Naive simplification of f(z)=sqrt(z-1)*sqrt(-1*(-z-1)) to F(z)=sqrt(z^2-1)results in a pair of functions that agree on only part of the complex plane. In this application, the enhanced ability of Maple 18 to find and display branch cuts of composite functions is used to explore the branch cuts and regions of agreement/disagreement of f 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 inherently prone to introducing spurious solutions, in which case the returned branch cut is not correct. One such instance in which a spurious solution arises is in the calculation of the branch cut for f; a best suggestion for dealing with such errors is found in the application.

Application: Branch Cuts for a Product of Two Square Roots

For those interested in learning more, the design for the new branch-cut facility in Maple 18 is inspired by the following paper:

England, M., Bradford, R., Davenport, J. H., and Wilson, D. 2013.  Understanding branch cuts of expressions. In: Carette, J., Aspinall, D., Lange, C., Sojka, P. and Windsteiger, W., eds.  Intelligent Computer Mathematics. Berlin: Springer, pp. 136-151. (Lecture Notes in Computer Science; 7961)

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