MathieuCharacteristicA

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

(Subscript[a, r])(q) is the characteristic value a for even Mathieu functions w(z)==Ce(a,q,z) with characteristic exponent r and parameter q, such that there exists a solution of the corresponding Mathieu differential equation w''(z)+(a-2 q cos(2 z)) w(z)==0 that is of the form w(z)==(e^(i r z)) f(z), where f(z) is an even function of z with period 2 π.

Specific values

Specialized values

Values at fixed points

General characteristics

Domain and analyticity

(Subscript[a, r])(q) is an analytical function of r, q which is defined in (ℂ^2).

Symmetries and periodicities

Parity

(Subscript[a, r])(q) is an even function.

Mirror symmetry

Periodicity

No periodicity

Branch points

Branch points locations: complicated

Branch cuts

Branch cut locations: complicated

Series representations

Generalized power series

Expansions at q==0

Representations through more general functions

Through other functions

Representations through equivalent functions

With related functions

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)

Copyright