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1: 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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Figure 28.2.1: Eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$ of Mathieu’s equation as functions of $q$ for $0\le q\le 10$, $n=0,1,2,3,4$ ($a$’s), $n=1,2,3,4$ ($b$’s). Magnify

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§28.2(vi) Eigenfunctions

… ►For simple roots $q$ of the corresponding equations (28.2.21) and (28.2.22), the functions are made unique by the normalizations …

2: 28.20 Definitions and Basic Properties

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28.20.1 $$w^{\prime \prime }-\left(a-2q\cosh \left(2z\right)\right)w=0,$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

A&S Ref:

20.1.2 (in slightly different form) 20.8.6

Referenced by:

§28.20(iii), §28.32(i), 1st item, 2nd item, item (b), §28.8(iv), §28.8(iv)

Permalink:

http://dlmf.nist.gov/28.20.E1

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Annotations for §28.20(i), §28.20 and Ch.28

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28.20.6 $${\mathrm{Fe}}_n(z,q)=\mp \mathrm{i}{\mathrm{fe}}_n(\pm \mathrm{i}z,q),$$ $n=0,1,\mathrm{\dots }$,

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Defines:

${\mathrm{Fe}}_n(z,q)$: modified Mathieu function

Symbols:

${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $n$: integer and $z$: complex variable

A&S Ref:

20.6.1 (in slightly different form) 20.6.2 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.20.E6

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Annotations for §28.20(ii), §28.20 and Ch.28

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28.20.7 $${\mathrm{Ge}}_n(z,q)={\mathrm{ge}}_n(\pm \mathrm{i}z,q),$$ $n=1,2,\mathrm{\dots }$.

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Defines:

${\mathrm{Ge}}_n(z,q)$: modified Mathieu function

Symbols:

${\mathrm{ge}}_n(z,q)$: second solution, Mathieu’s equation, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $n$: integer and $z$: complex variable

A&S Ref:

20.6.1 (in slightly different form) 20.6.2 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.20.E7

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Annotations for §28.20(ii), §28.20 and Ch.28

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28.20.17 $${\mathrm{Ie}}_n(z,h)={\mathrm{i}}^{-n}{\mathrm{Mc}}_n^{(1)}(z,\mathrm{i}h),$$

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Defines:

${\mathrm{Ie}}_n(z,h)$: modified Mathieu function

Symbols:

$\mathrm{i}$: imaginary unit, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer and $z$: complex variable

A&S Ref:

20.8.10 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.20.E17

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Annotations for §28.20(v), §28.20 and Ch.28

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28.20.18 $${\mathrm{Io}}_n(z,h)={\mathrm{i}}^{-n}{\mathrm{Ms}}_n^{(1)}(z,\mathrm{i}h),$$

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Defines:

${\mathrm{Io}}_n(z,h)$: modified Mathieu function

Symbols:

$\mathrm{i}$: imaginary unit, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer and $z$: complex variable

A&S Ref:

20.8.10 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.20.E18

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Annotations for §28.20(v), §28.20 and Ch.28

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3: 9.16 Physical Applications

… ►A quite different application is made in the study of the diffraction of sound pulses by a circular cylinder (Friedlander (1958)). … ►Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. …An application of Airy functions to the solution of this equation is given in Gramtcheff (1981). … ►Reference to many of these applications as well as to the theory of elasticity and to the heat equation are given in Vallée and Soares (2010): a book devoted specifically to the Airy and Scorer functions and their applications in physics. … ►Solutions of the Schrödinger equation involving the Airy functions are given for other potentials in Vallée and Soares (2010). …

4: 30.10 Series and Integrals

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5: 10.73 Physical Applications

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§10.73(ii) Spherical Bessel Functions

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6: 17.17 Physical Applications

… ►See Kassel (1995). …

7: 21.9 Integrable Equations

§21.9 Integrable Equations

… ►Typical examples of such equations are the Korteweg–de Vries equation …Here, and in what follows, $x,y$, and $t$ suffixes indicate partial derivatives. … ► … ►

8: 28 Mathieu Functions and Hill’s Equation

Chapter 28 Mathieu Functions and Hill’s Equation

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9: Simon Ruijsenaars

… ►His main research interests cover integrable systems, special functions, analytic difference equations, classical and quantum mechanics, and the relations between these areas. … ►

28 Mathieu Functions and Hill’s Equation

10: 10.36 Other Differential Equations

§10.36 Other Differential Equations

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10.36.1 $$z^2(z^2+{\nu }^2)w^{\prime \prime }+z(z^2+3{\nu }^2)w^{\prime }-\left({(z^2+{\nu }^2)}^2+z^2-{\nu }^2\right)w=0,$$ $w={𝒵}_{\nu }^{\prime }\left(z\right)$,

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Symbols:

${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.36.E1

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See also:

Annotations for §10.36 and Ch.10

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10.36.2 $$z^2{w}^{\prime \prime }+z(1\pm 2z)w^{\prime }+(\pm z-{\nu }^2)w=0,$$ $w={\mathrm{e}}^{\mp z}{𝒵}_{\nu }\left(z\right)$.

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Symbols:

$\mathrm{e}$: base of natural logarithm, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.41

Permalink:

http://dlmf.nist.gov/10.36.E2

Encodings:

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See also:

Annotations for §10.36 and Ch.10

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