Logarithm of a Function, a Well-Posed Inverse Problem

It poses the inverse problem that consists in finding the logarithm of a function. It shows that when the function is holomorphic in a simply connected domain , the solution at the inverse problem exists and is unique if a branch of the logarithm is fixed. In addition, it’s demonstrated that when the function is continuous in a domain , where is Hausdorff space and connected by paths. The solution of the problem exists and is unique if a branch of the logarithm is fixed and is stable; for what in this case, the inverse problem turns out to be well-posed.

• Logarithm Function; Inverse Problem; Stability

References

1. C. W. Groetsch, “Inverse Problems: Activities for Undergraduates,” The Mathematical Association of America, Ohio, 1999.

2. A. Browder, “Topology in the Complex Plane,” The American Mathematical Monthly, Vol. 107 No. 10, 2006, pp. 393-401.https://getinfo.de/app/Topology-in-the-Complex-Plane/id/BLSE%3ARN079983226

3. A. Hatcher, “Algebraic Topology,” Cambridge University Press, Cambridge, 2009.

4. L. V. Ahlfors, “Complex Analysis,” McGraw-Hill, New York, 1979.

5. A. Kirsch, “An Introduction to the Mathematical Theory of Inverse Problems,” 2nd Edition, Springer, Berlin, 2011.http://dx.doi.org/10.1007/978-1-4419-8474-6

6. E. Stein and R. Shakarchi, “Complex Analysis,” Princeton University Press, Princeton, 2009.