dt.$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.3.2

Referenced by:

§33.7, §33.7

Permalink:

http://dlmf.nist.gov/33.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.7 and Ch.33

…

20: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

… ►Let $q$ be a normal value (§28.12(i)) with respect to $\nu$, and $f(z)$ be a function that is analytic on a doubly-infinite open strip $S$ that contains the real axis. …