6) Applications of the residue theorem

Question 1

What ensures that the improper integral of a continuous function f:R→C converges

Answer 1

Question 2

What is the general method to evaluate the integral of a function over a symmetric interval [−R,R] using complex analysis

Answer 2

Question 3

How can Cauchy’s Residue Theorem be used to evaluate integrals involving trigonometric functions

Answer 3

Question 4

What does it mean for cot(πz) to be bounded on specific regions

Answer 4

There is a bound, independent of N, on cotπz for z ∈CN , i.e. there exists M > 0 such that for all N and all z ∈CN , we have |cotπz| ≤ M

Question 5

What is the general method for using Cauchy’s Residue Theorem to evaluate infinite sums

Answer 5