MathieuC

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Ce(a,q,z) is the even Mathieu function with characteristic value a and parameter q.  It is defined as the even in z  solution w(z)==Ce(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 Ce(a,0,z)==cos((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[a, r])(q)  with integer or rational numbers r,  which makes even solutions of the form (e^(i r z)) f(z) where f(z) is an even function of z with period 2π).

Specific values

Specialized values

For fixed a , z

General characteristics

Domain and analyticity

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

Symmetries and periodicities

Parity

Ce(a,q,z) is an even 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 more general functions

Through other functions

Involving spheroidal functions

Representations through equivalent functions

With related functions

Theorems

The odd and even solutions for the eigenmodes of an elliptical membrane

The odd and even  solutions for the eigenmodes  ψ(r,ϕ) of an elliptical membrane fixed at the boundary in elliptical coordinates  {x=c cosh(r)cos(ϕ), y=c sinh(r)  sin(ϕ)} with semi-axes a and b are given by

ψ(r,ϕ)∝Ce((Subscript[a, n])((OverTilde[q])),(OverTilde[q]),ϕ)  Ce((Subscript[a, n])((OverTilde[q])),(OverTilde[q]),i r)/;a==c cosh((Subscript[r, 0])),b=c sinh((Subscript[r, 0]))∧Ce((Subscript[a, n])((OverTilde[q])),(OverTilde[q]),i (Subscript[r, 0]))==0∧n∈ℕ,

ψ(r,ϕ)∝Se((Subscript[b, n])((OverTilde[q])),(OverTilde[q]),ϕ)  Se((Subscript[b, n])((OverTilde[q])),(OverTilde[q]),i r)/;a==c cosh((Subscript[r, 0])),b=c sinh((Subscript[r, 0]))∧Ce((Subscript[a, n])((OverTilde[q])),(OverTilde[q]),i (Subscript[r, 0]))==0∧n∈(SuperPlus[ℕ]).

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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