Mathieu equation

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21: 28.14 Fourier Series

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28.14.4 $$q{c}_{2m+2}-\left(a-{(\nu +2m)}^2\right)c_{2m}+q{c}_{2m-2}=0,$$ $a={\lambda }_{\nu }\left(q\right),c_{2m}=c_{2m}^{\nu }(q)$,

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $m$: integer, $q=h^2$: parameter, $\nu$: complex parameter, $a$: parameter and $c_{2m}(q)$: coefficients

Referenced by:

item (d), item (d)

Permalink:

http://dlmf.nist.gov/28.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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22: 28.29 Definitions and Basic Properties

… ►A generalization of Mathieu’s equation (28.2.1) is Hill’s equation ►

28.29.1 $$w^{\prime \prime }(z)+\left(\lambda +Q(z)\right)w=0,$$

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Symbols:

$z$: complex variable, $w(z)$: Mathieu’s equation solution, $\lambda$: parameter and $Q(z)$: function

Referenced by:

§28.29(ii), §28.29(ii), §28.29(ii), §28.29(iii), §28.29(iii), §28.29(iii)

Permalink:

http://dlmf.nist.gov/28.29.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(i), §28.29 and Ch.28

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28.29.7 $$w(z+\pi )={\mathrm{e}}^{\pi \mathrm{i}\nu }w(z),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\nu$: complex parameter and $w(z)$: Mathieu’s equation solution

Referenced by:

§28.29(ii)

Permalink:

http://dlmf.nist.gov/28.29.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(ii), §28.29 and Ch.28

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28.29.11 $$w(z+\pi )={(-1)}^{\nu }w(z)+cP(z),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $\nu$: complex parameter, $w(z)$: Mathieu’s equation solution and $c$: constant

Permalink:

http://dlmf.nist.gov/28.29.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(ii), §28.29 and Ch.28

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28.29.13 $$w(z+\pi )+w(z-\pi )=2\cos \left(\pi \nu \right)w(z).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $z$: complex variable, $\nu$: complex parameter and $w(z)$: Mathieu’s equation solution

Permalink:

http://dlmf.nist.gov/28.29.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(ii), §28.29 and Ch.28

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23: 28.32 Mathematical Applications

… ►The separated solutions $V(\xi ,\eta )=v(\xi )w(\eta )$ can be obtained from the modified Mathieu’s equation (28.20.1) for $v$ and from Mathieu’s equation (28.2.1) for $w$, where $a$ is the separation constant and $q=\frac{1}{4}{c}^2{k}^2$. … ►This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting $\zeta =\mathrm{i}\xi$, $z=\eta$ in (28.32.3)). … ►Let $u(\zeta )$ be a solution of Mathieu’s equation (28.2.1) and $K(z,\zeta )$ be a solution of …defines a solution of Mathieu’s equation, provided that (in the case of an improper curve) the integral converges with respect to $z$ uniformly on compact subsets of $ℂ$. … ►

24: 28.22 Connection Formulas

§28.22 Connection Formulas

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

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:

pMML, png, TeX

See also:

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

►Here ${\mathrm{me}}_{\nu }(0,h^2)$ $(\ne 0)$ is given by (28.14.1) with $z=0$, and ${\mathrm{M}}_{\nu }^{(1)}(0,h)$ is given by (28.24.1) with $j=1$, $z=0$, and $n$ chosen so that $|c_{2n}^{\nu }(h^2)|=\max (|c_{2\mathrm{\ell }}^{\nu }(h^2)|)$, where the maximum is taken over all integers $\mathrm{\ell }$. … ►

25: 28.35 Tables

§28.35 Tables

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Blanch and Clemm (1969) includes eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$ for $q=\rho {\mathrm{e}}^{\mathrm{i}\varphi }$, $\rho =0(.5)25$, $\varphi =5^{\circ }(5^{\circ }){90}^{\circ }$, $n=0(1)15$; 4D. Also $a_n\left(q\right)$ and $b_n\left(q\right)$ for $q=\mathrm{i}\rho$, $\rho =0(.5)100$, $n=0(2)14$ and $n=2(2)16$, respectively; 8D. Double points for $n=0(1)15$; 8D. Graphs are included.

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26: Simon Ruijsenaars

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28 Mathieu Functions and Hill’s Equation

27: 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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28: 28.28 Integrals, Integral Representations, and Integral Equations

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§28.28(i) Equations with Elementary Kernels

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28.28.1 $$w=\cosh z\cos t\cos \alpha +\sinh z\sin t\sin \alpha .$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\sin z$: sine function, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter

Permalink:

http://dlmf.nist.gov/28.28.E1

Encodings:

pMML, png, TeX

See also:

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

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28.28.2 $$\frac{1}{2\pi }{\int }_0^{2\pi }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{ce}}_n(t,h^2)dt={\mathrm{i}}^n{\mathrm{ce}}_n(\alpha ,h^2){\mathrm{Mc}}_n^{(1)}(z,h),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter

A&S Ref:

20.7.34 20.7.35 20.7.36 20.7.37

Permalink:

http://dlmf.nist.gov/28.28.E2

Encodings:

pMML, png, TeX

See also:

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

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28.28.3 $$\frac{1}{2\pi }{\int }_0^{2\pi }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{se}}_n(t,h^2)dt={\mathrm{i}}^n{\mathrm{se}}_n(\alpha ,h^2){\mathrm{Ms}}_n^{(1)}(z,h),$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter

Permalink:

http://dlmf.nist.gov/28.28.E3

Encodings:

pMML, png, TeX

See also:

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

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28.28.16 $${\int }_0^{\mathrm{\infty }}\sin \left(2h\cos y\cosh t\right){\mathrm{Ce}}_{2n}(t,h^2)dt=-\frac{\pi A_0^{2n}(h^2)}{2{\mathrm{ce}}_{2n}(\frac{1}{2}\pi ,h^2)}\left({\mathrm{ce}}_{2n}(y,h^2)\mp \frac{2}{\pi C_{2n}(h^2)}{\mathrm{fe}}_{2n}(y,h^2)\right),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, $\sin z$: sine function, $h$: parameter, $n$: integer, $y$: real variable, $A_m(q)$: Fourier coefficient and $C_n(q)$: factor

Permalink:

http://dlmf.nist.gov/28.28.E16

Encodings:

pMML, png, TeX

See also:

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

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29: Bibliography L

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The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.

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External Links:

ISSN 0002-9947, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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Algorithm 537: Characteristic values of Mathieu’s differential equation. ACM Trans. Math. Software 5 (1), pp. 112–117.

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External Links:

ISSN 0098-3500, Document, GAMS (module 537; package TOMS)

Referenced by:

2nd item, 2nd item.

Encodings:

BibTeX

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30: Gerhard Wolf

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28 Mathieu Functions and Hill’s Equation

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