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.
Examplesopen allclose all
Basic Examples (3)
Plot the HeunC function:
Series expansion of HeunC:
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:
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 :
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:
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
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:
Introduced in 2020