MAT334-2018F > MAT334--Lectures & Home Assignments

2.6 #4

(1/1)

celina q:
Can someone show me the process for this question?

Kris:
We consider the function $f(z)=\frac{e^{iaz}}{(z^2+1)(z^2+4)}$.
$(z^2+1)(z^2+4)=0\Rightarrow$$z=\pm i,$$\pm 2i.$ Only i, 2i are in the upper-half plane.
$Res(f,i)=\frac{e^{iai}}{2i(-1+4)}=\frac{e^{-a}}{6i}$    $Res(f,2i)=\frac{e^{ia2i}}{(-4+1)4i}=\frac{e^{-2a}}{-12i}$. $I=2\pi i(\frac{e^{-a}}{6i}-\frac{e^{-2a}}{12i})=\pi(\frac{e^{-a}}{3}-\frac{e^{-2a}}{6})$

Navigation

[0] Message Index

Go to full version