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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 DirectedInfinity[a].
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 (37)

Numerical Evaluation (10)

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 DirectedInfinity[a]:

Compute the elementwise values of an array:

Or compute the matrix HeunG function using MatrixFunction:

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 :

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