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HeunC
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
open allclose allBasic Examples (3)Summary of the most common use cases
https://wolfram.com/xid/0cg50zvk1-3kfjq
Plot the HeunC function:
https://wolfram.com/xid/0cg50zvk1-ftt82q
Series expansion of HeunC:
https://wolfram.com/xid/0cg50zvk1-z8evs7
Scope (26)Survey of the scope of standard use cases
Numerical Evaluation (8)
https://wolfram.com/xid/0cg50zvk1-uqurky
The precision of the output tracks the precision of the input:
https://wolfram.com/xid/0cg50zvk1-lw9h0n
HeunC can take one or more complex number parameters:
https://wolfram.com/xid/0cg50zvk1-64g5bd
https://wolfram.com/xid/0cg50zvk1-ft5oo1
HeunC can take complex number arguments:
https://wolfram.com/xid/0cg50zvk1-hunut5
Finally, HeunC can take all complex number input:
https://wolfram.com/xid/0cg50zvk1-56m4mo
Evaluate HeunC efficiently at high precision:
https://wolfram.com/xid/0cg50zvk1-2c7v5i
https://wolfram.com/xid/0cg50zvk1-yaawua
https://wolfram.com/xid/0cg50zvk1-22a9kq
https://wolfram.com/xid/0cg50zvk1-1knfqv
https://wolfram.com/xid/0cg50zvk1-7yjugj
Evaluate HeunC for points at branch cut to :
https://wolfram.com/xid/0cg50zvk1-vush9d
Specific Values (3)
Value of HeunC at origin:
https://wolfram.com/xid/0cg50zvk1-nuboa
Value of HeunC at regular singular point is indeterminate:
https://wolfram.com/xid/0cg50zvk1-124w4g
Values of HeunC in "logarithmic" cases, i.e. for nonpositive integer , are not determined:
https://wolfram.com/xid/0cg50zvk1-kqbmlj
https://wolfram.com/xid/0cg50zvk1-jzxssp
https://wolfram.com/xid/0cg50zvk1-i7w0vo
Visualization (5)
Plot the HeunC function:
https://wolfram.com/xid/0cg50zvk1-n742f
Plot the absolute value of the HeunC function for complex parameters:
https://wolfram.com/xid/0cg50zvk1-35sv9o
Plot HeunC as a function of its second parameter :
https://wolfram.com/xid/0cg50zvk1-vhxvag
Plot HeunC as a function of and :
https://wolfram.com/xid/0cg50zvk1-lpckuj
https://wolfram.com/xid/0cg50zvk1-8282mz
Plot the family of HeunC functions for different accessory parameter :
https://wolfram.com/xid/0cg50zvk1-0i98nd
https://wolfram.com/xid/0cg50zvk1-dnzkk3
Function Properties (1)
HeunC can be simplified to Hypergeometric1F1 function in the following case:
https://wolfram.com/xid/0cg50zvk1-ekd48u
Differentiation (2)
The -derivative of HeunC is HeunCPrime:
https://wolfram.com/xid/0cg50zvk1-6eb2k6
Higher derivatives of HeunC are calculated using HeunCPrime:
https://wolfram.com/xid/0cg50zvk1-baz2gr
Integration (3)
Indefinite integrals of HeunC are not expressed in elementary or other special functions:
https://wolfram.com/xid/0cg50zvk1-ecaem6
Definite numerical integral of HeunC:
https://wolfram.com/xid/0cg50zvk1-3rkya0
More integrals with HeunC:
https://wolfram.com/xid/0cg50zvk1-gjk5w4
https://wolfram.com/xid/0cg50zvk1-q3siwd
Series Expansions (4)
Taylor expansion for HeunC at regular singular origin:
https://wolfram.com/xid/0cg50zvk1-dux5ad
Coefficient of the first term in the series expansion of HeunC at :
https://wolfram.com/xid/0cg50zvk1-9rxgh1
Plot the first three approximations for HeunC around :
https://wolfram.com/xid/0cg50zvk1-ua3x3f
https://wolfram.com/xid/0cg50zvk1-yorvz
https://wolfram.com/xid/0cg50zvk1-hrtnwe
Series expansion for HeunC at any ordinary complex point:
https://wolfram.com/xid/0cg50zvk1-ukhgue
Applications (4)Sample problems that can be solved with this function
Solve the confluent Heun differential equation using DSolve:
https://wolfram.com/xid/0cg50zvk1-8yj5vx
https://wolfram.com/xid/0cg50zvk1-k9x9uv
https://wolfram.com/xid/0cg50zvk1-tnehim
Solve the initial value problem for the confluent Heun differential equation:
https://wolfram.com/xid/0cg50zvk1-z52b2
https://wolfram.com/xid/0cg50zvk1-r5dzq7
Plot the solution for different values of the accessory parameter q:
https://wolfram.com/xid/0cg50zvk1-y6b6ka
https://wolfram.com/xid/0cg50zvk1-cu90mb
Directly solve the confluent Heun differential equation:
https://wolfram.com/xid/0cg50zvk1-1sgwqv
https://wolfram.com/xid/0cg50zvk1-0ef07s
HeunC with specific parameters solves the Mathieu equation:
https://wolfram.com/xid/0cg50zvk1-38qf80
https://wolfram.com/xid/0cg50zvk1-q4vu8n
Construct the general solution of the Mathieu equation in terms of HeunC functions:
https://wolfram.com/xid/0cg50zvk1-77hx4t
https://wolfram.com/xid/0cg50zvk1-ooyx9j
Properties & Relations (3)Properties of the function, and connections to other functions
HeunC is analytic at the origin:
https://wolfram.com/xid/0cg50zvk1-usyc66
is a singular point of the HeunC function:
https://wolfram.com/xid/0cg50zvk1-zysugy
Except for this singular point, HeunC can be calculated at any finite complex :
https://wolfram.com/xid/0cg50zvk1-txs34a
The derivative of HeunC is HeunCPrime:
https://wolfram.com/xid/0cg50zvk1-eqbaum
Possible Issues (1)Common pitfalls and unexpected behavior
HeunC cannot be evaluated if is a nonpositive integer (so-called logarithmic cases):
https://wolfram.com/xid/0cg50zvk1-8vqyfj
https://wolfram.com/xid/0cg50zvk1-uuaubk
https://wolfram.com/xid/0cg50zvk1-ru2szc
Neat Examples (2)Surprising or curious use cases
Create a table of some special cases for HeunC :
https://wolfram.com/xid/0cg50zvk1-chq0g5
https://wolfram.com/xid/0cg50zvk1-2lhh82
Solve the spheroidal wave equation in its general form in terms of HeunC:
https://wolfram.com/xid/0cg50zvk1-3knr6b
Plot the absolute value of the general solution for different values of λ:
https://wolfram.com/xid/0cg50zvk1-z90lw6
https://wolfram.com/xid/0cg50zvk1-b53akh
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.
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.
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
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
]}
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
]}