Question

The consider the counter integral

∮ c e ᶦᵏᶻ/ a ² + z ² 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

Answer 1

To evaluate the contour integral using the residue theorem procedure, we split the integral into two parts: the integral along the semicircle and the integral along the real line. For the integral along the semicircle, we parameterize it and evaluate it using standard methods. For the integral along the real line, we use the residue theorem by finding the residue at the pole and multiplying it by 2πi. We then add the results of the two integrals to find the final result.

Step-by-step explanation:

To evaluate the contour integral ∮c eikz/(a²+z²) dz using the residue theorem procedure, we can use the fact that the function eikz/(a²+z²) has a pole at z = ai. The contour c is a semicircle centered at the origin with radius R. We can split the contour integral into two parts: the integral along the semicircle and the integral along the real line.

1. For the integral along the semicircle, we can parameterize it as z = Reiθ, where 0 ≤ θ ≤ π. The integral becomes ∫0ˡπ eik(Reθ)/(a²+(Reθ)²) (iReθ) dθ. We can simplify this by substituting u = Reθ and du = Rieθ dθ, giving us ∫0ˡπ eiku/(a²+u²) du. This integral can be evaluated using standard methods.

2. For the integral along the real line, we can use the fact that the function eikz/(a²+z²) is holomorphic except at the pole z = ai. Therefore, the residue at the pole is given by Residue = limz→ai (z-ai)eikz/(a²+z²). We can then use the residue theorem, which states that the contour integral is equal to 2πi times the sum of the residues inside the contour. In this case, the only residue is at z = ai, so the contour integral is equal to 2πi times the residue at ai.

We can then add the results of the integral along the semicircle and the integral along the real line to find the final result of the contour integral.