Question

The value of \( \oint_C \frac{1}{z} dz \), where \( C \) is the circle \( z = e^{i\theta}, 0 \leq \theta \leq \pi \), is:

• A. \( \pi i \)
• B. \( -\pi i \)
• C. \( 2\pi i \)
• D. \( 0 \)

Hint:

\[ \oint_C \frac{1}{z} dz = 2\pi i \] if \( C \) encloses the origin. A semicircle contour gives half this value.

Answer 1

The Correct Option is A

Step 1: Integral of \( \frac{1}{z} \) over a contour. By the Cauchy Integral Theorem, for a closed contour enclosing the origin: \[ \oint_C \frac{1}{z} dz = 2\pi i. \] Step 2: Consider the given semicircular contour. - Given contour \( C \) covers half of the full circle. - So, the integral is half of \( 2\pi i \), which gives: \[ \pi i. \] Step 3: Selecting the correct option. Since \( \pi i \) is correct, the answer is (A).