The FunctionAdvisor(branch_cuts, expression) command returns the branch cuts or segments of discontinuity of a given mathematical expression, if any, or the string "No branch cuts" when there are none. If the information necessary to perform the calculation is not available, FunctionAdvisor returns NULL, depending on the case including informative related messages.

The expression indicated can be just the name of a mathematical function, e.g. arcsin, or a mathematical function call like $\arcsin \left(x\right)$, or a whole mathematical expression, involving compositions between +, *, ^, possibly fractional powers, with every mathematical function call for which FunctionAdvisor knows the branch cuts, with no restrictions to the level of nesting in the expression.

CAVEAT 1: Depending on the mathematical expression, there is a situation where the program cannot guarantee an exhaustive list of branch cuts. This happens, typically, when a non-rational expression needs to be algebraically inverted, and the system can only invert it using RootOf, which means, basically, that the system does not know how many roots are around or exactly where they are located). Example: $\arctan \left(\frac{J_2\left(z\left(1+z\right)\right)}{-z^3+1}\right)$, where $J_2\left(z\left(1+z\right)\right)$ is the BesselJ function.

CAVEAT 2: Depending on the mathematical expression, the algebraic cuts returned may contain some spurious cuts in the sense that the expression is actually continuous over the those cuts. You can determine this by visualizing the expression cuts directly, by using the options plot = 3D or  plot = 32D (the spurious cuts are not visualized as discontinuities) and comparing with the 2D plot where these spurious cuts are shown together with the real discontinuities of the expression.

Note: In Maple, by convention, when you say, for instance, z <= 1, it is implicitly assumed that $\mathrm{\Im }$(z) = 0.

Regarding the 3D plots, they are generated with plotcompare, they are centered at the origin, and the options used by default are scale_range = Pi, orientation = [-115, 81, 7], and grid = [200, 200] in order to display the cuts sharply. Depending on the case, centering at the origin may not be appropriate - to override that use the option shift_range explained in plotcompare. Likely, the default value for orientation may not visualize the discontinuities optimally - you can then rotate the plot with the mouse or use the option orientation = [a, b, c] explained in plot3D options. The high value used for grid may slowdown the visualization - to override speeding up use the option grid = [a, b] with smaller values of a and b; typically 50 would suffice. In all cases the array of plots is assigned to the variable _P so that it can be accessed for subsequent manipulation.

The option 32D is as the option 3D but for orientation = [-90, 0, 0], style = surface and shading = zgrayscale, so that the 3D plot is viewed from above, projected over the 2D (x, y) plane, as a smooth surface and with black and white contrast. The resulting visualization is similar to the one obtained using plot = 2D, but it has the advantage that it is free of the spurious cuts mentioned above in CAVEAT 2, and helps visualizing the branch points by contrast (see the last example).

The algebraic calculation is by default performed using the method so-called Standard; other optional choices for method are RealVariables and ComplexVariable. These methods and their limitations are discussed in the reference (at the end). Briefly summarized, given a mathematical expression involving functions $f_i\left(p\left(z\right)\right)$ where $p\left(z\right)$ is rational or involve fractional powers of $z$,

Obtain the branch cuts $C_j\left(z\right)$ of $f_i\left(z\right)$

Set $z=iy+x$ and compute two expressions, $R\left(x\&comma;y\right)=\mathrm{\Re }\left(p\left(x\&comma;y\right)\right)$ and $\mathrm{I}\left(x\&comma;y\right)=\mathrm{\Im }\left(p\left(x\&comma;y\right)\right)$

For each $C_i\left(p\left(z\right)\right)$, set $p\left(x\&comma;y\right)=R+\mathrm{I}i$ and solve for $\left\{x\&comma;y\right\}$

Reintroduce $z$, resulting in algebraic expressions $G_k\left(z\right)=\mathrm{...}$ for each cut $C_j$ of $f_i\left(p\left(z\right)\right)$

Take for branch cuts of the given expression the union of the $G_k\left(z\right)$

This approach can produce spurious branch cuts due to the denesting of powers at the time of solving for $x$ and $y$ or because of the intersection of cuts of different parts of an expression. Some of these spurious cuts can be detected automatically and are removed before returning the sequence of cuts for the given expression.

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

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

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

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

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

$\left[\mathrm{BesselJ}\left(a\&comma;z\right)\&comma;a::\left(\neg \mathbb{Z}\right)\wedge z<0\right]$

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

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

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

$\left[\mathrm{arccot}\left(z\right)\&comma;z\in \mathrm{ComplexRange}\left(-\mathrm{\infty }\mathrm{I}\&comma;\mathrm{-I}\right)\vee z\in \mathrm{ComplexRange}\left(\mathrm{I}\&comma;\mathrm{\infty }\mathrm{I}\right)\right]$

$\mathrm{convert}\left(\&comma;\mathrm{relation}\right)$

$\left[\mathrm{arccot}\left(z\right)\&comma;\left(\mathrm{\Re }\left(z\right)=0\wedge -\mathrm{\infty }\le \mathrm{\Im }\left(z\right)\wedge \mathrm{\Im }\left(z\right)\le \mathrm{-1}\right)\vee \left(\mathrm{\Re }\left(z\right)=0\wedge 1\le \mathrm{\Im }\left(z\right)\wedge \mathrm{\Im }\left(z\right)\le \mathrm{\infty }\right)\right]$

$\mathrm{has}\left(\&comma;z\right)$

$\mathrm{false}$

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

$\left[\mathrm{arccot}\left(z\right)\&comma;z\in \mathrm{ComplexRange}\left(-\mathrm{\infty }\mathrm{I}\&comma;\mathrm{-I}\right)\vee z\in \mathrm{ComplexRange}\left(\mathrm{I}\&comma;\mathrm{\infty }\mathrm{I}\right)\right]$

$\mathrm{has}\left(\&comma;z\right)$

$\mathrm{true}$

$\ln \left(\mathrm{sqrt}\left(z^3\right)\right)$

$\frac{\ln \left(z^3\right)}{2}$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\&comma;\mathrm{plot}=3.\right)$

$\left[\frac{\ln \left(z^3\right)}{2}\&comma;z<0\&comma;\mathrm{\Re }\left(z\right)=-\frac{\sqrt{3}\mathrm{\Im }\left(z\right)}{3}\wedge \mathrm{\Im }\left(z\right)<0\&comma;\mathrm{\Re }\left(z\right)=\frac{\sqrt{3}\mathrm{\Im }\left(z\right)}{3}\wedge 0<\mathrm{\Im }\left(z\right)\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\&comma;\mathrm{plot}=2.\right)$

$\left[\frac{\ln \left(z^3\right)}{2}\&comma;z<0\&comma;\mathrm{\Re }\left(z\right)=-\frac{\sqrt{3}\mathrm{\Im }\left(z\right)}{3}\wedge \mathrm{\Im }\left(z\right)<0\&comma;\mathrm{\Re }\left(z\right)=\frac{\sqrt{3}\mathrm{\Im }\left(z\right)}{3}\wedge 0<\mathrm{\Im }\left(z\right)\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\&comma;\mathrm{plot}=32.\right)$

$\left[\frac{\ln \left(z^3\right)}{2}\&comma;z<0\&comma;\mathrm{\Re }\left(z\right)=-\frac{\sqrt{3}\mathrm{\Im }\left(z\right)}{3}\wedge \mathrm{\Im }\left(z\right)<0\&comma;\mathrm{\Re }\left(z\right)=\frac{\sqrt{3}\mathrm{\Im }\left(z\right)}{3}\wedge 0<\mathrm{\Im }\left(z\right)\right]$

$\arcsin \left(2z\mathrm{sqrt}\left(1-z^2\right)\right)$

$\arcsin \left(2z\sqrt{-z^2+1}\right)$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\&comma;\mathrm{plot}=3.\&comma;\mathrm{orientation}=\left[-118\&comma;50\&comma;-10\right]\right)$

$\left[\arcsin \left(2z\sqrt{-z^2+1}\right)\&comma;1<z\&comma;z<\mathrm{-1}\&comma;\mathrm{\Re }\left(z\right)=-\frac{\sqrt{4{\mathrm{\Im }\left(z\right)}^2+2}}{2}\&comma;\mathrm{\Re }\left(z\right)=\frac{\sqrt{4{\mathrm{\Im }\left(z\right)}^2+2}}{2}\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\&comma;\mathrm{plot}=2.\right)$

$\left[\arcsin \left(2z\sqrt{-z^2+1}\right)\&comma;1<z\&comma;z<\mathrm{-1}\&comma;\mathrm{\Re }\left(z\right)=-\frac{\sqrt{4{\mathrm{\Im }\left(z\right)}^2+2}}{2}\&comma;\mathrm{\Re }\left(z\right)=\frac{\sqrt{4{\mathrm{\Im }\left(z\right)}^2+2}}{2}\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\&comma;\mathrm{plot}=32.\right)$

$\left[\arcsin \left(2z\sqrt{-z^2+1}\right)\&comma;1<z\&comma;z<\mathrm{-1}\&comma;\mathrm{\Re }\left(z\right)=-\frac{\sqrt{4{\mathrm{\Im }\left(z\right)}^2+2}}{2}\&comma;\mathrm{\Re }\left(z\right)=\frac{\sqrt{4{\mathrm{\Im }\left(z\right)}^2+2}}{2}\right]$

$\arctan \left(z\right)+\arctan \left(z^2\right)-\arctan \left(\frac{z\left(1+z\right)}{1-z^3}\right)$

$\arctan \left(z\right)+\arctan \left(z^2\right)-\arctan \left(\frac{z\left(z+1\right)}{-z^3+1}\right)$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\&comma;\mathrm{plot}=3.\&comma;\mathrm{orientation}=\left[-110\&comma;67\&comma;24\right]\right)$

$\left[\arctan \left(z\right)+\arctan \left(z^2\right)-\arctan \left(\frac{z\left(z+1\right)}{-z^3+1}\right)\&comma;\mathrm{\Re }\left(z\right)=0\wedge 1\le \mathrm{\Im }\left(z\right)\&comma;\mathrm{\Re }\left(z\right)=0\wedge \mathrm{\Im }\left(z\right)\le \mathrm{-1}\&comma;\mathrm{\Re }\left(z\right)=\mathrm{\Im }\left(z\right)\wedge \frac{\sqrt{2}}{2}\le \mathrm{\Im }\left(z\right)\&comma;\mathrm{\Re }\left(z\right)=\mathrm{\Im }\left(z\right)\wedge \mathrm{\Im }\left(z\right)\le -\frac{\sqrt{2}}{2}\&comma;\mathrm{\Re }\left(z\right)=-\mathrm{\Im }\left(z\right)\wedge \frac{\sqrt{2}}{2}\le \mathrm{\Im }\left(z\right)\&comma;\mathrm{\Re }\left(z\right)=-\mathrm{\Im }\left(z\right)\wedge \mathrm{\Im }\left(z\right)\le -\frac{\sqrt{2}}{2}\&comma;\mathrm{\Re }\left(z\right)=\sqrt{-{\mathrm{\Im }\left(z\right)}^2+1}\wedge -\frac{\sqrt{2}}{2}\le \mathrm{\Im }\left(z\right)\wedge \mathrm{\Im }\left(z\right)<0\&comma;\mathrm{\Re }\left(z\right)=\sqrt{-{\mathrm{\Im }\left(z\right)}^2+1}\wedge \mathrm{\Im }\left(z\right)\le \frac{\sqrt{2}}{2}\wedge 0<\mathrm{\Im }\left(z\right)\&comma;\mathrm{\Re }\left(z\right)=-\sqrt{-{\mathrm{\Im }\left(z\right)}^2+1}\wedge \mathrm{-1}\le \mathrm{\Im }\left(z\right)\wedge \mathrm{\Im }\left(z\right)<-\frac{\sqrt{3}}{2}\&comma;\mathrm{\Re }\left(z\right)=-\sqrt{-{\mathrm{\Im }\left(z\right)}^2+1}\wedge \mathrm{\Im }\left(z\right)\le 1\wedge \frac{\sqrt{3}}{2}<\mathrm{\Im }\left(z\right)\&comma;\mathrm{\Re }\left(z\right)=-\sqrt{-{\mathrm{\Im }\left(z\right)}^2+1}\wedge \frac{\sqrt{2}}{2}\le \mathrm{\Im }\left(z\right)\wedge \mathrm{\Im }\left(z\right)<\frac{\sqrt{3}}{2}\&comma;\mathrm{\Re }\left(z\right)=-\sqrt{-{\mathrm{\Im }\left(z\right)}^2+1}\wedge \mathrm{\Im }\left(z\right)\le -\frac{\sqrt{2}}{2}\wedge -\frac{\sqrt{3}}{2}<\mathrm{\Im }\left(z\right)\right]$

$2\mathrm{arccosh}\left(\frac{3+2z}{3}\right)-2\mathrm{arccosh}\left(\frac{12+5z}{3\left(z+4\right)}\right)$

$2\mathrm{arccosh}\left(1+\frac{2z}{3}\right)-2\mathrm{arccosh}\left(\frac{12+5z}{3z+12}\right)$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\&comma;\mathrm{plot}=3.\right)$

$\left[2\mathrm{arccosh}\left(1+\frac{2z}{3}\right)-2\mathrm{arccosh}\left(\frac{12+5z}{3z+12}\right)\&comma;z<-\frac{9}{2}\&comma;z\le 0\wedge \mathrm{-3}<z\&comma;\mathrm{-4}<z\wedge z<\mathrm{-3}\right]$

$-2\mathrm{arccosh}\left(2\left(z+3\right)\mathrm{sqrt}\left(\frac{z+3}{27\left(z+4\right)}\right)\right)$

$2\mathrm{arccosh}\left(1+\frac{2z}{3}\right)-2\mathrm{arccosh}\left(\frac{12+5z}{3z+12}\right)-2\mathrm{arccosh}\left(2\left(z+3\right)\sqrt{\frac{z+3}{27z+108}}\right)$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\&comma;\mathrm{plot}=3.\&comma;\mathrm{shift\_range}=-3\&comma;\mathrm{scale\_range}=2\&comma;\mathrm{orientation}=\left[-124\&comma;20\&comma;14\right]\right)\&colon;$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\&comma;\mathrm{plot}=32.\&comma;\mathrm{shift\_range}=-3\&comma;\mathrm{scale\_range}=2\right)\&colon;$

$\ln \left(z+1\right)-\ln \left(z-1\right)$

$\ln \left(z+1\right)-\ln \left(z-1\right)$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\&comma;\mathrm{plot}=2.\right)$

$\left[\ln \left(z+1\right)-\ln \left(z-1\right)\&comma;\mathrm{-1}\le z\wedge z<1\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\&comma;\mathrm{plot}=3.\&comma;\mathrm{orientation}=\left[112\&comma;80\&comma;15\right]\right)\&colon;$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\&comma;\mathrm{plot}=32.\right)\&colon;$

England M., Bradford R., Davenport J.H., Wilson, D. "Understanding branch cuts of expressions". http://opus.bath.ac.uk/32511/

Cheb-Terrab, E.S. "The function wizard project: A Computer Algebra Handbook of Special Functions". Proceedings of the Maple Summer Workshop, University of Waterloo, Ontario, Canada, 2002.