Classroom Tips and Techniques: Branch Cuts for a Product of Two Square-Roots

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1 Classroom Tips and Techniques: Branch Cuts for a Product of Two Square-Roots Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft Introduction Naive simplification of to results in a pair of functions that agree on only part of the complex plane. The enhanced ability of Maple 18 to find and display branch cuts of composite functions is used in this article to explore the branch cuts and regions of agreement/disagreement of and F. The algorithm by which Maple calculates branch cuts for square-root functions involves squaring (to remove the square root) and solving appropriate equations and inequalities. Unfortunately, this process is prone to introducing spurious solutions, in which case the returned branch cut is not correct. Such an error arises when Maple calculates the branch cut for ; a best suggestion for dealing with such errors is found below. The new branch-cut facility in Maple is based on a paper listed in the Reference section at the bottom of the help page?branch_cuts. Specifically, the paper is "Understanding branch cuts of expressions", England M., Bradford R., Davenport J.H., Wilson, D.; it can be found here: uk/32511/. The basic idea behind the algorithm is illustrated below for. Branch Cuts Prior to Maple 17, the branch_cuts option to the FunctionAdvisor command would only provide an analytic representation of branch cuts for a select class of functions known to Maple. For example, the branch cut for the complex logarithm would be found via the following command. Maple can now find branch cuts of expressions involving members of the class of known functions. For example, the branch cut for is found via the following command.

2 A careful reading of the return shows that the cut is along the imaginary axis, and if, then it corresponds to. The FunctionAdvisor command will draw the branch cut in the plane, and will also draw graphs of the real and imaginary parts of, as shown in Figures 1 and 2, respectively. Figure 1 Branch cuts for Figure 2 Real and imaginary parts of To make rotating the graphs in Figure 2 easier, the grid density has actually been reduced from its higher default. The discontinuities in the graph of the imaginary part show as nearly vertical segments, and correspond to the branch cuts drawn as curves (here, lines) in the complex plane.

3 Essentially, the branch-cut code makes use of the known cut for, namely,, and computes where the conditions are satisfied. To see this in Maple, change the default symbol for from to by means of the following task template. Tools_ Tasks_ Browse: Algebra_ Complex Arithmetic_ Set Imaginary Unit Notation for Imaginary Unit Access Settings (Click the "Access Settings" button, select as the symbol for the imaginary unit, and click "OK." Alternatively, execute the command interface(imaginaryunit = i).) The branch cut for is found by solving the system, where. The calculations for this are found in Table 1, where an analytical solution of the system is obtained. Write Context Menu: Assign Function assign as function h Obtain., where Context Menu: Solve_ Solve solve Table 1 Calculating the branch cut for Figure 3 shows an alternate graphical approach to solving the system, while Figure 4 shows a graph of the analytic solution of the system.

4 Figure 3 Graphical solution of the system Figure 4 Graph of the analytic solution of the system Branch Cut for Maple determines the branch cut for analytically, and draws it in Figure 5 based on the analytic solution of the system. This squaring operation introduces spurious segments for the cut, as can be inferred from the separate graphs of the real and imaginary parts of shown in Figure 6. Write Context Menu: Assign Function assign as function f Apply the FunctionAdvisor command to generate Figures 5 and 6

5 Figure 5 Incorrect branch cut for Figure 6 Real and imaginary parts of Figure 6 suggests that the branch cut is along the interval on the real axis. That the discontinuity in does not extend to the left of on the real axis is shown by the equality of the following limits. Alternatively, the function can be represented analytically as

6 , where,,. Since the factor is real and positive, the discontinuity in must be in the exponential term. Using Maple, the imaginary part of this term can be expressed as I (1) imaginary part e (2) (3) assign as function E Determining where is continuous and discontinuous is not a simple task. Figure 6 suggests the animation in Figure 7 wherein is graphed as a function of with as an animation parameter.

7 y: Figure 7 As passes through zero, the discontinuity in can be seen for Branch Cut for Maple determines the branch cut for analytically, and draws it in Figure 8 based on the analytic solution of the system. Because, Maple obtains the correct analytic representation of the branch cut, with Figure 9 corroborating the result in Figure 8. Write Context Menu: Assign Function assign as function F Apply the FunctionAdvisor command to generate Figures 8 and 9

8 Figure 8 Branch cut for Figure 9 Real and imaginary parts of Maple correctly determines the branch cut by solving the system (4) to obtain (5)

9 (5) A careful interpretation of this solution shows the branch cut to be the union of the imaginary axis and the real interval, in agreement with Figures 8 and 9 Indeed, if, then ; and if, then, which is satisfied for all real. Behavior across the Branch Cuts Figures animate a circle in the -plane, and the images of the circle under and. In particular, the circles are chosen to cross the axes is such a way as to detect graphically any discontinuity arising from a branch cut. In Figure 10,, a circle of radius with center at, is drawn in black. At, the images under both and are discontinuous. Both functions have a branch cut on the interval. Figure 10 Circle in -plane (black); images under (red), (green) In Figure 11,, a circle of radius with center at, is drawn in black. As, the circle closes, but the images under both and show discontinuities. Both functions have a branch cut on the interval.

10 Figure 11 Circle in -plane (black); images under (red), (green) In Figure 12,, a circle of radius with center at, is drawn in black. The images of this circle under both and are continuous. Neither function has a branch cut to the left of on the real axis. Figure 12 Circle in -plane (black); images under (red), (green) In Figure 13,, a circle of radius with center at, is drawn in black. At the image under is discontinuous but the image under is not. At the image under is again discontinuous, but the image under is not. In other words,, the naive simplification of, has a branch cut along the imaginary axis, but does not. (Note the small black square that traces out the green trajectory.)

11 Figure 13 Circle in -plane (black); images under (red), (green) Comparing the Values of and The function and its naive simplification have different branch cuts. But that is not the only difference between these two functions. Figures 14 and 15 show respectively the real and imaginary parts of the difference. Both figures suggest that and agree in the right-half plane. Figure 14 Real part of Figure 15 Imaginary part of The calculations in Table 2 support the further conclusions that if, then, and

12 results in results in Table 2 Values of along portions of the real and imaginary axes Consequently, and agree in the right-half plane, and along the -axis where, and along the imaginary axis where. Legal Notice: Maplesoft, a division of Waterloo Maple Inc Maplesoft and Maple are trademarks of Waterloo Maple Inc. This application may contain errors and Maplesoft is not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact Maplesoft for permission if you wish to use this application in for-profit activities.

Classroom Tips and Techniques: Nonlinear Curve Fitting. Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft

Classroom Tips and Techniques: Nonlinear Curve Fitting Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft Introduction I was recently asked for help in fitting a nonlinear curve