Solving Complex Integration Involving Bessel, Singularities

[KayBox]

Well, here it is. I am at a loss as to how to approach this. I understand I can use the residue theorem for the poles at a and b, those are not the problem. I have heard that you can expand the function in a Laurent series and look at certain terms for the c term , but I don't fully understand this. Any help is appreciated. I attached a picture of what the question looks like. Many thanks in advance!

 

[KayBox]

sorry about that, here is the attachment