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.
Examplesopen allclose all
Basic Examples (3)
Plot the double-confluent Heun function:
Series expansion of HeunD:
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:
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 :
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:
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