gives the odd Mathieu function with characteristic value a and parameter q.


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The Mathieu functions satisfy the equation .
  • For certain special arguments, MathieuS automatically evaluates to exact values.
  • MathieuS can be evaluated to arbitrary numerical precision.
  • MathieuS automatically threads over lists.


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Basic Examples  (4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Scope  (15)

Numerical Evaluation  (4)

Evaluate numerically to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments and parameters:

Evaluate MathieuS efficiently at high precision:

MathieuS threads elementwise over lists:

Specific Values  (3)

Simple exact values are generated automatically:

Find a local maximum as the root of in the maximum's neighborhood:

MathieuS is an odd function:

Visualization  (3)

Plot the MathieuS function:

Plot the real part of MathieuS for and :

Plot the imaginary part of MathieuS for and :

Plot the real part of MathieuS for and :

Plot the imaginary part of MathieuS for and :

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for and :

Plot higher derivatives for and :

Mathieu functions are the solutions to the differential equation :

Series Expansions  (2)

Taylor expansion:

Plot the first three approximations for MathieuS around :

Taylor expansion of MathieuS at a generic point:

Applications  (2)

Solve the Schrödinger equation with periodic potential:

By the Bloch theorem, solutions are bounded provided is within an energy band. The energy gap corresponds to a range of where MathieuCharacteristicExponent has a non-vanishing imaginary part:

Solve the Laplace equation in an ellipse using separation of variables:

This finds a zero:

This plots an eigenfunction. It vanishes at the ellipse boundary:

Possible Issues  (1)

Machine-precision input is insufficient to give a correct answer:

Neat Examples  (1)

Phase space plots of the Mathieu function:

Introduced in 1996