Question

If a complex function \( f(z) \) is analytic everywhere inside a closed contour \( C \) with anti-clockwise direction, then which of the following statements are correct?

• A. \( \oint_C \cos z \, dz = 0 \)
• B. \( \oint_C \sec z \, dz \neq 0 \)
• C. \( \oint_C z^n \, dz = 0 \)
• D. \( \oint_C e^z \, dz = 0 \)

Hint:

Cauchy's Integral Theorem states that if a function is analytic inside and on a closed contour, its integral is zero. However, if a function has poles inside the contour, then we apply Residue Theorem.

Answer 1

The Correct Option is A

Applying Contour Integration Theorems.
By Cauchy's Integral Theorem (CIT): \[ \oint_C f(z) dz = 0, \quad \text{if } f(z) \text{ is analytic inside and on } C. \] Checking the options:
1. \( \cos z \) is entire (analytic everywhere), so \( \oint_C \cos z dz = 0 \). % Option (A) is correct.
2. \( \sec z = \frac{1}{\cos z} \) has poles where \( \cos z = 0 \) (e.g., \( z = \frac{\pi}{2}, -\frac{\pi}{2} \)). - If a pole exists inside \( C \), the integral is nonzero. % Option (B) is correct.
3. \( z^n \) is analytic for all \( n \neq -1 \), so \( \oint_C z^n dz = 0 \) for \( n \neq -1 \). % Option (C) is correct.
4. \( e^z \) is entire (analytic everywhere), so \( \oint_C e^z dz = 0 \). % Option (D) is correct.