Jordan's Lemma

Jordan’s Lemma describes the behaviour of a contour integral on the semicircular arc or radius (excluding the real axis) in the upper half of the complex plane as the radius of the semicircle tends to infinity.

Jordan's Lemma

If the only singularities of F(z) are simple andasthen

ifand |F (z)| → 0 as R → ∞.

Proof: Sinceis the semi-circleand

Nowin the upper half plane, andtends to zero faster than any power of R increases ifso by splitting the semicircular contour into three parts,

we obtainandd similarly forand forsince as stated,tends to 0 faster than any power ofincreases as rightarrow infinity , so

Whenifor a rational function ofwith the degree of the denominator higher than the degree of the numerator can all be used.

Ifthen a modification is needed. The degree of the denominator must be at least two greater than the degree of the nureator.

Ifthen we can still apply the Lemma by taking our contour in the lower half plane.