# The Limit of a Complex Integral

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##

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

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.

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

Nevermind. You're not enclosing the real axis.

jbunniii