gives the confluent Heun function.


  • 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.


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Basic Examples  (3)

Evaluate numerically:

Plot the HeunC function:

Series expansion of HeunC:

Scope  (26)

Numerical Evaluation  (8)

Evaluate to high precision:

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

HeunC can take one or more complex number parameters:

HeunC can take complex number arguments:

Finally, HeunC can take all complex number input:

Evaluate HeunC efficiently at high precision:

Lists and matrices:

Evaluate HeunC for points at branch cut to :

Specific Values  (3)

Value of HeunC at origin:

Value of HeunC at regular singular point is indeterminate:

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

Visualization  (5)

Plot the HeunC function:

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

Plot HeunC as a function of its second parameter :

Plot HeunC as a function of and :

Plot the family of HeunC functions for different accessory parameter :

Function Properties  (1)

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

Differentiation  (2)

The -derivative of HeunC is HeunCPrime:

Higher derivatives of HeunC are calculated using HeunCPrime:

Integration  (3)

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

Definite numerical integral of HeunC:

More integrals with HeunC:

Series Expansions  (4)

Taylor expansion for HeunC at regular singular origin:

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

Plot the first three approximations for HeunC around :

Series expansion for HeunC at any ordinary complex point:

Applications  (3)

Solve the confluent Heun differential equation using DSolve:

Plot the solution:

Directly solve the confluent Heun differential equation:

HeunC with specific parameters solves the Mathieu equation:

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

Properties & Relations  (3)

HeunC is analytic at the origin:

is a singular point of the HeunC function:

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

The derivative of HeunC is HeunCPrime:

Possible Issues  (1)

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

Neat Examples  (2)

Create a table of some special cases for HeunC :

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

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

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.


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. Retrieved from https://reference.wolfram.com/language/ref/HeunC.html


@misc{reference.wolfram_2022_heunc, author="Wolfram Research", title="{HeunC}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HeunC.html}", note=[Accessed: 03-December-2022 ]}


@online{reference.wolfram_2022_heunc, organization={Wolfram Research}, title={HeunC}, year={2020}, url={https://reference.wolfram.com/language/ref/HeunC.html}, note=[Accessed: 03-December-2022 ]}