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21: 33.6 Power-Series Expansions in $\rho$

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33.6.1 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\sum _{k=\mathrm{\ell }+1}^{\mathrm{\infty }}{A}_k^{\mathrm{\ell }}(\eta ){\rho }^k,$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

A&S Ref:

14.1.4, 14.1.5

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

►

33.6.2 $$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\sum _{k=\mathrm{\ell }+1}^{\mathrm{\infty }}k{A}_k^{\mathrm{\ell }}(\eta ){\rho }^{k-1},$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

A&S Ref:

14.1.4, 14.1.5

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

… ►

33.6.5 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )=\frac{{\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}}{(2\mathrm{\ell }+1)!\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k}{{\left(2\mathrm{\ell }+2\right)}_{k}k!}{(\mp 2\mathrm{i}\rho )}^{a+k}\left(\ln \left(\mp 2\mathrm{i}\rho \right)+\psi \left(a+k\right)-\psi \left(1+k\right)-\psi \left(2\mathrm{\ell }+2+k\right)\right)-\sum _{k=1}^{2\mathrm{\ell }+1}\frac{(2\mathrm{\ell }+1)!(k-1)!}{(2\mathrm{\ell }+1-k)!{\left(1-a\right)}_k}{(\mp 2\mathrm{i}\rho )}^{a-k}\right),$$

ⓘ

Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $a$: variable

A&S Ref:

14.1.14 (equivalent)

Referenced by:

§33.13, §33.6, §33.6, Erratum (V1.0.19) for Equation (33.6.5)

Permalink:

http://dlmf.nist.gov/33.6.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.0.19):

On the right-hand side the factor in the denominator was incorrectly written as $\mathrm{\Gamma }\left(-\mathrm{\ell }+\mathrm{i}\eta \right)$. This has been corrected to $\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)$.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.6 and Ch.33

…

22: 33.9 Expansions in Series of Bessel Functions

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33.9.1 $$F_{\mathrm{\ell }}(\eta ,\rho )=\rho \sum _{k=0}^{\mathrm{\infty }}{a}_k{𝗃}_{\mathrm{\ell }+k}\left(\rho \right),$$

ⓘ

Symbols:

$F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $a_k$: coefficients

A&S Ref:

14.4.5

Referenced by:

§33.9(i), §33.9(i)

Permalink:

http://dlmf.nist.gov/33.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.9(i), §33.9 and Ch.33

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33.9.3 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\frac{(2\mathrm{\ell }+1)!}{{(2\eta )}^{2\mathrm{\ell }+1}}{\rho }^{-\mathrm{\ell }}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{b}_k{t}^{k/2}{I}_k\left(2\sqrt{t}\right),$$ $\eta >0$,

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $t$: parameter and $b_k$: coefficient

A&S Ref:

14.4.1

Referenced by:

§33.9(ii), §33.9(ii)

Permalink:

http://dlmf.nist.gov/33.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.9(ii), §33.9 and Ch.33

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33.9.4 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\frac{(2\mathrm{\ell }+1)!}{{(2\left|\eta \right|)}^{2\mathrm{\ell }+1}}{\rho }^{-\mathrm{\ell }}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{b}_k{t}^{k/2}{J}_k\left(2\sqrt{t}\right),$$ .

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!$: factorial (as in $n!$), $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $t$: parameter and $b_k$: coefficient

Referenced by:

§33.9(ii), §33.9(ii)

Permalink:

http://dlmf.nist.gov/33.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.9(ii), §33.9 and Ch.33

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33.9.6 $$G_{\mathrm{\ell }}(\eta ,\rho )\sim \frac{{\rho }^{-\mathrm{\ell }}}{(\mathrm{\ell }+\frac{1}{2}){\lambda }_{\mathrm{\ell }}(\eta )C_{\mathrm{\ell }}\left(\eta \right)}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{(-1)}^k{b}_k{t}^{k/2}{K}_k\left(2\sqrt{t}\right),$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\sim$: asymptotic equality, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $t$: parameter, $b_k$: coefficient and ${\lambda }_{\mathrm{\ell }}(\eta )$: coefficient

A&S Ref:

14.4.2

Referenced by:

§33.9(ii)

Permalink:

http://dlmf.nist.gov/33.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.9(ii), §33.9 and Ch.33

…

23: 28.28 Integrals, Integral Representations, and Integral Equations

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§28.28(i) Equations with Elementary Kernels

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28.28.23 $$\frac{2}{\pi }{\int }_0^{\pi }{𝒞}_{2\mathrm{\ell }+2}^{(j)}(2hR)\sin \left((2\mathrm{\ell }+2)\varphi \right){\mathrm{se}}_{2m+2}(t,h^2)dt={(-1)}^{\mathrm{\ell }+m}{B}_{2\mathrm{\ell }+2}^{2m+2}(h^2){\mathrm{Ms}}_{2m+2}^{(j)}(z,h).$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $\sin z$: sine function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function and $B_m(q)$: Fourier coefficient

A&S Ref:

20.7.28 (in different form and only for $\mathrm{\ell }=0$)

Permalink:

http://dlmf.nist.gov/28.28.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(ii), §28.28 and Ch.28

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§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order

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28.28.49 $${\widehat{\alpha }}_{n,m}^{(c)}=\frac{1}{2\pi }{\int }_0^{2\pi }\cos t{\mathrm{ce}}_n(t,h^2){\mathrm{ce}}_m(t,h^2)dt={(-1)}^{p+1}\frac{2}{\mathrm{i}\pi }\frac{{\mathrm{ce}}_n(0,h^2){\mathrm{ce}}_m(0,h^2)}{h{\mathrm{Dc}}_0(n,m,0)}.$$

ⓘ

Defines:

${\widehat{\alpha }}_{n,m}$ (locally)

Symbols:

${\mathrm{Dc}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $m$: integer, $h$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.28.E49

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iv), §28.28 and Ch.28

►

§28.28(v) Compendia

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24: 33.12 Asymptotic Expansions for Large $\eta$

… ►When $\mathrm{\ell }=0$ and $\eta >0$, the outer turning point is given by ${\rho }_{\mathrm{tp}}(\eta ,0)=2\eta$; compare (33.2.2). … ►

33.12.2 $$\genfrac{}{}{0pt}{}{F_0(\eta ,\rho )}{G_0(\eta ,\rho )}\sim {\pi }^{1/2}{(2\eta )}^{1/6}\left\{\genfrac{}{}{0pt}{}{\mathrm{Ai}\left(x\right)}{\mathrm{Bi}\left(x\right)}\left(1+\frac{B_1}{\mu }+\frac{B_2}{{\mu }^2}+\mathrm{\cdots }\right)+\genfrac{}{}{0pt}{}{{\mathrm{Ai}}^{\prime }\left(x\right)}{{\mathrm{Bi}}^{\prime }\left(x\right)}\left(\frac{A_1}{\mu }+\frac{A_2}{{\mu }^2}+\mathrm{\cdots }\right)\right\},$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\rho$: nonnegative real variable, $\eta$: real parameter, $x$: variable, $\mu$: variable, $A_j$: coefficients and $B_j$: coefficients

A&S Ref:

14.4.9

Permalink:

http://dlmf.nist.gov/33.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

►

33.12.3 $$\genfrac{}{}{0pt}{}{F_0^{\prime }(\eta ,\rho )}{G_0^{\prime }(\eta ,\rho )}\sim -{\pi }^{1/2}{(2\eta )}^{-1/6}\left\{\genfrac{}{}{0pt}{}{\mathrm{Ai}\left(x\right)}{\mathrm{Bi}\left(x\right)}\left(\frac{B_1^{\prime }+x{A}_1}{\mu }+\frac{B_2^{\prime }+x{A}_2}{{\mu }^2}+\mathrm{\cdots }\right)+\genfrac{}{}{0pt}{}{{\mathrm{Ai}}^{\prime }\left(x\right)}{{\mathrm{Bi}}^{\prime }\left(x\right)}\left(\frac{B_1+A_1^{\prime }}{\mu }+\frac{B_2+A_2^{\prime }}{{\mu }^2}+\mathrm{\cdots }\right)\right\},$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\rho$: nonnegative real variable, $\eta$: real parameter, $x$: variable, $\mu$: variable, $A_j$: coefficients and $B_j$: coefficients

A&S Ref:

14.4.10

Permalink:

http://dlmf.nist.gov/33.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

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33.12.6 $$\genfrac{}{}{0pt}{}{F_0(\eta ,2\eta )}{3^{-1/2}{G}_0(\eta ,2\eta )}\sim \frac{\mathrm{\Gamma }\left(\frac{1}{3}\right){\omega }^{1/2}}{2\sqrt{\pi }}\left(1\mp \frac{2}{35}\frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\frac{1}{{\omega }^4}-\frac{8}{2025}\frac{1}{{\omega }^6}\mp \frac{5792}{\mathrm{46\hspace{0.17em}06875}}\frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\frac{1}{{\omega }^{10}}-\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\eta$: real parameter and $\omega$: factor

Permalink:

http://dlmf.nist.gov/33.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

►

33.12.7 $$\genfrac{}{}{0pt}{}{F_0^{\prime }(\eta ,2\eta )}{3^{-1/2}{G}_0^{\prime }(\eta ,2\eta )}\sim \frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{2\sqrt{\pi }{\omega }^{1/2}}\left(\pm 1+\frac{1}{15}\frac{\mathrm{\Gamma }\left(\frac{1}{3}\right)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\frac{1}{{\omega }^2}\pm \frac{2}{14175}\frac{1}{{\omega }^6}+\frac{1436}{\mathrm{23\hspace{0.17em}38875}}\frac{\mathrm{\Gamma }\left(\frac{1}{3}\right)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\frac{1}{{\omega }^8}\pm \mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\eta$: real parameter and $\omega$: factor

Referenced by:

§33.12(i)

Permalink:

http://dlmf.nist.gov/33.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

…

25: 28.1 Special Notation

… ► ►►

${\mathrm{Ce}}_{\nu }(z,q)$,	${\mathrm{Se}}_{\nu }(z,q)$,	${\mathrm{Fe}}_n(z,q)$,	${\mathrm{Ge}}_n(z,q)$,
…

►The functions ${\mathrm{Mc}}_n^{(j)}(z,h)$ and ${\mathrm{Ms}}_n^{(j)}(z,h)$ are also known as the radial Mathieu functions. … ►

$f_{o,n}(h).$

… ►The radial functions ${\mathrm{Mc}}_n^{(j)}(z,h)$ and ${\mathrm{Ms}}_n^{(j)}(z,h)$ are denoted by ${\mathrm{Mc}}_n^{(j)}(z,q)$ and ${\mathrm{Ms}}_n^{(j)}(z,q)$, respectively.

26: 33.11 Asymptotic Expansions for Large $\rho$

… ►

33.11.1 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )\sim {\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k{\left(b\right)}_k}{k!{(\pm 2\mathrm{i}\rho )}^k},$$

ⓘ

Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $a$: coefficient and $b$: coefficient

A&S Ref:

14.5.9

Referenced by:

Erratum (V1.0.19) for Equation (33.11.1), Erratum (V1.0.22) for Equation (33.11.1)

Permalink:

http://dlmf.nist.gov/33.11.E1

Encodings:

pMML, png, TeX

Errata (effective with 1.0.22):

Previously this formula was expressed as an equality. Since this formula expresses an asymptotic expansion, it has been corrected by using instead an $\sim$ relation.

Reported 2019-01-29 by Gergő Nemes

Errata (effective with 1.0.19):

Originally the factor in the denominator on the right-hand side was written incorrectly as ${(\mp 2\mathrm{i}\rho )}^k$. This has been corrected to ${(\pm 2\mathrm{i}\rho )}^k$.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.11 and Ch.33

… ►

33.11.4 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )={\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}}(f(\eta ,\rho )\pm \mathrm{i}g(\eta ,\rho )),$$

ⓘ

Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $f(\eta ,\rho )$: function and $g(\eta ,\rho )$: function

Permalink:

http://dlmf.nist.gov/33.11.E4

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.22):

The arguments of some of the functions in these equations were included to improve clarity of the presentation.

See also:

Annotations for §33.11 and Ch.33

…

27: Bibliography Y

… ►

$L$-squared discretizations of the continuum: Radial kinetic energy and the Coulomb Hamiltonian. Phys. Rev. A 11 (4), pp. 1144–1156.

ⓘ

Referenced by:

§18.39(iv), §18.39(iv), §18.39(iv), §18.40(ii).

Encodings:

BibTeX

…

28: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

§28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

… ►

28.24.2 $${\epsilon }_s{\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}\frac{A_{2\mathrm{\ell }}^{2m}(h^2)}{A_{2s}^{2m}(h^2)}\left(J_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }+s}^{(j)}(h{\mathrm{e}}^z)+J_{\mathrm{\ell }+s}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }-s}^{(j)}(h{\mathrm{e}}^z)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{e}$: base of natural logarithm, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, ${\epsilon }_s$: constants and $A_m(q)$: Fourier coefficient

A&S Ref:

20.6.7

Permalink:

http://dlmf.nist.gov/28.24.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

►

28.24.3 $${\mathrm{Mc}}_{2m+1}^{(j)}(z,h)={(-1)}^m\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}\frac{A_{2\mathrm{\ell }+1}^{2m+1}(h^2)}{A_{2s+1}^{2m+1}(h^2)}\left(J_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }+s+1}^{(j)}(h{\mathrm{e}}^z)+J_{\mathrm{\ell }+s+1}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }-s}^{(j)}(h{\mathrm{e}}^z)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{e}$: base of natural logarithm, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

A&S Ref:

20.6.8

Permalink:

http://dlmf.nist.gov/28.24.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

►

28.24.4 $${\mathrm{Ms}}_{2m+1}^{(j)}(z,h)={(-1)}^m\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}\frac{B_{2\mathrm{\ell }+1}^{2m+1}(h^2)}{B_{2s+1}^{2m+1}(h^2)}\left(J_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }+s+1}^{(j)}(h{\mathrm{e}}^z)-J_{\mathrm{\ell }+s+1}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }-s}^{(j)}(h{\mathrm{e}}^z)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{e}$: base of natural logarithm, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

A&S Ref:

20.6.9

Permalink:

http://dlmf.nist.gov/28.24.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

… ►For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).

29: 33.16 Connection Formulas

… ►

33.16.1 $$F_{\mathrm{\ell }}(\eta ,\rho )=\frac{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{{(-2\eta )}^{\mathrm{\ell }+1}}f(1/{\eta }^2,\mathrm{\ell };-\eta \rho ),$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $!$: factorial (as in $n!$), $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Referenced by:

§33.18

Permalink:

http://dlmf.nist.gov/33.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.16(i), §33.16 and Ch.33

►

33.16.2 $$G_{\mathrm{\ell }}(\eta ,\rho )=\frac{\pi {(-2\eta )}^{\mathrm{\ell }}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}h(1/{\eta }^2,\mathrm{\ell };-\eta \rho ),$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Referenced by:

§33.18

Permalink:

http://dlmf.nist.gov/33.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.16(i), §33.16 and Ch.33

… ►

33.16.4 $$f(ϵ,\mathrm{\ell };r)={\left(\frac{2}{\pi \tau }\frac{1-{\mathrm{e}}^{-2\pi /\tau }}{A(ϵ,\mathrm{\ell })}\right)}^{1/2}{F}_{\mathrm{\ell }}(-1/\tau ,\tau r),$$

ⓘ

Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function and $\tau$: parameter

Referenced by:

§33.16(iv), item 1

Permalink:

http://dlmf.nist.gov/33.16.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.16(ii), §33.16 and Ch.33

►

33.16.5 $$h(ϵ,\mathrm{\ell };r)={\left(\frac{2}{\pi \tau }\frac{A(ϵ,\mathrm{\ell })}{1-{\mathrm{e}}^{-2\pi /\tau }}\right)}^{1/2}{G}_{\mathrm{\ell }}(-1/\tau ,\tau r).$$

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Symbols:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function and $\tau$: parameter

Permalink:

http://dlmf.nist.gov/33.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §33.16(ii), §33.16 and Ch.33

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33.16.6 $$f(ϵ,\mathrm{\ell };r)={(-1)}^{\mathrm{\ell }+1}{\left(\frac{2}{\pi \tau }\frac{{\mathrm{e}}^{2\pi /\tau }-1}{A(ϵ,\mathrm{\ell })}\right)}^{1/2}{F}_{\mathrm{\ell }}(1/\tau ,-\tau r),$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function and $\tau$: parameter

Permalink:

http://dlmf.nist.gov/33.16.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.16(ii), §33.16 and Ch.33

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30: Bibliography V

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A Fortran computer program for calculating the oblate spheroidal radial functions of the first and second kind and their first derivatives. NRL Report No. 6959 Naval Res. Lab.  Washingtion, D.C..

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Referenced by:

§30.18(iii).

Encodings:

BibTeX

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Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives. Quart. Appl. Math. 60 (3), pp. 589–599.

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Notes:

For software, see Van Buren (website)

External Links:

ISSN 0033-569X, MathReview (Alan M. Cohen), ZentralBlatt

Referenced by:

§30.16(iii).

Encodings:

BibTeX

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Improved calculation of prolate spheroidal radial functions of the second kind and their first derivatives. Quart. Appl. Math. 62 (3), pp. 493–507.

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Notes:

For software, see Van Buren (website)

External Links:

ISSN 0033-569X, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§30.16(iii), §30.16(iii).

Encodings:

BibTeX

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