For the Cauchy Residue Theorem, the integral oint_C f(Z),dZ where f has isolated singularities z_1,z_2,ldots,z_n inside C equals:

Home › MCQs › SSC Pure Mathematics › Question #1291

SSC Pure Mathematics QUESTION #1291

Question 1

For the Cauchy Residue Theorem, the integral \(\oint_C f(Z)\,dZ\) where \(f\) has isolated singularities \(z_1,z_2,\ldots,z_n\) inside \(C\) equals:

Sum of residues
2\(\pi i\) times sum of residues✔️
\(\pi i\) times sum of residues
Twice the sum of residues

Correct Answer Explanation

Cauchy's Residue Theorem: \(\oint_C f(Z)\,dZ=2\pi i\sum_{k=1}^n\text{Res}(f,z_k)\), where the sum is over all isolated singularities inside the positively oriented simple closed contour \(C\). The factor \(2\pi i\) is essential and always present.

More Options