integer parameters

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1: 14.34 Software

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Adams and Swarztrauber (1997). Integer parameters. Fortran.

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Braithwaite (1973). Integer parameters. Fortran.

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Delic (1979a). Integer parameters. Fortran.

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Gil and Segura (1998). Integer parameters. Fortran.

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Lozier and Smith (1981). Integer parameters. Fortran.

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2: 28.11 Expansions in Series of Mathieu Functions

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28.11.3 $$1=2\sum _{n=0}^{\mathrm{\infty }}{A}_0^{2n}(q){\mathrm{ce}}_{2n}(z,q),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

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28.11.4 $$\cos 2mz=\sum _{n=0}^{\mathrm{\infty }}{A}_{2m}^{2n}(q){\mathrm{ce}}_{2n}(z,q),$$ $m\ne 0$,

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

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28.11.5 $$\cos (2m+1)z=\sum _{n=0}^{\mathrm{\infty }}{A}_{2m+1}^{2n+1}(q){\mathrm{ce}}_{2n+1}(z,q),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

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28.11.6 $$\sin (2m+1)z=\sum _{n=0}^{\mathrm{\infty }}{B}_{2m+1}^{2n+1}(q){\mathrm{se}}_{2n+1}(z,q),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

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28.11.7 $$\sin (2m+2)z=\sum _{n=0}^{\mathrm{\infty }}{B}_{2m+2}^{2n+2}(q){\mathrm{se}}_{2n+2}(z,q).$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

3: 28.23 Expansions in Series of Bessel Functions

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28.23.6 $${\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m}(0,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}{A}_{2\mathrm{\ell }}^{2m}(h^2){𝒞}_{2\mathrm{\ell }}^{(j)}(2h\cosh z),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: 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:

20.6.12

Referenced by:

§28.23

Permalink:

http://dlmf.nist.gov/28.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.7 $${\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m}(\frac{1}{2}\pi ,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{A}_{2\mathrm{\ell }}^{2m}(h^2){𝒞}_{2\mathrm{\ell }}^{(j)}(2h\sinh z),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

►

28.23.8 $${\mathrm{Mc}}_{2m+1}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m+1}(0,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}{A}_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\cosh z),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.9 $${\mathrm{Mc}}_{2m+1}^{(j)}(z,h)={(-1)}^{m+1}{\left({\mathrm{ce}}_{2m+1}^{\prime }(\frac{1}{2}\pi ,h^2)\right)}^{-1}\coth z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}(2\mathrm{\ell }+1)A_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\sinh z),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.10 $${\mathrm{Ms}}_{2m+1}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{se}}_{2m+1}^{\prime }(0,h^2)\right)}^{-1}\tanh z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}(2\mathrm{\ell }+1)B_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\cosh z),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

…

4: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

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28.19.2 $$f(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{f}_n{\mathrm{me}}_{\nu +2n}(z,q),$$

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Defines:

$f(z)$: function (locally)

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $\nu$: complex parameter and $f_n$: coefficients

Referenced by:

§28.19

Permalink:

http://dlmf.nist.gov/28.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.19 and Ch.28

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28.19.3 $$f_n=\frac{1}{\pi }{\int }_0^{\pi }f(z){\mathrm{me}}_{\nu +2n}(-z,q)dz.$$

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Defines:

$f_n$: coefficients (locally)

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, $n$: integer, $z$: complex variable, $\nu$: complex parameter and $f(z)$: function

Permalink:

http://dlmf.nist.gov/28.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.19 and Ch.28

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28.19.4 $${\mathrm{e}}^{\mathrm{i}\nu z}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{-2n}^{\nu +2n}(q){\mathrm{me}}_{\nu +2n}(z,q),$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/28.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.19, §28.19 and Ch.28

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5: 28.4 Fourier Series

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28.4.9 $$2{\left(A_0^{2n}(q)\right)}^2+\sum _{m=1}^{\mathrm{\infty }}{\left(A_{2m}^{2n}(q)\right)}^2=1,$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $n$: integer and $A_m(q)$: Fourier coefficient

A&S Ref:

20.5.4

Referenced by:

item (d)

Permalink:

http://dlmf.nist.gov/28.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(iii), §28.4 and Ch.28

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28.4.10 $$\sum _{m=0}^{\mathrm{\infty }}{\left(A_{2m+1}^{2n+1}(q)\right)}^2=1,$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $n$: integer and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(iii), §28.4 and Ch.28

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28.4.11 $$\sum _{m=0}^{\mathrm{\infty }}{\left(B_{2m+1}^{2n+1}(q)\right)}^2=1,$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $n$: integer and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(iii), §28.4 and Ch.28

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28.4.12 $$\sum _{m=0}^{\mathrm{\infty }}{\left(B_{2m+2}^{2n+2}(q)\right)}^2=1.$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $n$: integer and $B_m(q)$: Fourier coefficient

Referenced by:

item (d)

Permalink:

http://dlmf.nist.gov/28.4.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(iii), §28.4 and Ch.28

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28.4.17 $$A_{2m}^{2n}(-q)={(-1)}^{n-m}{A}_{2m}^{2n}(q),$$

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Symbols:

$m$: integer, $q=h^2$: parameter, $n$: integer and $A_m(q)$: Fourier coefficient

A&S Ref:

20.8.5

Permalink:

http://dlmf.nist.gov/28.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(v), §28.4 and Ch.28

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6: 28.28 Integrals, Integral Representations, and Integral Equations

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28.28.20 $$\frac{2}{\pi }{\int }_0^{\pi }{𝒞}_{2\mathrm{\ell }}^{(j)}(2hR)\cos \left(2\mathrm{\ell }\varphi \right){\mathrm{ce}}_{2m}(t,h^2)dt={\epsilon }_{\mathrm{\ell }}{(-1)}^{\mathrm{\ell }+m}{A}_{2\mathrm{\ell }}^{2m}(h^2){\mathrm{Mc}}_{2m}^{(j)}(z,h),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function, ${\epsilon }_{\mathrm{\ell }}$: constants and $A_m(q)$: Fourier coefficient

A&S Ref:

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

Permalink:

http://dlmf.nist.gov/28.28.E20

Encodings:

pMML, png, TeX

See also:

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

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28.28.36 $$\frac{\sinh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\cos t{\mathrm{se}}_n(t,h^2){\mathrm{se}}_m(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{p+1}\mathrm{i}h{\widehat{\alpha }}_{n,m}^{(s)}{\mathrm{Ds}}_0(n,m,z),$$

ⓘ

Symbols:

${\mathrm{Ds}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $n$: integer, $z$: complex variable and ${\widehat{\alpha }}_{n,m}$

Permalink:

http://dlmf.nist.gov/28.28.E36

Encodings:

pMML, png, TeX

See also:

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

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28.28.37 $$\frac{\cosh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\sin t{\mathrm{se}}_n^{\prime }(t,h^2){\mathrm{se}}_m(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{p+1}\mathrm{i}h{\widehat{\alpha }}_{n,m}^{(s)}{\mathrm{Ds}}_1(n,m,z),$$

ⓘ

Symbols:

${\mathrm{Ds}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $n$: integer, $z$: complex variable and ${\widehat{\alpha }}_{n,m}$

Permalink:

http://dlmf.nist.gov/28.28.E37

Encodings:

pMML, png, TeX

See also:

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

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28.28.38 $${\widehat{\alpha }}_{n,m}^{(s)}=\frac{1}{2\pi }{\int }_0^{2\pi }\cos t{\mathrm{se}}_n(t,h^2){\mathrm{se}}_m(t,h^2)dt={(-1)}^p\frac{2}{\mathrm{i}\pi }\frac{{\mathrm{se}}_n^{\prime }(0,h^2){\mathrm{se}}_m^{\prime }(0,h^2)}{h{\mathrm{Ds}}_2(n,m,0)}.$$

ⓘ

Symbols:

${\mathrm{Ds}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{se}}_n(z,q)$: 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, $n$: integer and ${\widehat{\alpha }}_{n,m}$

Permalink:

http://dlmf.nist.gov/28.28.E38

Encodings:

pMML, png, TeX

See also:

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

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28.28.41 $$\frac{\cosh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\sin t{\mathrm{se}}_n(t,h^2){\mathrm{ce}}_m(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{p+1}\mathrm{i}h{\widehat{\beta }}_{n,m}{\mathrm{Dsc}}_0(n,m,z),$$

ⓘ

Symbols:

${\mathrm{Dsc}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $n$: integer, $z$: complex variable and ${\widehat{\beta }}_{n,m}$

Permalink:

http://dlmf.nist.gov/28.28.E41

Encodings:

pMML, png, TeX

See also:

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

…

7: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

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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

►

28.24.5 $${\mathrm{Ms}}_{2m+2}^{(j)}(z,h)={(-1)}^m\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}\frac{B_{2\mathrm{\ell }+2}^{2m+2}(h^2)}{B_{2s+2}^{2m+2}(h^2)}\left(J_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }+s+2}^{(j)}(h{\mathrm{e}}^z)-J_{\mathrm{\ell }+s+2}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }-s}^{(j)}(h{\mathrm{e}}^z)\right),$$

ⓘ

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:

20.6.10

Permalink:

http://dlmf.nist.gov/28.24.E5

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).

8: 8.28 Software

… ►

§8.28(vi) Generalized Exponential Integral for Real Argument and Integer Parameter

…

9: 31.14 General Fuchsian Equation

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

ⓘ

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

ⓘ

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

►

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

ⓘ

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

…

10: 28.22 Connection Formulas

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28.22.1 $${\mathrm{Mc}}_m^{(1)}(z,h)=\sqrt{\frac{2}{\pi }}\frac{1}{g_{e,m}(h){\mathrm{ce}}_m(0,h^2)}{\mathrm{Ce}}_m(z,h^2),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors

A&S Ref:

20.6.15 (in different form)

Permalink:

http://dlmf.nist.gov/28.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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28.22.2 $${\mathrm{Ms}}_m^{(1)}(z,h)=\sqrt{\frac{2}{\pi }}\frac{1}{g_{o,m}(h){\mathrm{se}}_m^{\prime }(0,h^2)}{\mathrm{Se}}_m(z,h^2),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, ${\mathrm{Se}}_{\nu }(z,q)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors

Permalink:

http://dlmf.nist.gov/28.22.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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28.22.5 $$g_{e,2m}(h)={(-1)}^m\sqrt{\frac{2}{\pi }}\frac{{\mathrm{ce}}_{2m}(\frac{1}{2}\pi ,h^2)}{A_0^{2m}(h^2)},$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $h$: parameter, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.22.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

►

28.22.6 $$g_{e,2m+1}(h)={(-1)}^{m+1}\sqrt{\frac{2}{\pi }}\frac{{\mathrm{ce}}_{2m+1}^{\prime }(\frac{1}{2}\pi ,h^2)}{h{A}_1^{2m+1}(h^2)},$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $h$: parameter, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.22.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

►

28.22.7 $$g_{o,2m+1}(h)={(-1)}^m\sqrt{\frac{2}{\pi }}\frac{{\mathrm{se}}_{2m+1}(\frac{1}{2}\pi ,h^2)}{h{B}_1^{2m+1}(h^2)},$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $h$: parameter, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.22.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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