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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Summer 2023 • Question 3

Question

Math 6220 • Complex Analysis Summer 2023 • Question 3

Prove the following version of Morera’s theorem. If \(f:\mathbb {D}\to \mathbb {C}\) is a continuous function such that for every rectangle \(R\subset \mathbb {D}\)

\[ \int _{\partial R} f(z)\,dz=0 \]

then \(f\) is holomorphic. Note: It does not follow from the assumption in any obvious way that integrals over triangles are 0.

Raw LaTeX

Prove the following version of Morera's theorem. If \(f:\mathbb{D}\to\mathbb{C}\) is a
continuous function such that for every rectangle \(R\subset\mathbb{D}\)

\[
\int_{\partial R} f(z)\,dz=0
\]

then \(f\) is holomorphic. Note: It does not follow from the assumption in
any obvious way that integrals over triangles are 0.

Topic Tags

Cauchy Integral Theory and Morera's Theorem Complex Integration and Real Integral Evaluation Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms