# HeunCPrime HeunCPrime[q,α,γ,δ,ϵ,z]

gives the -derivative of the HeunC function.

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• HeunCPrime belongs to the Heun class of functions.
• For certain special arguments, HeunCPrime automatically evaluates to exact values.
• HeunCPrime can be evaluated for arbitrary complex parameters.
• HeunCPrime can be evaluated to arbitrary numerical precision.
• HeunCPrime automatically threads over lists.

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot the HeunCPrime function:

Series expansion of HeunCPrime:

## Scope(24)

### Numerical Evaluation(8)

Evaluate to high precision:

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

HeunCPrime can take one or more complex number parameters:

HeunCPrime can take complex number arguments:

Finally, HeunCPrime can take all complex number input:

Evaluate HeunCPrime efficiently at high precision:

Lists and matrices:

Evaluate HeunCPrime for points at branch cut to :

### Specific Values(3)

Value of HeunCPrime at the origin:

Value of HeunCPrime at regular singular point is indeterminate:

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

### Visualization(5)

Plot the HeunCPrime function:

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

Plot HeunCPrime as a function of its second parameter :

Plot HeunCPrime as a function of and :

Plot the family of HeunCPrime functions for different accessory parameter :

### Differentiation(1)

The derivatives of HeunCPrime are calculated using the HeunC function:

### Integration(3)

Integral of HeunCPrime gives back HeunC:

Definite numerical integral of HeunCPrime:

More integrals with HeunCPrime:

### Series Expansions(4)

Taylor expansion for HeunCPrime at regular singular origin:

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

Plots of the first three approximations for HeunCPrime around :

Series expansion for HeunCPrime at any ordinary complex point:

## Applications(1)

Use the HeunCPrime function to calculate the derivatives of HeunC:

## Properties & Relations(3)

HeunCPrime is analytic at the origin: is a singular point of the HeunCPrime function:

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

HeunCPrime is the derivative of HeunC:

## Possible Issues(1)

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