Mathieu functions

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31: 28.4 Fourier Series

§28.4 Fourier Series

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§28.4(ii) Recurrence Relations

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§28.4(iii) Normalization

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§28.4(v) Change of Sign of $q$

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§28.4(vi) Behavior for Small $q$

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32: 28.7 Analytic Continuation of Eigenvalues

… ►As functions of $q$, $a_n\left(q\right)$ and $b_n\left(q\right)$ can be continued analytically in the complex $q$-plane. … ►All the $a_{2n}\left(q\right)$, $n=0,1,2,\mathrm{\dots }$, can be regarded as belonging to a complete analytic function (in the large). … ►

28.7.4 $$\sum _{n=0}^{\mathrm{\infty }}\left(b_{2n+2}\left(q\right)-{(2n+2)}^2\right)=0.$$

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Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.7 and Ch.28

33: 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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34: Bibliography N

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On simplified asymptotic formulas for a class of Mathieu functions. SIAM J. Math. Anal. 15 (6), pp. 1205–1213.

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External Links:

ISSN 0036-1410, Document, MathReview (Harold Exton), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

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On a simplified asymptotic formula for the Mathieu function of the third kind. SIAM J. Math. Anal. 18 (6), pp. 1616–1629.

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External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

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Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview, ZentralBlatt

Referenced by:

§28.1, §28.26(i), 6th item, Notations S, Notations S.

Encodings:

BibTeX

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35: 29.5 Special Cases and Limiting Forms

… ►where ${\mathrm{ce}}_m(z,\theta )$ and ${\mathrm{se}}_m(z,\theta )$ are Mathieu functions; see §28.2(vi).

36: 28.6 Expansions for Small $q$

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§28.6(ii) Functions ${\mathrm{ce}}_n$ and ${\mathrm{se}}_n$

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28.6.21 $$2^{1/2}{\mathrm{ce}}_0(z,q)=1-\frac{1}{2}q\cos 2z+\frac{1}{32}{q}^2\left(\cos 4z-2\right)-\frac{1}{128}{q}^3\left(\frac{1}{9}\cos 6z-11\cos 2z\right)+\mathrm{\cdots },$$

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Symbols:

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

A&S Ref:

20.2.27

Referenced by:

§28.6(ii)

Permalink:

http://dlmf.nist.gov/28.6.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(ii), §28.6 and Ch.28

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28.6.22 $${\mathrm{ce}}_1(z,q)=\cos z-\frac{1}{8}q\cos 3z+\frac{1}{128}{q}^2\left(\frac{2}{3}\cos 5z-2\cos 3z-\cos z\right)-\frac{1}{1024}{q}^3\left(\frac{1}{9}\cos 7z-\frac{8}{9}\cos 5z-\frac{1}{3}\cos 3z+2\cos z\right)+\mathrm{\cdots },$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/28.6.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(ii), §28.6 and Ch.28

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28.6.24 $${\mathrm{ce}}_2(z,q)=\cos 2z-\frac{1}{4}q\left(\frac{1}{3}\cos 4z-1\right)+\frac{1}{128}{q}^2\left(\frac{1}{3}\cos 6z-\frac{76}{9}\cos 2z\right)+\mathrm{\cdots },$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/28.6.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(ii), §28.6 and Ch.28

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28.6.25 $${\mathrm{se}}_2(z,q)=\sin 2z-\frac{1}{12}q\sin 4z+\frac{1}{128}{q}^2\left(\frac{1}{3}\sin 6z-\frac{4}{9}\sin 2z\right)+\mathrm{\cdots }.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/28.6.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(ii), §28.6 and Ch.28

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37: Bibliography E

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Algorithm 934: Fortran 90 subroutines to compute Mathieu functions for complex values of the parameter. ACM Trans. Math. Softw. 40 (1), pp. 8:1–8:19.

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External Links:

ISSN 0098-3500, FullText, Document, MathReview Entry, ZentralBlatt

Referenced by:

4th item, §28.36(iii).

Encodings:

BibTeX

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Algorithm 861: Fortran 90 subroutines for computing the expansion coefficients of Mathieu functions using Blanch’s algorithm. ACM Trans. Math. Software 32 (4), pp. 622–634.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §28.36(iii).

Encodings:

BibTeX

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38: Bibliography Z

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Mathieu functions for purely imaginary parameters. J. Comput. Appl. Math. 236 (17), pp. 4513–4524.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview (Bartolomeu D. B. Figueiredo), ZentralBlatt

Referenced by:

§28.36(iii), §28.36(iii).

Encodings:

BibTeX

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39: 28.31 Equations of Whittaker–Hill and Ince

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28.31.3 $$w^{\prime \prime }+\xi \sin \left(2z\right)w^{\prime }+(\eta -p\xi \cos \left(2z\right))w=0.$$

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Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable, $w(z)$: Mathieu’s equation solution, $\eta$: variable and $\xi$: variable

Referenced by:

§28.31(ii)

Permalink:

http://dlmf.nist.gov/28.31.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.31(ii), §28.31 and Ch.28

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28.31.4 $$w_{e,s}(z)=\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{A}_{2\mathrm{\ell }+s}\cos (2\mathrm{\ell }+s)z,$$ $s=0,1$,

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Symbols:

$\cos z$: cosine function, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $A$: constant

Referenced by:

§28.31(ii)

Permalink:

http://dlmf.nist.gov/28.31.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.31(ii), §28.31 and Ch.28

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28.31.5 $$w_{o,s}(z)=\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{B}_{2\mathrm{\ell }+s}\sin (2\mathrm{\ell }+s)z,$$ $s=1,2$,

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Symbols:

$\sin z$: sine function, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $B$: constant

Referenced by:

§28.31(ii)

Permalink:

http://dlmf.nist.gov/28.31.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.31(ii), §28.31 and Ch.28

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28.31.18 $$w^{\prime \prime }+\left(\eta -\frac{1}{8}{\xi }^2-(p+1)\xi \cos \left(2z\right)+\frac{1}{8}{\xi }^2\cos \left(4z\right)\right)w=0,$$

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Symbols:

$\cos z$: cosine function, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\eta$: change of variable

Permalink:

http://dlmf.nist.gov/28.31.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §28.31(iii), §28.31 and Ch.28

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40: Bibliography F

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Algebraic methods to compute Mathieu functions. J. Phys. A 34 (17), pp. 3541–3551.

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External Links:

ISSN 0305-4470, Document, Link, MathReview (Aleksander Denisiuk), ZentralBlatt

Referenced by:

§28.6(i), §28.6(i), §28.6(ii), §28.6(ii), §28.8(i), §28.8(i), §28.8(ii), §28.8(ii), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii).

Encodings:

BibTeX

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Characteristic values of the integral equation satisfied by the Mathieu functions and its application to a system with chirality-pair interaction on a one-dimensional lattice. Phys. A 190 (3-4), pp. 346–362.

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External Links:

ISSN 0378-4371, Document, MathReview

Referenced by:

6th item.

Encodings:

BibTeX

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