Question

Let $f(z)$ be continuous in a simply connected region $G$ and suppose $\oint_c f(z)dz = 0$ around every simple closed curve $c$. Then

• A. $f(z)$ is not analytic in $G$.
• B. $f(z) = 0$ for all $z \in G$.
• C. $f(z)$ is analytic in $G$.
• D. $f(z)$ is only continuous in $G$.

Hint:

Cauchy's Theorem: Analytic $\implies \oint_C f(z)dz = 0$. Morera's Theorem: Continuous + $\oint_C f(z)dz = 0 \implies$ Analytic. They are exact converses of each other!

Answer 1

The Correct Option is C