Jordan's lemma

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The path in the z-plane used for Jordan's Lemma. Th

Jordan's Lemma in complex analysis is a powerful tool used frequently when evaluating contour integrals, and real integrals from -∞ to +∞.

Consider a function of the form:

$f(z)=e^{iaz} g(z)\,$

Jordan's Lemma states that

$\lim_{R \to \infty} \int_{C_1} f(z)\, dz = 0 \quad$

where C₁ is the semicircular path of radius R, centred on the origin. The equation of this curve is $z = R e i θ$ .

Jordan's Lemma only holds if

$\mbox{If } a>0, \quad |g \left(z\right)| \to 0 \mbox{ as } R \to \infty$
$\mbox{If } a=0, \quad |g \left(z\right)| \to 0 \mbox{ faster than } 1/z \mbox{ as } R \to \infty$

The maximum is taken over the semicircle in the upper half plane. If $a$ is negative, then the lower half plane must be used.

For example, $g(z)=\frac{1}{1+z^2}$ is a valid function for Jordan's Lemma, but $g(z)=\frac{1}{z}$ is not.

[edit] Applications of Jordan's Lemma

A typical application of Jordan's lemma is one where we want to evaluate an integral, which can be written in the form as shown below, along the real axis. In this case, the integral can be easily evaluated (using the residue theorem) by summing the residues in the upper or lower half planes depending on the sign of a.

Now consider the integral around the closed contour C, composed of C₁ and C₂. By breaking the path C into two parts, we get:

$\oint_{C} f(z)\, dz= \int_{C_1}f(z)\,dz + \int_{C_2} f(z)\,dz$

Because z on the real axis is x, and the path is a straight line, we can now write the second integral as a definite integral with respect to x:

$\oint_{C} f(z)\, dz= \int_{C_1}f(z)\,dz + \int_{-R}^{R} f(x)\,dx$

We also know Jordan's Lemma that the first integral is zero as R tends to infinity, so

$\oint_{C} f(z)\, dz= \int_{-\infty}^{\infty} f(x)\,dx$

The left-hand side is given by the residue theorem, so we can now evaluate the right-hand side for functions f(x) that would otherwise be very difficult to calculate. Note that below, the LHS and RHS have been switched:

$\int_{-\infty}^\infty f(x) dx\,=2\pi i \sum \operatorname{Res\ }{\left[ f(x) \right]}$

where $\operatorname{Res\ }{\left[ f(x) \right]}$ is a residue of $f(x) \,$ and the summations is taken over all residues in the upper half plane (but not on the real axis) when a is positive and in the lower half plane (but not the real axis) when a is negative.