modified Mathieu functions

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11: 28.26 Asymptotic Approximations for Large $q$

§28.26 Asymptotic Approximations for Large $q$

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28.26.1 $${\mathrm{Mc}}_m^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fc}}_m(z,h)-\mathrm{i}{\mathrm{Gc}}_m(z,h)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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28.26.2 $$\mathrm{i}{\mathrm{Ms}}_{m+1}^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fs}}_m(z,h)-\mathrm{i}{\mathrm{Gs}}_m(z,h)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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§28.26(ii) Uniform Approximations

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12: 28.36 Software

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§28.36(iii) Mathieu and Modified Mathieu Functions

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13: 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

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28.24.11 $${\mathrm{Ko}}_{2m+2}(z,h)=\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}\frac{B_{2\mathrm{\ell }+2}^{2m+2}(h^2)}{B_{2s+2}^{2m+2}(h^2)}\left(I_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right)K_{\mathrm{\ell }+s+2}\left(h{\mathrm{e}}^z\right)-I_{\mathrm{\ell }+s+2}\left(h{\mathrm{e}}^{-z}\right)K_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^z\right)\right),$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, ${\mathrm{Ko}}_n(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.24.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

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28.24.12 $${\mathrm{Ke}}_{2m+1}(z,h)=\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}\frac{B_{2\mathrm{\ell }+1}^{2m+1}(h^2)}{B_{2s+1}^{2m+1}(h^2)}\left(I_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right)K_{\mathrm{\ell }+s+1}\left(h{\mathrm{e}}^z\right)-I_{\mathrm{\ell }+s+1}\left(h{\mathrm{e}}^{-z}\right)K_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^z\right)\right),$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, ${\mathrm{Ke}}_n(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.24.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

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28.24.13 $${\mathrm{Ko}}_{2m+1}(z,h)=\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}\frac{A_{2\mathrm{\ell }+1}^{2m+1}(h^2)}{A_{2s+1}^{2m+1}(h^2)}\left(I_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right)K_{\mathrm{\ell }+s+1}\left(h{\mathrm{e}}^z\right)+I_{\mathrm{\ell }+s+1}\left(h{\mathrm{e}}^{-z}\right)K_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^z\right)\right).$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, ${\mathrm{Ko}}_n(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $A_m(q)$: Fourier coefficient

Referenced by:

§28.24, item (a)

Permalink:

http://dlmf.nist.gov/28.24.E13

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).

14: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

§28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

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28.25.1 $${\mathrm{M}}_{\nu }^{(3,4)}(z,h)\sim \frac{{\mathrm{e}}^{\pm \mathrm{i}\left(2h\cosh z-\left(\frac{1}{2}\nu +\frac{1}{4}\right)\pi \right)}}{{\left(\pi h(\cosh z+1)\right)}^{\frac{1}{2}}}\sum _{m=0}^{\mathrm{\infty }}\frac{D_m^{\pm }}{{\left(\mp 4\mathrm{i}h(\cosh z+1)\right)}^m},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.1 (for 3 and for integer $\nu$) 20.9.3 (for 4 and for integer $\nu$)

Referenced by:

§28.25

Permalink:

http://dlmf.nist.gov/28.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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15: 28.32 Mathematical Applications

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16: 28.8 Asymptotic Expansions for Large $q$

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Barrett’s Expansions

… ►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions ${\mathrm{me}}_{\nu }(z,q)$ (§28.12(ii)) and modified Mathieu functions ${\mathrm{M}}_{\nu }^{(j)}(z,h)$ (§28.20(iii)). …

17: Bibliography V

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Accurate calculation of the modified Mathieu functions of integer order. Quart. Appl. Math. 65 (1), pp. 1–23.

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Notes:

For software, see Van Buren (website)

External Links:

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

Referenced by:

§28.36(iii).

Encodings:

BibTeX

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18: Bibliography S

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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Uniform asymptotic expansions of modified Mathieu functions. J. Reine Angew. Math. 247, pp. 1–17.

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External Links:

MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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19: Bibliography K

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Tables of values of the modified Mathieu functions. Math. Comp. 14, pp. 118–129.

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External Links:

FullText, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

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20: Bibliography B

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Asymptotic expansions of the modified Bessel function of the third kind of imaginary order. SIAM J. Appl. Math. 15, pp. 1315–1323.

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External Links:

ISSN 0036-1399, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.45.

Encodings:

BibTeX

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Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.

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External Links:

ISSN 0080-4614, FullText, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

1st item.

Encodings:

BibTeX

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Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

2nd item, 1st item.

Encodings:

BibTeX

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A note on Mathieu functions. Proc. Nederl. Akad. Wetensch. 51 (7), pp. 891–893=Indagationes Math. 10, 319–321 (1948).

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External Links:

MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§28.7.

Encodings:

BibTeX

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