# HeunC

HeunC[q,α,γ,δ,ϵ,z]

gives the confluent Heun function.

# 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

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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(4)

Solve the confluent Heun differential equation using DSolve:

Plot the solution:

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

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

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: