Branches and Branch Cuts - Maple Programming Help

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Branches and Branch Cuts

for the Inverse Trig and Hyperbolic Functions

 Introduction This example worksheet supplements the Branch Cuts Tutor.  Access the Branch Cuts Tutor from the Tools menu, under Tutors>Complex Variables.   Properties of the inverse trig and hyperbolic functions in Maple depend on Maple's choices of branch cuts and principal branches.  In this example worksheet, we show how to determine principal branches and branch cuts for these twelve functions and assemble the information used in the Branch Cuts Tutor.



The FunctionAdvisor command provides access to the information Maple has stored for nearly all its special functions.  In particular, it can be queried for the branch cuts of a function.  For example, applying it to the arctangent function, we find



$\left[{\mathrm{arctan}}{}\left({z}\right){,}{z}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{∈}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{ComplexRange}}{}\left({-}{\mathrm{∞}}{}{I}{,}{-}{I}\right){,}{z}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{∈}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{ComplexRange}}{}\left({I}{,}{\mathrm{∞}}{}{I}\right)\right]{,}\left[{\mathrm{arctan}}{}\left({y}{,}{x}\right){,}{\mathrm{ComplexRange}}{}\left({-}{\mathrm{∞}}{}{I}{,}{-}{I}{}\left|{y}\right|\right){,}{\mathrm{ComplexRange}}{}\left({I}{}\left|{y}\right|{,}{\mathrm{∞}}{}{I}\right){,}{\mathrm{And}}{}\left({\mathrm{ℑ}}{}\left({y}\right){=}{0}{,}{x}{=}{0}\right)\right]$

There are a number of issues to deal with here.  First, note that information for both $\mathrm{arctan}\left(z\right)$ and $\mathrm{arctan}\left(y,x\right)$ has been returned.  To focus on just the first, issue the command as



 $\left[{\mathrm{arctan}}{}\left({z}\right){,}{z}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{∈}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{ComplexRange}}{}\left({-}{\mathrm{∞}}{}{I}{,}{-}{I}\right){,}{z}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{∈}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{ComplexRange}}{}\left({I}{,}{\mathrm{∞}}{}{I}\right)\right]$ (2.1)



Second, note that the "ComplexRange" construction yields to a conversion to "relation" as in



$\mathrm{convert}\left(,\mathrm{relation}\right)$

 $\left[{\mathrm{arctan}}{}\left({z}\right){,}{\mathrm{And}}{}\left({\mathrm{ℜ}}{}\left({z}\right){=}{0}{,}{-}{\mathrm{∞}}{\le }{\mathrm{ℑ}}{}\left({z}\right){,}{\mathrm{ℑ}}{}\left({z}\right){\le }{-}{1}\right){,}{\mathrm{And}}{}\left({\mathrm{ℜ}}{}\left({z}\right){=}{0}{,}{1}{\le }{\mathrm{ℑ}}{}\left({z}\right){,}{\mathrm{ℑ}}{}\left({z}\right){\le }{\mathrm{∞}}\right)\right]$ (2.2)



Third, to enter an underscore in Math mode, the escape character (\) should be typed first.  Otherwise, Math mode interprets the underscore as a signal to lower the cursor for a subscript.

Finally, note that the "And" construction can be simplified.  We show how to do this after we delete $\mathrm{arctan}\left(z\right)$ from (2.2) with

$\mathrm{subsop}\left(1=\mathrm{NULL},\right)$

 $\left[{\mathrm{And}}{}\left({\mathrm{ℜ}}{}\left({z}\right){=}{0}{,}{-}{\mathrm{∞}}{\le }{\mathrm{ℑ}}{}\left({z}\right){,}{\mathrm{ℑ}}{}\left({z}\right){\le }{-}{1}\right){,}{\mathrm{And}}{}\left({\mathrm{ℜ}}{}\left({z}\right){=}{0}{,}{1}{\le }{\mathrm{ℑ}}{}\left({z}\right){,}{\mathrm{ℑ}}{}\left({z}\right){\le }{\mathrm{∞}}\right)\right]$ (2.3)



Since (2.3) is a list containing two instances of "And", we must map the conversion process onto the list.  Thus, we have



$\mathrm{map}\left(\mathrm{convert},,\mathrm{list}\right)$

$\left[\left[{\mathrm{ℜ}}{}\left({z}\right){=}{0}{,}{-}{\mathrm{∞}}{\le }{\mathrm{ℑ}}{}\left({z}\right){,}{\mathrm{ℑ}}{}\left({z}\right){\le }{-}{1}\right]{,}\left[{\mathrm{ℜ}}{}\left({z}\right){=}{0}{,}{1}{\le }{\mathrm{ℑ}}{}\left({z}\right){,}{\mathrm{ℑ}}{}\left({z}\right){\le }{\mathrm{∞}}\right]\right]$



Table 1 contains the result of all such manipulations for the inverse trig, and inverse hyperbolic functions.

 $\left[\begin{array}{cc}\mathrm{arcsin}& \left[z\le -1,1\le z\right]\\ \mathrm{arccos}& \left[z\le -1,1\le z\right]\\ \mathrm{arctan}& \left[\left[\mathrm{\Re }\left(z\right)=0,-\infty \le \mathrm{\Im }\left(z\right),\mathrm{\Im }\left(z\right)\le -1\right],\left[\mathrm{\Re }\left(z\right)=0,1\le \mathrm{\Im }\left(z\right),\mathrm{\Im }\left(z\right)\le \infty \right]\right]\\ \mathrm{arccot}& \left[\left[\mathrm{\Re }\left(z\right)=0,-\infty \le \mathrm{\Im }\left(z\right),\mathrm{\Im }\left(z\right)\le -1\right],\left[\mathrm{\Re }\left(z\right)=0,1\le \mathrm{\Im }\left(z\right),\mathrm{\Im }\left(z\right)\le \infty \right]\right]\\ \mathrm{arcsec}& \left[\left[-1\le z,z<0\right],\left[0 Table 1   Modified output for FunctionAdvisor applied to each inverse trig and hyperbolic function

Table 1 was generated as a Maple matrix, using a number of Maple commands to make the desired modifications.  The final results are about the best that can be obtained using a basic set of commands.  Of course, it would have been possible to typeset (by hand) a more readable version, but instead, we have captured the information in a more visual way, in Figure 4, below.

Visualizing a Branch Cut



Figures 1 and 2 are respectively graphs of the real and imaginary parts of the function $w=\mathrm{arcsin}\left(z\right)$, where .



$\mathrm{plot3d}\left(\mathrm{ℜ}\left(\mathrm{arcsin}\left(x+I\cdot y\right)\right),x=-3..3,y=-3..3,\mathrm{axes}=\mathrm{box},\mathrm{orientation}=\left[-50,65\right]\right)$

${}$

$\mathrm{plot3d}\left(\mathrm{ℑ}\left(\mathrm{arcsin}\left(x+I\cdot y\right)\right),x=-3..3,y=-3..3,\mathrm{axes}=\mathrm{box},\mathrm{orientation}=\left[-50,65\right]\right)$

${}$

Figure 1   Real part of $w=\mathrm{arcsin}\left(z\right)$

Figure 2   Imaginary part of $w=\mathrm{arcsin}\left(z\right)$



In Figure 2, a discontinuity is apparent along the real axis, and corresponds to the information produced by the FunctionAdvisor, namely, that the branch cuts are along $z\le -1$ and $z\ge 1$.



The branches Command



The branches command produces a schematic of the principal branches of $z=f\left(w\right)$, where $f$ is one of the twelve functions in Table 1.  For example, Figure 3 displays the result of applying the branches command to $\mathrm{arcsin}\left(w\right)$.

This schematic is drawn in the range space, using the notation $z=\mathrm{arcsin}\left(w\right)$, the axis labels being hard-coded with $\mathrm{ℜ}\left(z\right)$ and $\mathrm{ℑ}\left(z\right)$.  Unfortunately, this notation contradicts the usage in the FunctionAdvisor where the functions are given as $w=f\left(z\right)$.

$\mathrm{branches}\left(\mathrm{arcsin}\right)$



Figure 3   Application of the branches command to the arcsin function



The magenta boundaries indicate the continuity of the function as the boundary of the principal branch is approached.  It is unfortunate that the boundary of the branch to the left of the principal branch uses the color red.  It may be difficult to distinguish between these two colors when they are contiguous.  Similarly, the branch to the right of the principal branch uses green, and this abuts a line segment drawn in cyan, two nearly identical colors. The branches command allows the user to impose alternate labels, and to be consistent with the branch-cut information generated by the FunctionAdvisor command.   Furthermore, the user can control the colors used and the labels on the axes.  We use these options for the composite tool in the Branch Cuts Tutor, also given in Figure 4 below.



Compilation of Branch Information



In addition to the information shown in Figures 1, 2, and 3, it is also useful to see the image, under $w=f\left(z\right)$, of the branch cuts of the principal branch.  It is also useful to see a graph of the branch cuts themselves -- it's a lot easier to comprehend the visual than it is to interpret the analytic information in Table 1.

Figure 4 is a composite of all the information that Maple can provide about principal branches and their branch cuts.  On the left, there are graphs of the real and imaginary parts of $w=f\left(z\right)$, where $f$ is one of the inverse functions selected by clicking on a radio button in the display.  In the central column are two graphs drawn in the $w$-plane, the upper one generated by the improved branches command; and the lower one being just the image in the $w$-plane of the cut under the mapping $w=f\left(z\right)$.  The graph on the upper right shows the cut itself in the $z$-plane, and this is color-coded to the graphs of the images of the cuts in the $w$-plane.  This graph is interactive -- dragging the Click and Drag Manipulator with the mouse causes the image of a point in the $z$-plane to appear as a red dot in the $w$-plane. (To select this indicator, click on the graph and make the selection from the plotting toolbar at the top of the worksheet.  Alternatively, from the context menu for the plot, choose Manipulator_Click and Drag.)

Finally, at the bottom of Figure 4 the branch cuts are given in interval notation.



Branches of ${\mathbit{w}}{\mathbf{=}}{\mathbit{f}}\left({\mathbit{z}}\right)$, the Inverse Trig and Hyperbolic Functions

       

Branch Cut(s):

Figure 4   Composite Maple information on the branches and branch cuts of the inverse trig and hyperbolic functions