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:

Compute the elementwise values of an array:

Or compute the matrix MathieuS function using MatrixFunction:

Simple exact values are generated automatically:

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

MathieuS is an odd function:

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 :

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

is neither nondecreasing nor nonincreasing:

MathieuS is neither non-negative nor non-positive:

MathieuS 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 MathieuS around :

Taylor expansion of MathieuS at a generic point: