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HeunC[q,α,γ,δ,ϵ,z]

gives the confluent Heun function.

Details

  • HeunC belongs to the Heun class of functions and occurs in quantum mechanics, mathematical physics and applications.
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • HeunC[q,α,γ,δ,ϵ,z] satisfies the confluent Heun differential equation .
  • The HeunC function is the regular solution of the confluent Heun equation that satisfies the condition HeunC[q,α,γ,δ,ϵ,0]=1.
  • HeunC has a branch cut discontinuity in the complex plane running from to .
  • For certain special arguments, HeunC automatically evaluates to exact values.
  • HeunC can be evaluated for arbitrary complex parameters.
  • HeunC can be evaluated to arbitrary numerical precision.
  • HeunC automatically threads over lists.

Examples

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Basic Examples  (3)Summary of the most common use cases

Evaluate numerically:

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Plot the HeunC function:

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Series expansion of HeunC:

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Scope  (26)Survey of the scope of standard use cases

Numerical Evaluation  (8)

Evaluate to high precision:

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

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HeunC can take one or more complex number parameters:

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HeunC can take complex number arguments:

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Finally, HeunC can take all complex number input:

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Evaluate HeunC efficiently at high precision:

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Lists and matrices:

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Evaluate HeunC for points at branch cut to :

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Specific Values  (3)

Value of HeunC at origin:

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Value of HeunC at regular singular point is indeterminate:

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

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Visualization  (5)

Plot the HeunC function:

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

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Plot HeunC as a function of its second parameter :

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Plot HeunC as a function of and :

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Plot the family of HeunC functions for different accessory parameter :

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Function Properties  (1)

HeunC can be simplified to Hypergeometric1F1 function in the following case:

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Differentiation  (2)

The -derivative of HeunC is HeunCPrime:

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Higher derivatives of HeunC are calculated using HeunCPrime:

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Integration  (3)

Indefinite integrals of HeunC are not expressed in elementary or other special functions:

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Definite numerical integral of HeunC:

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More integrals with HeunC:

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Series Expansions  (4)

Taylor expansion for HeunC at regular singular origin:

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

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Plot the first three approximations for HeunC around :

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Series expansion for HeunC at any ordinary complex point:

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Applications  (4)Sample problems that can be solved with this function

Solve the confluent Heun differential equation using DSolve:

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Plot the solution:

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Solve the initial value problem for the confluent Heun differential equation:

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Plot the solution for different values of the accessory parameter q:

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Directly solve the confluent Heun differential equation:

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HeunC with specific parameters solves the Mathieu equation:

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Construct the general solution of the Mathieu equation in terms of HeunC functions:

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Properties & Relations  (3)Properties of the function, and connections to other functions

HeunC is analytic at the origin:

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is a singular point of the HeunC function:

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

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The derivative of HeunC is HeunCPrime:

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Possible Issues  (1)Common pitfalls and unexpected behavior

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

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Neat Examples  (2)Surprising or curious use cases

Create a table of some special cases for HeunC :

Solve the spheroidal wave equation in its general form in terms of HeunC:

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Plot the absolute value of the general solution for different values of λ:

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

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2024_heunc, author="Wolfram Research", title="{HeunC}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HeunC.html}", note=[Accessed: 07-January-2025 ]}

Copy to clipboard.
@misc{reference.wolfram_2024_heunc, author="Wolfram Research", title="{HeunC}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HeunC.html}", note=[Accessed: 07-January-2025 ]}

BibLaTeX

@online{reference.wolfram_2024_heunc, organization={Wolfram Research}, title={HeunC}, year={2020}, url={https://reference.wolfram.com/language/ref/HeunC.html}, note=[Accessed: 07-January-2025 ]}

Copy to clipboard.
@online{reference.wolfram_2024_heunc, organization={Wolfram Research}, title={HeunC}, year={2020}, url={https://reference.wolfram.com/language/ref/HeunC.html}, note=[Accessed: 07-January-2025 ]}