MathieuSPrime

MathieuSPrime[a,q,z]

gives the derivative with respect to z of the odd Mathieu function with characteristic value a and parameter q.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For certain special arguments, MathieuSPrime automatically evaluates to exact values.
  • MathieuSPrime can be evaluated to arbitrary numerical precision.
  • MathieuSPrime 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 at the origin:

Scope  (19)

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 MathieuSPrime efficiently at high precision:

MathieuSPrime threads elementwise over lists:

Compute the elementwise values of an array:

Or compute the matrix MathieuSPrime function using MatrixFunction:

Specific Values  (3)

Simple exact values are generated automatically:

Find a zero of MathieuSPrime:

MathieuSPrime is an even function:

Visualization  (2)

Plot the MathieuSPrime function:

Plot the real part of MathieuSPrime for and :

Plot the imaginary part of MathieuSPrime for and :

Function Properties  (4)

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

is neither nondecreasing nor nonincreasing:

MathieuSPrime is neither non-negative nor non-positive:

MathieuSPrime is neither convex nor concave:

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for and :

Plot higher derivatives for and :

MathieuSPrime is the derivative of MathieuS:

Series Expansions  (2)

Taylor expansion:

Plot the first three approximations for MathieuSPrime around :

Taylor expansion of MathieuSPrime at a generic point:

Applications  (1)

Mathieu functions arise as solutions of the Laplace equation in an ellipse:

This defines the square of the gradient (the local kinetic energy of a vibrating membrane):

This finds a zero:

This plots the absolute value of the gradient of an eigenfunction:

Neat Examples  (1)

Phase space plots of the Mathieu function:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_mathieusprime, author="Wolfram Research", title="{MathieuSPrime}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/MathieuSPrime.html}", note=[Accessed: 25-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_mathieusprime, organization={Wolfram Research}, title={MathieuSPrime}, year={1996}, url={https://reference.wolfram.com/language/ref/MathieuSPrime.html}, note=[Accessed: 25-May-2025 ]}