Question

A contour integral is defined as \[ I_n = \oint_C \frac{dz}{(z - n)^2 + \pi^2} \] where \( n \) is a positive integer and \( C \) is the closed contour, as shown in the figure, consisting of the line from \( -100 \) to \( 100 \) and the semicircle traversed in the counter-clockwise sense. The value of \( \sum_{n=1}^5 I_n \) (in integer) is \(\underline{\hspace{2cm}}\). 

Hint:

To evaluate contour integrals, use the residue theorem and sum the residues at the poles inside the contour.

Answer 1

Correct Answer: 5

The given contour integral can be evaluated using the residue theorem. The integrand has simple poles at \( z = n \pm i\pi \) for each integer \( n \). For each pole, we calculate the residue and sum the contributions to the total integral. By applying the residue theorem, we obtain the value of \( \sum_{n=1}^5 I_n \) as \( 5 \). Thus, the value of \( \sum_{n=1}^5 I_n \) is \( 5 \).