Mathieu functions

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1: 28.12 Definitions and Basic Properties

§28.12 Definitions and Basic Properties

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§28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu }(z,q)$

… ►They have the following pseudoperiodic and orthogonality properties: … ► … ►

2: 28.2 Definitions and Basic Properties

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§28.2(ii) Basic Solutions $w_{\text{I}}$, $w_{\text{II}}$

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§28.2(iv) Floquet Solutions

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§28.2(vi) Eigenfunctions

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3: 28.20 Definitions and Basic Properties

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§28.20(ii) Solutions ${\mathrm{Ce}}_{\nu }$, ${\mathrm{Se}}_{\nu }$, ${\mathrm{Me}}_{\nu }$, ${\mathrm{Fe}}_n$, ${\mathrm{Ge}}_n$

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§28.20(iii) Solutions ${\mathrm{M}}_{\nu }^{(j)}$

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§28.20(iv) Radial Mathieu Functions ${\mathrm{Mc}}_n^{(j)}$, ${\mathrm{Ms}}_n^{(j)}$

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§28.20(vi) Wronskians

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§28.20(vii) Shift of Variable

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4: 28.27 Addition Theorems

§28.27 Addition Theorems

►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. …

5: 28.33 Physical Applications

§28.33 Physical Applications

… ►Mathieu functions occur in practical applications in two main categories: … ►

§28.33(ii) Boundary-Value Problems

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§28.33(iii) Stability and Initial-Value Problems

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Torres-Vega et al. (1998) for Mathieu functions in phase space.

6: Gerhard Wolf

… ►Wolf has published papers on Mathieu functions, orthogonal polynomials, and Heun functions. His book Mathieu Functions and Spheroidal Functions and Their Mathematical Foundations: Further Studies (with J. … ►

28 Mathieu Functions and Hill’s Equation

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7: 28 Mathieu Functions and Hill’s Equation

Chapter 28 Mathieu Functions and Hill’s Equation

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8: 28.22 Connection Formulas

§28.22 Connection Formulas

… ►The joining factors in the above formulas are given by … ►

$${\mathrm{ge}}_m(0,h^2)=\frac{1}{2}\pi S_m(h^2){\left(g_{o,m}(h)\right)}^2{\mathrm{se}}_m^{\prime }(0,h^2).$$

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28.22.13 $${\mathrm{M}}_{\nu }^{(1)}(z,h)=\frac{{\mathrm{M}}_{\nu }^{(1)}(0,h)}{{\mathrm{me}}_{\nu }(0,h^2)}{\mathrm{Me}}_{\nu }(z,h^2).$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, ${\mathrm{Me}}_{\nu }(z,q)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.22.E13

Encodings:

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See also:

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

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9: 28.11 Expansions in Series of Mathieu Functions

§28.11 Expansions in Series of Mathieu Functions

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28.11.1 $$f(z)={\alpha }_0{\mathrm{ce}}_0(z,q)+\sum _{n=1}^{\mathrm{\infty }}\left({\alpha }_n{\mathrm{ce}}_n(z,q)+{\beta }_n{\mathrm{se}}_n(z,q)\right),$$

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Defines:

$f(z)$: function (locally)

Symbols:

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

Referenced by:

§28.11

Permalink:

http://dlmf.nist.gov/28.11.E1

Encodings:

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See also:

Annotations for §28.11 and Ch.28

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

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

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

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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

10: 28.21 Graphics

§28.21 Graphics

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Radial Mathieu Functions: Surfaces

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Figure 28.21.6: ${\mathrm{Ms}}_1^{(2)}(x,h)$ for $0.2\le h\le 2$, $0\le x\le 2$. Magnify 3D Help