return the singularities of the linear ODE satisfied by a given Appell or Heun function
any of the 10 Heun or 4 Appell functions.
The Singularities command accepts one of the Heun or Appell functions and returns the singularities of the linear ODE behind the given function. In doing so, the last argument - say z - is considered a symbol, the independent variable of the linear ODE behind the function, regardless of its value in the given F.
The location of these singularities is relevant for the numerical evaluation of the function or mathematical expression: any series solution around an expansion point (the origin or a regular singularity) has for radius of convergence the distance between the expansion point and the singularity closest to that expansion point.
The Singularities command is complementary to the GenerateRecurrence command in that the singularity closest to the origin indicates the radius of convergence of the recurrence returned by GenerateRecurrence.
Initialization: Load the package and set the display of special functions in output to typeset mathematical notation (textbook notation):
Consider the HeunGPrime function
The singularities of HG are
How are these singularities computed? By first computing the linear ODE behind the function, then computing the ODE's singularities:
PDEtools:-dpolyformfz = HeunGPrimea, q, alpha, beta, gamma, delta, z, no_Fn
So a recurrence around the origin would have for radius of convergence
The singularities behind the general case of AppellF4:
In the output above we see, for instance, that when z1=1, at least one of the singularities disappears. Let's check that
So the whole set of singularities collapsed. The AppellF2 function has less complicated singularities
but the situation at z1=1 is similar, only one finite singularity beyond the origin, though in this case equal to 1, as is the case of all the 10 Heun functions,
The MathematicalFunctions[Evalf][Singularities] command was introduced in Maple 2017.
For more information on Maple 2017 changes, see Updates in Maple 2017.
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