The FunctionAdvisor(branch_points, math_function) command returns the branch points of the function, if any, or the string "No branch points". If the requested information is not available, the FunctionAdvisor command returns NULL.

A branch point of an analytic function is a point in the complex plane which can be mapped to different values of the argument of the function. Also, the function is not analytic in the neighborhood of a branch point, so these points are not isolated singularities.

For example, the origin is a branch point for the square root function because this function will change sign along any path that encircles the origin.  To see this, write the square root function in polar notation $\sqrt{z}=\sqrt{r}{\&ExponentialE;}^{\frac{\mathrm{I}}{2}\mathrm{\theta }}$ where $r=\left|z\right|$ and the angle $\mathrm{\theta }$ lies in $0\le \mathrm{\theta }<2\mathrm{\pi }$ . At $\mathrm{\theta }=0$, $\sqrt{z}=\sqrt{r}$. As $\mathrm{\theta }$ approaches $2\mathrm{\pi }$, the same point is approached, but now, $\sqrt{z}\to \sqrt{r}{\&ExponentialE;}^{\mathrm{\pi }I}=-\sqrt{r}$ .  Note that this example does not define the branch cut of the square root function as the positive real axis.  The branch cut of the square root function is the negative real axis because the range of the argument function is $\left(-\mathrm{\pi },\mathrm{\pi }\right]$ .

In general, branch points are related to the presence of branch cuts and the related multivaluedness of the function in the complex plane. Branch points frequently signal the endpoints of a branch cut line or segment.

For the isolated singularities of a mathematical function, see singularities. For computing the singular points of an expression, see singular.

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_points}\&comma;\arcsin \right)$

$\left[\arcsin \left(z\right)\&comma;z\in \left[\mathrm{-1}\&comma;1\&comma;\mathrm{\infty }+\mathrm{\infty }\mathrm{I}\right]\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\arcsin \right)$

$\left[\arcsin \left(z\right)\&comma;z\le \mathrm{-1}\vee 1\le z\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_points}\&comma;\mathrm{BesselK}\right)$

$\left[\mathrm{BesselK}\left(a\&comma;z\right)\&comma;z\in \left[0\&comma;\mathrm{\infty }+\mathrm{\infty }\mathrm{I}\right]\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_points}\&comma;\exp \right)$

$\left[{\&ExponentialE;}^z\&comma;''No\; branch\; points''\right]$