Suppose the following conditions are met: 1) a function $f(z)$ is analytic at all points on the upper half plane of the complex plane that are outside of $|z|=R_0$, where $R_0$ is the radius of a circle centered on the origin. ^24cd25 2) We define a contour $C_R$ that's given as $z=Re^{i\phi}$ where $\phi \in (0,\pi)$, meaning that the contour is a semicircle of radius $R$ that lies on the upper half of the complex plane. ^1358c3 3) $\forall z$ on the contour there is a positive constant $f_{max}$ such that $|f(z)|\leq f_{max}$ and $\lim_{R\rightarrow \infty} f_{max} =0.$ ^3ea7e2 Then the [contour integral](Contour%20integrals.md) over the semicircle as it expands to infinity is: $\lim_{R\rightarrow \infty} \int_{C_R}dz f(z)e^{iaz} = 0$ where $a$ is any real positive constant. ^16a424 Over the lower half of the complex plane we use $e^{-iaz}$ and $|f(z)|>f_{max}$.  --- # Proofs and examples ## Proof of Jordan's lemma     #MathematicalFoundations/Analysis/ComplexAnalysis/Functions #MathematicalFoundations/Analysis/ComplexAnalysis/Integrals