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):

$\mathrm{with}\left(\mathrm{MathematicalFunctions}:-\mathrm{Evalf}\right)\;\mathrm{Typesetting}:-\mathrm{EnableTypesetRule}\left(\mathrm{Typesetting}:-\mathrm{SpecialFunctionRules}\right)\:$

$\left\{\mathrm{Add}\,\mathrm{Evalb}\,\mathrm{Zoom}\,\mathrm{QuadrantNumbers}\,\mathrm{Singularities}\,\mathrm{GenerateRecurrence}\,\mathrm{PairwiseSummation}\right\}$

$\mathrm{HG}≔\mathrm{FunctionAdvisor}\left(\mathrm{syntax}\,\mathrm{HeunGPrime}\right)$

$\mathrm{HG}≔\mathrm{HG}\prime \left(a\,q\,\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,z\right)$

$\mathrm{Singularities}\left(\mathrm{HG}\right)$

$\left[0.\,a\,\frac{q}{\mathrm{\alpha }\mathrm{\beta }}\,1.\right]$

$\mathrm{PDEtools}:-\mathrm{dpolyform}\left(f\left(z\right) \= \mathrm{HeunGPrime}\left(a\, q\, \mathrm{alpha}\, \mathrm{beta}\, \mathrm{gamma}\, \mathrm{delta}\, z\right)\, \mathrm{no\_Fn}\right)$

$\left[\frac{{\ⅆ}^2}{\ⅆz^2}f\left(z\right)=\frac{\left(-\mathrm{\beta }\mathrm{\alpha }\left(\mathrm{\beta }+\mathrm{\alpha }+3\right)z^3+\left(\mathrm{\beta }{\mathrm{\alpha }}^2+\left({\mathrm{\beta }}^2+\left(\left(\mathrm{\delta }+\mathrm{\gamma }+1\right)a-\mathrm{\delta }+2\right)\mathrm{\beta }+q\right)\mathrm{\alpha }+q\left(\mathrm{\beta }+4\right)\right)z^2+\left(\left(-a\mathrm{\beta }\mathrm{\gamma }-q\right)\mathrm{\alpha }-\left(\mathrm{\beta }+\left(\mathrm{\delta }+\mathrm{\gamma }+2\right)a-\mathrm{\delta }+3\right)q\right)z+aq\left(\mathrm{\gamma }+1\right)\right)\left(\frac{\ⅆ}{\ⅆz}f\left(z\right)\right)}{z\left(-\mathrm{\alpha }\mathrm{\beta }z+q\right)\left(z-1\right)\left(-z+a\right)}+\frac{\left(-q^2+\left(2\left(\mathrm{\alpha }+1\right)\left(\mathrm{\beta }+1\right)z-\mathrm{\alpha }-\mathrm{\beta }+\left(-\mathrm{\delta }-\mathrm{\gamma }\right)a+\mathrm{\delta }-1\right)q+\mathrm{\alpha }\left(-\left(\mathrm{\alpha }+1\right)\left(\mathrm{\beta }+1\right)z^2+\mathrm{\gamma }a\right)\mathrm{\beta }\right)f\left(z\right)}{z\left(-\mathrm{\alpha }\mathrm{\beta }z+q\right)\left(z-1\right)\left(-z+a\right)}\right]\&where\left[f\left(z\right)\ne 0\right]$

$\mathrm{DEtools}:-\mathrm{singularities}\left(\mathrm{op}\left(\left[1\,1\right]\,\right)\right)$

$\mathrm{regular}=\left\{0\,1\,a\,\mathrm{\infty }\,\frac{q}{\mathrm{\alpha }\mathrm{\beta }}\right\},\mathrm{irregular}=\varnothing$

$\mathrm{radius\_of\_convergence}≔\min \left(\mathrm{map}\left(\mathrm{abs}\,\mathrm{remove}\left(\mathrm{`=`}\,\,0\right)\right)\right)$

$\mathrm{radius\_of\_convergence}≔\min \left(1.\,\left|a\right|\,\left|\frac{q}{\mathrm{\alpha }\mathrm{\beta }}\right|\right)$

$\mathrm{F4}≔\mathrm{FunctionAdvisor}\left(\mathrm{syntax}\,\mathrm{AppellF4}\right)$

$\mathrm{F4}≔F_4\left(a\,b\,\mathrm{c\_\_1}\,\mathrm{c\_\_2}\,\mathrm{z\_\_1}\,\mathrm{z\_\_2}\right)$

$\mathrm{Singularities}\left(\mathrm{F4}\right)$

$\left[0\,\frac{\left(\mathrm{z\_\_1}-1\right)\left(a+b-\mathrm{c\_\_1}+1\right)\left(a+b-\mathrm{c\_\_1}-2\mathrm{c\_\_2}+3\right)}{\left(\mathrm{c\_\_1}-1-b+a\right)\left(-\mathrm{c\_\_1}+1-b+a\right)}\,\mathrm{z\_\_1}+1-2\sqrt{\mathrm{z\_\_1}}\,\mathrm{z\_\_1}+1+2\sqrt{\mathrm{z\_\_1}}\,\mathrm{\infty }+\mathrm{\infty }\mathrm{I}\right]$

$\mathrm{Singularities}\left(\mathrm{AppellF4}\left(a\,b\,\mathrm{c\_\_1}\,\mathrm{c\_\_2}\,1\,\mathrm{z\_\_2}\right)\right)$

$\left[0\,4\,\mathrm{\infty }+\mathrm{\infty }\mathrm{I}\right]$

$\mathrm{F2}≔\mathrm{FunctionAdvisor}\left(\mathrm{syntax}\,\mathrm{AppellF2}\right)$

$\mathrm{F2}≔F_2\left(a\,\mathrm{b\_\_1}\,\mathrm{b\_\_2}\,\mathrm{c\_\_1}\,\mathrm{c\_\_2}\,\mathrm{z\_\_1}\,\mathrm{z\_\_2}\right)$

$\mathrm{Singularities}\left(\mathrm{F2}\right)$

$\left[0\,1-\mathrm{z\_\_1}\,1\,\mathrm{\infty }+\mathrm{\infty }\mathrm{I}\right]$

$\mathrm{Singularities}\left(\mathrm{AppellF2}\left(a\,\mathrm{b\_\_1}\,\mathrm{b\_\_2}\,\mathrm{c\_\_1}\,\mathrm{c\_\_2}\,1\,\mathrm{z\_\_2}\right)\right)$

$\left[0\,1\,\mathrm{\infty }+\mathrm{\infty }\mathrm{I}\right]$

The MathematicalFunctions[Evalf][Singularities] command was introduced in Maple 2017.

For more information on Maple 2017 changes, see Updates in Maple 2017.