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The 192 Solutions of the Heun Equation

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A compilation of the 192 solutions to the Heun differential equation

Details

A nested Association containing the 192 solutions of the Heun differential equation, keyed by the expansion point of the corresponding Fuchs-Frobenius solution, characteristic exponent, and corresponding signed disjoint cycle permutation of the singular points at 0, 1, a, and ∞, corresponding to an appropriate Möbius transformation of the arguments.
The solutions are expressed in terms of HeunG, and are stored as pure functions with inactivated contents.

(4 elements)

Examples

Basic Examples (6) 

In[1]:=
Heun192Solutions = ResourceData[\!\(\* TagBox["\"\<The 192 Solutions of the Heun Equation\>\"", #& , BoxID -> "ResourceTag-The 192 Solutions of the Heun Equation-Input", AutoDelete->True]\)];

List the 24 Fuchs-Frobenius solutions corresponding to the expansion point z=0, with characteristic exponent 1-γ:

In[2]:=
sols20 = Activate[ Through[Values[Heun192Solutions[0][1 - \[FormalGamma]]][a, q, \[Alpha], \[Beta], \[Gamma], \[Delta], z]]]
Out[2]=
In[3]:=
Length[sols20]
Out[3]=

Verify that they are all solutions to the Heun ODE:

In[4]:=
heunODE = w''[z] + (\[Gamma]/z + \[Delta]/( z - 1) + (\[Alpha] + \[Beta] + 1 - \[Gamma] - \[Delta])/( z - a)) w'[z] + (\[Alpha] \[Beta] z - q)/(z (z - 1) (z - a)) w[z];
In[5]:=
heunODE /. Map[{w -> Function @@ {z, #}} &, sols20] // Simplify
Out[5]=

All 24 solutions are numerically equivalent, away from the branch cuts:

In[6]:=
sols20 /. Thread[{a, q, \[Alpha], \[Beta], \[Gamma], \[Delta], z} -> {4.3 + 0.1 I, -0.2, 1.3, 0.12, -0.14, 4.32, 0.12}]
Out[6]=

The solutions are no longer numerically equivalent for arguments lying on a branch cut:

In[7]:=
sols20 /. Thread[{a, q, \[Alpha], \[Beta], \[Gamma], \[Delta], z} -> {4.3 + 0.1 I, -0.2, 1.3, 0.12, -0.14, 4.32, Sign[4.3 + 0.1 I] 6}]
Out[7]=

Use DSolveValue to return a general solution to the Heun equation:

In[8]:=
DSolveValue[heunODE == 0, w[z], z]
Out[8]=

Use two of the Fuchs-Frobenius solutions at z=0 to produce the same general solution:

In[9]:=
{C[1], C[2]} . Activate[Through[{Heun192Solutions[0][0][[1]], Heun192Solutions[0][1 - \[FormalGamma]][[1]]}[a, q, \[Alpha], \[Beta], \[Gamma], \[Delta], z]]]
Out[9]=

Visualizations (2) 

Show one of the solutions at infinity:

In[10]:=
sinf = Heun192Solutions[\[Infinity]][\[FormalBeta]][[1]][a, q, \[Alpha], \[Beta], \[Gamma], \[Delta], z] // Activate
Out[10]=

Plot this solution for a particular choice of parameters:

In[11]:=
Plot[Evaluate[ sinf /. Thread[{a, q, \[Alpha], \[Beta], \[Gamma], \[Delta]} -> {4.3, -0.2, 1.3, 0.12, -0.14, 4.32}]], {z, 10, 50}]
Out[11]=

Wolfram Research, "The 192 Solutions of the Heun Equation" from the Wolfram Data Repository (2022)  

Data Resource History

  • Date Created:

Source Metadata

  • Title: The 192 Solutions of the Heun Equation
  • Creator: Robert S. Maier
  • Publisher: Mathematics of Computation
  • Date: 28 November 2006
  • Language: English
  • Citation:
    • Maier R.S., "The 192 solutions of the Heun equation." Mathematics of Computation, vol. 76, no. 258, 811-843, 2007.
      DOI: 10.1090/S0025-5718-06-01939-9

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Publisher Information

  • Prepared for the Wolfram Data Repository By: Jan Mangaldan
  • Publisher of Record: Wolfram Research