Jordan's Lemma is a small but important mathematical result that is useful in contour integration. In complex analysis we often wish to integrate functions around large semicircles in the complex plane, and Jordan's lemma provides useful information about how these integrals behave.

#### Statement of result

Let

```J = integral(g(z)exp(ikz)dz, z=Rexp(iθ), θ=0...π)
```

where k>0, and |g(z)| tends to zero as |z| gets large. Then J tends to zero 0 as R tends to infinity.

#### Proof

First define

```M(R) = max(|g(Rexp(iθ))|, θ=0...π)
```

Since |g(z)| tends to zero as |z| gets large, we deduce that M(R) tends to zero as |z| tends to zero. Hence, using the identity sin(θ) >= 2θ/π for 0 <= θ <= π/2, we have

```|J| ≤ integral(|g(Rexp(iθ)|exp(-kRsin(θ))Rdθ, θ=0...π)
≤ R M(R) integral(exp(-kRsin(θ))dθ, θ=0...π)
= 2R M(R) integral(exp(-kRsin(θ))dθ, θ=0...π/2)
≤ 2R M(R) integral(exp(-kR 2θ/π)dθ, θ=0...π/2)
= R M(R) π/(kR) (1 - exp(-kR))
≤ M(R) π/k,
```

which tends to zero as R increases, as required.