The Limit of a Complex Integral

[Bachelier]

Though it is not homework I posted this here, hopefully it'll get more action. Thanks

given \int_{\Lambda} \frac{e^{i w}}{w^2 + 1} \mathrm{d}w where $w \in \mathbb{C}$ with $Im(w) \geqslant 0$ and $|w| = \xi$

want to evaluate the behavior of the Integral as $\xi \rightarrow \infty$

So I have \left|\int_{\Lambda} \frac{e^{i w}}{w^2 + 1} \mathrm{d}w \right| \leqslant \int_{\Lambda} \left|\frac{e^{iw}}{w^2 + 1} \right| \mathrm{d}w

we have $|exp(iw)| \leqslant 1$

so we're left with the denominator. We can get:

$\left| \frac{1}{w^2 + 1} \right| \leqslant \frac{1}{\xi^2 - i^2}$

Hence the integral is:

\left|\int_{\Lambda} \frac{e^{i w}}{w^2 + 1} \mathrm{d}w \right| \leqslant \frac{ \pi \xi}{\xi^2 + 1}

Thus it goes to $0$ if $\xi$ goes to $\infty$

 

[Jorriss]

What are you integrating over? By saying |w| > 0 IMw > 0 are you indicating we are integrating over a semicircle?

 

[Bachelier]

What are you integrating over? By saying |w| > 0 IMw > 0 are you indicating we are integrating over a semicircle?

Exactly, my region $\Lambda$ is where the Imaginary part of $w$ is positive. So we are in the upper half-circle.

 

[Bachelier]

And I am going to let $\xi$ explode, to use a Physicist term.

 

[Jorriss]

Nevermind. You're not enclosing the real axis.

 

[jbunniii]

Your estimate and conclusion look fine to me.