MathieuS

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Se(a,q,z) is the odd Mathieu function with characteristic value a and parameter q.  It is defined as the odd in z  solution w(z)==Se(a,q,z) of the Mathieu differential equation w''(z)+(a-2 q cos(2 z)) w(z)==0, which satisfies for q==0 relation Se(a,0,z)==sin((Sqrt[a]) z). It is analytical function in the variables a, q and z.  It is a periodical only in z function for special values of parameter a (so called characteristic values a==(Subscript[b, r])(q)  with integer or rational numbers r,  which makes odd solutions of the form (e^(i r z)) f(z) where f(z) is an odd function of z with period 2π).

Specific values

Specialized values

For fixed a , z

General characteristics

Domain and analyticity

Se(a,q,z) is an analytical function of a, q, z which is defined in (ℂ^3).

Symmetries and periodicities

Parity

Se(a,q,z) is an odd function with respect to z.

Mirror symmetry

Periodicity

No periodicity

Series representations

Generalized power series

Expansions at generic point z==(Subscript[z, 0])

Expansions at z==0

Expansions at q==0

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Differentiation

Low-order differentiation

With respect to z

Integration

Definite integration

Involving the direct function

Operations

Limit operation

Representations through equivalent functions

With related functions

Other information

History

– E. L. Mathieu (1868, 1873)
– H. Weber (1869)
– G.W. Hill (1877)
– E. Heine (1878)
– G. Floquet (1883)
– R. C. Maclaurin (1898)
– J. Dougall (1916, 1926)

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