# HeunG

HeunG[a,q,α,β,γ,δ,z]

gives the general Heun function.

# Details • HeunG belongs to the Heun class of functions, directly generalizes the Hypergeometric2F1 function and occurs in quantum mechanics, mathematical physics and applications.
• Mathematical function, suitable for both symbolic and numerical manipulation.
• HeunG[a,q,α,β,γ,δ,z] satisfies the general Heun differential equation .
• The HeunG function is the regular solution of the general Heun equation that satisfies the condition HeunG[a,q,α,β,γ,δ,0]=1.
• HeunG has one branch cut discontinuity in the complex plane running from to and one running from to .
• For certain special arguments, HeunG automatically evaluates to exact values.
• HeunG can be evaluated for arbitrary complex parameters.
• HeunG can be evaluated to arbitrary numerical precision.
• HeunG automatically threads over lists.
• HeunG[a,q,α,β,γ,δ,z] specializes to Hypergeometric2F1[α,β,γ,z] if and or and .

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot the HeunG function:

Series expansion of HeunG:

## Scope(36)

### Numerical Evaluation(9)

Evaluate to high precision:

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

HeunG can take one or more complex number parameters:

HeunG can take complex number arguments:

Finally, HeunG can take all complex number input:

Evaluate HeunG efficiently at high precision:

Lists and matrices:

Evaluate HeunG for points at branch cut to :

Evaluate HeunG for points on a branch cut from to :

### Specific Values(8)

Value of HeunG at origin:

Value of HeunG at the regular singular point is indeterminate:

Value of HeunG at the regular singular point is indeterminate:

Values of HeunG in "logarithmic" cases, for nonpositive integer , are not determined:

Value of HeunG is indeterminate if :

HeunG automatically evaluates to the Hypergeometric2F1 function if and :

HeunG automatically evaluates to the Hypergeometric2F1 function if and :

HeunG automatically evaluates to simpler functions for certain parameters:

### Visualization(5)

Plot the HeunG function:

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

Plot HeunG as a function of its third parameter :

Plot HeunG as a function of and :

Plot the family of HeunG functions for different accessory parameters :

### Function Properties(3)

Hypergeometric2F1 is a special case of HeunG:

HeunG can be simplified to the Hypergeometric2F1 function with nonlinear argument:

HeunG can be simplified to rational functions in special cases:

### Differentiation(4)

The -derivative of HeunG is HeunGPrime:

Higher derivatives of HeunG are calculated using HeunGPrime:

Derivatives of HeunG for specific cases of parameters:

Higher derivatives of HeunG involving specific cases of parameters:

### Integration(3)

Indefinite integrals of HeunG cannot be expressed in elementary or other special functions:

Definite numerical integrals of HeunG:

More integrals with HeunG:

### Series Expansions(4)

Taylor expansion for HeunG at regular singular origin:

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

Plot the first three approximations for HeunG around :

Series expansion for HeunG at any ordinary complex point:

## Applications(5)

Solve the general Heun differential equation using DSolve:

Plot the solution for different initial conditions:

Solve the initial value problem:

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

Solve the Lamé differential equation in terms of HeunG:

Plot the absolute value of the solution for different h:

Stationary 1D Schrödinger equation for this infinite potential well is solved in terms of HeunG:

Plot the potential:

The fundamental solution of the Schrödinger equation in terms of HeunG:

Verify this solution by direct substitution:

The general form of a second-order Fuchsian equation with four regular singularities at and exponent parameters , subject to the constraint :

Construct two linearly independent solutions in terms of HeunG:

Verify that these solutions satisfy the Fuchsian equation:

## Properties & Relations(6)

HeunG is analytic at the origin: and are singular points of the HeunG function:

Except for these two singular points, HeunG can be calculated at any finite complex :

The derivative of HeunG is HeunGPrime:

HeunG is symmetric in the parameters and :

Four equivalent expressions for HeunG, corresponding to parameter transformations that leave the argument and singular point invariant:

Use Series to show that the series expansions of the last three expressions at agree with that of the first:

Six equivalent expressions for HeunG, corresponding to argument transformations that leave the parameters and invariant:

Use Series to show that the series expansions of the last five expressions at agree with that of the first:

## Possible Issues(2)

HeunG is not defined if is a nonpositive integer (so-called logarithmic cases):

HeunG is undefined when :

## Neat Examples(1)

Create a table of some special cases for HeunG :