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

Step 1: Concept
This question states the hypotheses of Morera's Theorem, which serves as a converse to Cauchy's Integral Theorem.

Step 2: Key Formulas and Approach
Morera's Theorem: Let $f(z)$ be a continuous complex-valued function on a connected open domain $G$. If $\oint_C f(z) dz = 0$ for every simple closed curve $C$ lying in $G$, then $f(z)$ is analytic throughout $G$.

Step 3: Step-by-step Explanation

• Since $\oint_C f(z) dz = 0$ for all simple closed contours $C \subset G$, path-independence holds.
• Fix a point $z_0 \in G$ and define an antiderivative function: \[ F(z) = \int_{z_0}^z f(w) dw \]
• Because the integral is independent of path, $F(z)$ is well-defined and complex-differentiable on $G$ with $F'(z) = f(z)$.
• By the property of analytic functions, any function that possesses a derivative is infinitely differentiable.
• Since $F(z)$ is analytic, its derivative $F'(z) = f(z)$ must also be analytic in $G$.
• Therefore, $f(z)$ is analytic in $G$.

Step 4: Final Answer
By Morera's theorem, $f(z)$ is analytic in $G$. Thus, Option (C) is correct.