I have this problem with a complex integral and I'm having a lot of difficulty solving it:
Show that for R and U both greater than 2a, \exists C > 0, independent of R,U,k and a, such that $$\int_{L_{-R,U}\cup L_{R,U}} \lvert f(z)\rvert\,\lvert dz\rvert \leqslant \frac{C}{kR}.$$
Where a > 0, k...
I have been trying to show that
$$\lim_{U\rightarrow\infty}\int_C \frac{ze^{ikz}}{z^2+a^2}dz = 0 $$
Where $$R>2a$$ and $$k>0$$ And C is the curve, defined by $$C = {x+iU | -R\le x\le R}$$
I have tried by using the fact that
$$|\int_C \frac{ze^{ikz}}{z^2+a^2}dz| \le\int_C...