Evaluate numerically to high precision:

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

Evaluate for complex arguments and parameters:

Evaluate MathieuC efficiently at high precision:

MathieuC threads elementwise over lists:

Compute the elementwise values of an array:

Or compute the matrix MathieuC function using MatrixFunction:

Simple exact values are generated automatically:

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

MathieuC is an even function:

Heun functions can be reduced to Mathieu functions:

Plot the MathieuC function:

Plot the real part of MathieuC for and :

Plot the imaginary part of MathieuC for and :

Plot the real part of MathieuC for and :

Plot the imaginary part of MathieuC for and :

MathieuC has singularities and discontinuities when the characteristic exponent is an integer:

is neither nondecreasing nor nonincreasing:

MathieuC is neither non-negative nor non-positive:

MathieuC is neither convex nor concave:

First derivative:

Higher derivatives:

Plot higher derivatives for and :

Plot higher derivatives for and :

Mathieu functions are the solutions to the differential equation :

Taylor expansion:

Plot the first three approximations for MathieuC around :

Taylor expansion of MathieuC at a generic point: