What types of real integrands accept contour integration?

[Al-Layth]

I am trying to learn contour integration, to help in evaluating some definite real integrands.

But even after examining the process of the technique, i cannot see how a person would know if this technique would be useful/ or possible for a specific integrand.

What types of real integrand accept contour integration?

 

[BigBeachBanana]

A couple of cases I can recall

1. Contour integration can be used to evaluate integrals of trigonometric functions because they can be expressed in terms of complex exponentials using Euler's formula, and the resulting integrand can be written as a complex function that is "nicer" in the complex plane
2. Integrals involve rational functions because the integrand can be written as a complex function with poles at the roots in the denominator, and the contour of integration can be chosen to avoid these poles.

See Wiki for some examples.

 

[Al-Layth]

A couple of cases I can recall

1. Contour integration can be used to evaluate integrals of trigonometric functions because they can be expressed in terms of complex exponentials using Euler's formula, and the resulting integrand can be written as a complex function that is "nicer" in the complex plane
2. Integrals involve rational functions because the integrand can be written as a complex function with poles at the roots in the denominator, and the contour of integration can be chosen to avoid these poles.

See Wiki for some examples.

Thx for the precise answer, i really appreciate answers like this

a couple more questions:
(1) - Is there a way to evaluate an integrand containing hyperbolics with contour integration?
(2) - [i don't expect it but just in case we can] Can we use contour integration to find the antiderivative of a real integrand?

 

[BigBeachBanana]

Thx for the precise answer, i really appreciate answers like this

a couple more questions:
(1) - Is there a way to evaluate an integrand containing hyperbolics with contour integration?
(2) - [i don't expect it but just in case we can] Can we use contour integration to find the antiderivative of a real integrand?

(1) Yes.
(2) Have you looked at the Wiki's examples?