HeunCPrime

HeunCPrime[q,α,γ,δ,ϵ,z]

gives the -derivative of the HeunC function.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • HeunCPrime belongs to the Heun class of functions.
  • For certain special arguments, HeunCPrime automatically evaluates to exact values.
  • HeunCPrime can be evaluated for arbitrary complex parameters.
  • HeunCPrime can be evaluated to arbitrary numerical precision.
  • HeunCPrime automatically threads over lists.

Examples

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Basic Examples  (3)

Evaluate numerically:

Plot the HeunCPrime function:

Series expansion of HeunCPrime:

Scope  (25)

Numerical Evaluation  (9)

Evaluate to high precision:

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

HeunCPrime can take one or more complex number parameters:

HeunCPrime can take complex number arguments:

Finally, HeunCPrime can take all complex number input:

Evaluate HeunCPrime efficiently at high precision:

Lists and matrices:

Evaluate HeunCPrime for points at branch cut to :

Compute the elementwise values of an array:

Or compute the matrix HeunCPrime function using MatrixFunction:

Specific Values  (3)

Value of HeunCPrime at the origin:

Value of HeunCPrime at regular singular point is indeterminate:

Values of HeunCPrime in "logarithmic" cases, i.e. for nonpositive integer , are not determined:

Visualization  (5)

Plot the HeunCPrime function:

Plot the absolute value of the HeunCPrime function for complex parameters:

Plot HeunCPrime as a function of its second parameter :

Plot HeunCPrime as a function of and :

Plot the family of HeunCPrime functions for different accessory parameter :

Differentiation  (1)

The derivatives of HeunCPrime are calculated using the HeunC function:

Integration  (3)

Integral of HeunCPrime gives back HeunC:

Definite numerical integral of HeunCPrime:

More integrals with HeunCPrime:

Series Expansions  (4)

Taylor expansion for HeunCPrime at regular singular origin:

Coefficient of the first term in the series expansion of HeunCPrime at :

Plots of the first three approximations for HeunCPrime around :

Series expansion for HeunCPrime at any ordinary complex point:

Applications  (1)

Use the HeunCPrime function to calculate the derivatives of HeunC:

Properties & Relations  (3)

HeunCPrime is analytic at the origin:

is a singular point of the HeunCPrime function:

Except for this singular point, HeunCPrime can be calculated at any finite complex :

HeunCPrime is the derivative of HeunC:

Possible Issues  (1)

HeunCPrime cannot be evaluated if is a nonpositive integer (so-called logarithmic cases):

Wolfram Research (2020), HeunCPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/HeunCPrime.html.

Text

Wolfram Research (2020), HeunCPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/HeunCPrime.html.

CMS

Wolfram Language. 2020. "HeunCPrime." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HeunCPrime.html.

APA

Wolfram Language. (2020). HeunCPrime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HeunCPrime.html

BibTeX

@misc{reference.wolfram_2024_heuncprime, author="Wolfram Research", title="{HeunCPrime}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HeunCPrime.html}", note=[Accessed: 21-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_heuncprime, organization={Wolfram Research}, title={HeunCPrime}, year={2020}, url={https://reference.wolfram.com/language/ref/HeunCPrime.html}, note=[Accessed: 21-December-2024 ]}