Semicircular contour theorem: Difference between revisions

Latest revision as of 19:18, 18 May 2008

Statement

Version where the integral over the contour tends to zero

Suppose $f$ is a meromorphic function on an open subset of $C$ containing the closed upper half-plane. Further, suppose there exists $k > 1$ and a constant $M$ such that for all sufficiently large $R > 0$ , we have:

$| f (z) | \leq \frac{M}{| z |^{k}}$

for $z$ in the upper half-plane. Then, if $γ_{R}$ denotes the semicircular arc of radius $R$ centered at zero, then:

$lim_{R \to \infty} | \int_{γ_{R}} f (z) d z | = 0$

In particular, if $f$ has no poles on the real axis, and it has finitely many poles $z_{1}, z_{2}, \dots, z_{n}$ in the upper half-plane, we get:

$P V \int_{- \infty}^{\infty} f (x) d x = 2 π i \sum_{j} r e s (f; z_{j})$

@@ Line 1: / Line 1: @@
 {{wikilocal}}
+==Statement==
+===Version where the integral over the contour tends to zero===
 Suppose <math>f</math> is a [[meromorphic function]] on an open subset of <math>\mathbb{C}</math> containing the closed upper half-plane. Further, suppose there exists <math>k > 1</math> and a constant <math>M</math> such that for all sufficiently large <math>R > 0</math>, we have:
@@ Line 11: / Line 15: @@
 In particular, if <math>f</math> has no poles on the real axis, and it has finitely many poles <math>z_1, z_2, \ldots, z_n</math> in the upper half-plane, we get:
-<math>\operatorname{PV} \int_{-\infty}^\infty f(x) \, dx = \sum_j \operatorname{res}(f;z_j)</math>
+<math>\operatorname{PV} \int_{-\infty}^\infty f(x) \, dx = 2\pi i \sum_j \operatorname{res}(f;z_j)</math>