Jordan's lemma

In complex analysis, Jordan’s lemma is a theorem that used with Cauchy’s residue theorem to evaluate contour integrals, especially in the context of the principal value integral.

Suppose $f(z)=\frac{p(z)}{q(z)}$ where $p,q$ are polynomials and the degree of $q>p+1$. Then:

$$\underset{R\to \infty }{\lim }{\int }_{C_R^{+}}{e}^{i\alpha z}\frac{p(z)}{q(z)}dz=\underset{R\to \infty }{\lim }{\int }_{C_R^{+}}{e}^{i\alpha z}f(z)dz=0$$

for $\alpha >0$. This is not true if $\alpha <0$, in which case we must take the negative semicircle to close the contour:

$$\underset{R\to \infty }{\lim }{\int }_{C_R^{-}}{e}^{i\alpha z}\frac{p(z)}{q(z)}dz=0$$

We think about this because the exponential would grow infinitely for certain cases.

THIS IS IMPORTANT. DO NOT TURN YOUR BRAIN OFF AND ONLY CHOOSE THE UPPER SEMICIRCLE. THINK BEFORE YOU INTEGRATE.

When we do $C_R^{-}$, we actually work clockwise (think about this for a second).

Related theorem

For a similar theorem, we have the degree of $q>p+2$, then:

$$\underset{R\to \infty }{\lim }{\int }_{C_R^{+}}f(z)dz=0$$