MathieuCharacteristicExponent

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

r(a,q) is the characteristic exponent for Mathieu functions (e^(i r(a,q) z))f(z) with 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(a,q) z)) f(z), where f(z) has period 2π, with characteristic value a and parameter q.

Specific values

Specialized values

Values at fixed points

General characteristics

Domain and analyticity

r(a,q) is an analytical function of a and q which is defined over (ℂ^2).

Symmetries and periodicities

Parity

Mirror symmetry

Periodicity

No periodicity

Branch points

Branch points locations: complicated

Branch cuts

Branch cut locations: complicated

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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