The GATE Grind

GATE 2025 IN – Question 49

Engineering Mathematics · Complex variables · 2 marks · Multiple select

Choose the correct statements about Cauchy’s theorem for $\oint_C f(z)\,dz$, where $C$ is a simple closed path in a simply connected domain $D$.

  1. Cauchy’s theorem cannot directly conclude $\oint_C f(z)\,dz=0$ when $f(z)=1/z^2$ and $C$ is the unit circle.
  2. If $f(z)$ is analytic in $D$, then $\oint_C f(z)\,dz=0$ for any simple closed path $C$ in $D$.
  3. The function $f(z)$ must be analytic in $D$ to conclude $\oint_C f(z)\,dz=0$ for any simple closed path $C$ in $D$.
  4. $\oint_C f(z)\,dz\ne0$ when $f(z)=1/z^2$, since it is not analytic at $z=0$.

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Show answer and explanation

Correct answer: (A) Cauchy’s theorem cannot directly conclude $\oint_C f(z)\,dz=0$ when $f(z)=1/z^2$ and $C$ is the unit circle.; (B) If $f(z)$ is analytic in $D$, then $\oint_C f(z)\,dz=0$ for any simple closed path $C$ in $D$.

Explanation

The function 1/z² is not analytic inside the unit circle, so Cauchy’s theorem cannot directly be invoked there, making A true.

Analyticity throughout the stated simply connected domain is sufficient for every closed-path integral to vanish, making B true.

A singular function may still have zero integral, as 1/z² does by its zero residue, so D is false; the unqualified necessity in C is not the theorem. **A,B**.