singular - Maple Programming Help

singular

find singularities of an expression

 Calling Sequence singular(expr) singular(expr, vars, range)

Parameters

 expr - algebraic expression vars - (optional) variable or set of variables range - (optional) a numeric range, to return only the singularities inside it

Description

 • The function singular outputs an expression sequence representing the singularities  of expr.
 • If two arguments are given and the second argument, vars, is a name or a set of them, the expression, expr, is considered to be a function in vars. If singular is called with only one argument, then expr is considered as a function in the variables returned by the command indets(expr, name).
 • If a numeric range is given as second or third argument, only the singularities found within this range, if any, will be returned.
 • Each singular point is represented by a set of equations, the left-hand side of the equations being the variables.
 • The singular function will return non-removable as well as removable singularities.  For instance, $\frac{x-2}{\mathrm{sin}\left(x-2\right)}$ will report a singularity at $x=2$.
 • The power of singular to find singularities is basically that of solve. For example, some zeros that solve cannot find may result in singularities that singular will not find.
 • The singular function may return expressions prefixed by _Z or _N, representing the integers and positive integers, respectively.

Examples

 > $\mathrm{singular}\left(xy+\frac{1}{xy},x\right)$
 $\left\{{x}{=}{0}\right\}{,}\left\{{x}{=}{\mathrm{\infty }}\right\}{,}\left\{{x}{=}{-}{\mathrm{\infty }}\right\}$ (1)
 > $\mathrm{singular}\left(\frac{\mathrm{ln}\left(x\right)}{{x}^{2}-1}\right)$
 $\left\{{x}{=}{-1}\right\}{,}\left\{{x}{=}{0}\right\}{,}\left\{{x}{=}{1}\right\}$ (2)
 > $\mathrm{singular}\left(\frac{x}{x-y}\right)$
 $\left\{{x}{=}{y}{,}{y}{=}{y}\right\}$ (3)
 > $\mathrm{singular}\left(\mathrm{tan}\left(x\right)\right)$
 $\left\{{x}{=}\frac{{1}}{{2}}{}{\mathrm{\pi }}{+}{\mathrm{_Z1~}}{}{\mathrm{\pi }}\right\}$ (4)
 > $\mathrm{singular}\left(\mathrm{tan}\left(x\right),1..10\right)$
 $\left\{{x}{=}\frac{{\mathrm{\pi }}}{{2}}\right\}{,}\left\{{x}{=}\frac{{3}{}{\mathrm{\pi }}}{{2}}\right\}{,}\left\{{x}{=}\frac{{5}{}{\mathrm{\pi }}}{{2}}\right\}$ (5)

The range must have endpoints that are of numeric type. Otherwise an error is returned.

 > $\mathrm{singular}\left(\mathrm{tan}\left(x\right),1..\mathrm{\pi }\right)$
 > $\mathrm{singular}\left(\mathrm{tan}\left(x\right),1..\mathrm{evalf}\left(\mathrm{\pi }\right)\right)$
 $\left\{{x}{=}\frac{{\mathrm{\pi }}}{{2}}\right\}$ (6)
 > $\mathrm{singular}\left(\mathrm{\Psi }\left(\frac{1}{x}\right)\right)$
 $\left\{{x}{=}{0}\right\}{,}\left\{{x}{=}{\mathrm{\infty }}\right\}{,}\left\{{x}{=}{-}{\mathrm{\infty }}\right\}{,}\left\{{x}{=}{-}\frac{{1}}{{\mathrm{_NN1~}}}\right\}$ (7)