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Initialization: Load the package and set the display of special functions in output to typeset mathematical notation (textbook notation):
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Consider the HeunGPrime function
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The singularities of are
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How are these singularities computed? By first computing the linear ODE behind the function, then computing the ODE's singularities:
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| (5) |
So a recurrence around the origin would have for radius of convergence
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The singularities behind the general case of AppellF4:
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In the output above we see, for instance, that when , at least one of the singularities disappears. Let's check that
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| (9) |
So the whole set of singularities collapsed. The AppellF2 function has less complicated singularities
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but the situation at 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,
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| (12) |