Singularities - Maple Help

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MathematicalFunctions[Evalf]

 Singularities
 return the singularities of the linear ODE satisfied by a given Appell or Heun function

 Calling Sequence Singularities(F)

Parameters

 F - any of the 10 Heun or 4 Appell functions.

Description

 • 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.

Examples

 Initialization: Load the package and set the display of special functions in output to typeset mathematical notation (textbook notation):
 > $\mathrm{with}\left(\mathrm{MathematicalFunctions}:-\mathrm{Evalf}\right);\mathrm{Typesetting}:-\mathrm{EnableTypesetRule}\left(\mathrm{Typesetting}:-\mathrm{SpecialFunctionRules}\right):$
 $\left\{{\mathrm{Add}}{,}{\mathrm{Evalb}}{,}{\mathrm{Zoom}}{,}{\mathrm{QuadrantNumbers}}{,}{\mathrm{Singularities}}{,}{\mathrm{GenerateRecurrence}}{,}{\mathrm{PairwiseSummation}}\right\}$ (1)

Consider the HeunGPrime function

 > $\mathrm{HG}≔\mathrm{FunctionAdvisor}\left(\mathrm{syntax},\mathrm{HeunGPrime}\right)$
 ${\mathrm{HG}}{≔}{\mathrm{HG}}{\prime }{}\left({a}{,}{q}{,}{\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\gamma }}{,}{\mathrm{\delta }}{,}{z}\right)$ (2)

The singularities of $\mathrm{HG}$ are

 > $\mathrm{Singularities}\left(\mathrm{HG}\right)$
 $\left[{0.}{,}{a}{,}\frac{{q}}{{\mathrm{\alpha }}{}{\mathrm{\beta }}}{,}{1.}\right]$ (3)

How are these singularities computed? By first computing the linear ODE behind the function, then computing the ODE's singularities:

 >
 $\left[\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right){=}\frac{\left({-}{\mathrm{\beta }}{}{\mathrm{\alpha }}{}\left({\mathrm{\beta }}{+}{\mathrm{\alpha }}{+}{3}\right){}{{z}}^{{3}}{+}\left({\mathrm{\beta }}{}{{\mathrm{\alpha }}}^{{2}}{+}\left({{\mathrm{\beta }}}^{{2}}{+}\left(\left({\mathrm{\delta }}{+}{\mathrm{\gamma }}{+}{1}\right){}{a}{-}{\mathrm{\delta }}{+}{2}\right){}{\mathrm{\beta }}{+}{q}\right){}{\mathrm{\alpha }}{+}{q}{}\left({\mathrm{\beta }}{+}{4}\right)\right){}{{z}}^{{2}}{+}\left(\left({-}{a}{}{\mathrm{\beta }}{}{\mathrm{\gamma }}{-}{q}\right){}{\mathrm{\alpha }}{-}\left({\mathrm{\beta }}{+}\left({\mathrm{\delta }}{+}{\mathrm{\gamma }}{+}{2}\right){}{a}{-}{\mathrm{\delta }}{+}{3}\right){}{q}\right){}{z}{+}{a}{}{q}{}\left({\mathrm{\gamma }}{+}{1}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right)\right)}{{z}{}\left({-}{\mathrm{\alpha }}{}{\mathrm{\beta }}{}{z}{+}{q}\right){}\left({z}{-}{1}\right){}\left({-}{z}{+}{a}\right)}{+}\frac{\left({-}{{q}}^{{2}}{+}\left({2}{}\left({\mathrm{\alpha }}{+}{1}\right){}\left({\mathrm{\beta }}{+}{1}\right){}{z}{-}{\mathrm{\alpha }}{-}{\mathrm{\beta }}{+}\left({-}{\mathrm{\delta }}{-}{\mathrm{\gamma }}\right){}{a}{+}{\mathrm{\delta }}{-}{1}\right){}{q}{+}{\mathrm{\alpha }}{}\left({-}\left({\mathrm{\alpha }}{+}{1}\right){}\left({\mathrm{\beta }}{+}{1}\right){}{{z}}^{{2}}{+}{\mathrm{\gamma }}{}{a}\right){}{\mathrm{\beta }}\right){}{f}{}\left({z}\right)}{{z}{}\left({-}{\mathrm{\alpha }}{}{\mathrm{\beta }}{}{z}{+}{q}\right){}\left({z}{-}{1}\right){}\left({-}{z}{+}{a}\right)}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left[{f}{}\left({z}\right){\ne }{0}\right]$ (4)
 > $\mathrm{DEtools}:-\mathrm{singularities}\left(\mathrm{op}\left(\left[1,1\right],\right)\right)$
 ${\mathrm{regular}}{=}\left\{{0}{,}{1}{,}{a}{,}{\mathrm{\infty }}{,}\frac{{q}}{{\mathrm{\alpha }}{}{\mathrm{\beta }}}\right\}{,}{\mathrm{irregular}}{=}{\varnothing }$ (5)

So a recurrence around the origin would have for radius of convergence

 > $\mathrm{radius_of_convergence}≔\mathrm{min}\left(\mathrm{map}\left(\mathrm{abs},\mathrm{remove}\left(\mathrm{=},,0\right)\right)\right)$
 ${\mathrm{radius_of_convergence}}{≔}{\mathrm{min}}{}\left({1.}{,}\left|{a}\right|{,}\left|\frac{{q}}{{\mathrm{\alpha }}{}{\mathrm{\beta }}}\right|\right)$ (6)

The singularities behind the general case of AppellF4:

 > $\mathrm{F4}≔\mathrm{FunctionAdvisor}\left(\mathrm{syntax},\mathrm{AppellF4}\right)$
 ${\mathrm{F4}}{≔}{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ (7)
 > $\mathrm{Singularities}\left(\mathrm{F4}\right)$
 $\left[{0}{,}\frac{\left(\mathrm{z__1}{-}{1}\right){}\left({a}{+}{b}{-}\mathrm{c__1}{+}{1}\right){}\left({a}{+}{b}{-}\mathrm{c__1}{-}{2}{}\mathrm{c__2}{+}{3}\right)}{\left(\mathrm{c__1}{-}{1}{-}{b}{+}{a}\right){}\left({-}\mathrm{c__1}{+}{1}{-}{b}{+}{a}\right)}{,}\mathrm{z__1}{+}{1}{-}{2}{}\sqrt{\mathrm{z__1}}{,}\mathrm{z__1}{+}{1}{+}{2}{}\sqrt{\mathrm{z__1}}{,}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}\right]$ (8)

In the output above we see, for instance, that when ${z}_{1}=1$, at least one of the singularities disappears. Let's check that

 > $\mathrm{Singularities}\left(\mathrm{AppellF4}\left(a,b,\mathrm{c__1},\mathrm{c__2},1,\mathrm{z__2}\right)\right)$
 $\left[{0}{,}{4}{,}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}\right]$ (9)

So the whole set of singularities collapsed. The AppellF2 function has less complicated singularities

 > $\mathrm{F2}≔\mathrm{FunctionAdvisor}\left(\mathrm{syntax},\mathrm{AppellF2}\right)$
 ${\mathrm{F2}}{≔}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ (10)
 > $\mathrm{Singularities}\left(\mathrm{F2}\right)$
 $\left[{0}{,}{1}{-}\mathrm{z__1}{,}{1}{,}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}\right]$ (11)

but the situation at ${z}_{1}=1$ 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,

 > $\mathrm{Singularities}\left(\mathrm{AppellF2}\left(a,\mathrm{b__1},\mathrm{b__2},\mathrm{c__1},\mathrm{c__2},1,\mathrm{z__2}\right)\right)$
 $\left[{0}{,}{1}{,}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}\right]$ (12)

Compatibility

 • The MathematicalFunctions[Evalf][Singularities] command was introduced in Maple 2017.
 • For more information on Maple 2017 changes, see Updates in Maple 2017.

 See Also