gives the double-confluent Heun function.


  • HeunD 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.
  • HeunD[q,α,γ,δ,ϵ,z] satisfies the double-confluent Heun differential equation .
  • The HeunD function is the power-series solution of the double-confluent Heun equation that satisfies the conditions and .
  • For certain special arguments, HeunD automatically evaluates to exact values.
  • HeunD can be evaluated for arbitrary complex parameters.
  • HeunD can be evaluated to arbitrary numerical precision.
  • HeunD automatically threads over lists.


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Basic Examples  (3)

Evaluate numerically:

Plot the double-confluent Heun function:

Series expansion of HeunD:

Scope  (24)

Numerical Evaluation  (8)

Evaluate to high precision:

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

HeunD can take one or more complex number parameters:

HeunD can take complex number arguments:

Finally, HeunD can take all complex number input:

Evaluate HeunD efficiently at high precision:

Lists and matrices:

Evaluate HeunD for points on the real negative axis, bypassing irregular singular origin:

Specific Values  (2)

Value of HeunD at unit point:

Value of HeunD at irregular singular origin is undetermined:

Visualization  (5)

Plot the HeunD function:

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

Plot HeunD as a function of its second parameter :

Plot HeunD as a function of and :

Plot the family of HeunD functions for different accessory parameter :

Differentiation  (2)

The -derivative of HeunD is HeunDPrime:

Higher derivatives of HeunD are calculated using HeunDPrime:

Integration  (3)

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

Definite numerical integral of HeunD:

More integrals with HeunD:

Series Expansions  (4)

Taylor expansion for HeunD at regular point :

Coefficient of the second term in the series expansion of HeunD at :

Plot the first three approximations for HeunD around :

Series expansion for HeunD at any ordinary complex point:

Applications  (3)

Solve the double-confluent Heun differential equation using DSolve:

Plot the solution:

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

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

Directly solve the double-confluent Heun differential equation:

Properties & Relations  (3)

HeunD is analytic at the point :

Origin is a singular point of the HeunD function:

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

The derivative of HeunD is HeunDPrime:

Possible Issues  (1)

HeunD diverges for big arguments:

Introduced in 2020