MathieuC

MathieuC[a,q,z]

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

Details

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

Examples

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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 about the origin:

Scope  (21)

Numerical Evaluation  (5)

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:

Specific Values  (4)

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:

Visualization  (3)

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 :

Function Properties  (4)

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:

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 MathieuC around :

Taylor expansion of MathieuC at a generic point:

Applications  (3)

This differential equation is solved in terms of MathieuC and MathieuS functions:

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:

Wolfram Research (1996), MathieuC, Wolfram Language function, https://reference.wolfram.com/language/ref/MathieuC.html.

Text

Wolfram Research (1996), MathieuC, Wolfram Language function, https://reference.wolfram.com/language/ref/MathieuC.html.

CMS

Wolfram Language. 1996. "MathieuC." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MathieuC.html.

APA

Wolfram Language. (1996). MathieuC. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MathieuC.html

BibTeX

@misc{reference.wolfram_2024_mathieuc, author="Wolfram Research", title="{MathieuC}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/MathieuC.html}", note=[Accessed: 21-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_mathieuc, organization={Wolfram Research}, title={MathieuC}, year={1996}, url={https://reference.wolfram.com/language/ref/MathieuC.html}, note=[Accessed: 21-December-2024 ]}