Question

Consider the contour integral

∮ eᶦᵏᶻ/c​a²+z²​ dz
where a>0,k>0 and i is the complex number, with the contour c being a semicircle.
Use the residue theorem procedure, as outlined in to evaluate this contour integral, and that the final answer is

Answer 1

The student's question pertains to evaluating a contour integral using the residue theorem in the field of complex analysis, specifically the integral of ei­kz/(a²+z²) over a semicircular contour in the complex plane.

Step-by-step explanation:

The student is asking about evaluating a contour integral in the complex plane using the residue theorem. The integral given is ∮ ei­kz/(a²+z²) dz, with the contour being a semicircle. This question belongs to the field of complex analysis, a branch of mathematics.

To solve this integral using the residue theorem, one needs to consider the residues of the function ei­kz/(a²+z²) at its poles within the contour. Due to the presence of the term a²+z² in the denominator, poles occur at z = ai and z = -ai. Since the contour is a semicircle which likely lies in the upper half of the complex plane, only the pole at z = ai would be enclosed by the contour.

The next step is to calculate the residue at z = ai and then multiply it by 2πi to get the integral's value over the semi-circular contour. This is only if the contour is closed with a line segment on the real axis, forming a closed path in the upper half-plane. If the contour does not include the pole, the integral over the contour would be zero due to Cauchy's integral theorem for analytic functions.