On the Residue Theorem


On the Residue TheoremMathworldPlanetmath Swapnil Sunil Jain December 26, 2006

On the Residue Theorem

The Residue Theorem

If γ is a simply closed contour and f is analytic within the region bounded by γ except for some finite number of poles z0,z1,,zn then

γf(z)𝑑z = 2πik=0nResz=zkf(z)

where Resz=zkf(z) is the reside of f(z) at zk.

Calculating Residues

The Residue of f(z) at a particular pole p depends on the characteristicPlanetmathPlanetmath of the pole.

For a single pole p, Resz=pf(z)=limzp[(z-p)f(z)]

For a double pole p, Resz=pf(z)=limzp[ddz(z-p)2f(z)]

For a n-tuple pole p, Resz=pf(z)=limzp[1(n-1)!d(n-1)dz(n-1)(z-p)nf(z)]

Evaluation of Real-Valued Definite Integrals

We can use the Residue theorem to evaluate real-valued definite integral of the form

02πf(sin(nθ),cos(nθ))𝑑θ (1)

If we let z=eiθ, then dzdθ=ieiθ=iz which implies that dθ=dziz. Then using the identityPlanetmathPlanetmath cos(nθ)=12(zn+z-n) and sin(nθ)=12i(zn-z-n), we can re-write (1) as

γg(z)dziz (2)

where g(z)=f(12i(zn-z-n),12(zn+z-n)) and γ is a contour that traces the unit circle. Then, by the Residue theorem, (2) is equal to

2πik=0nResz=zk(g(z)iz)=2πiik=0nResz=zk(g(z)z)=2πk=0nResz=zk(g(z)z)

where zk are the poles of g(z)z.

Title On the Residue Theorem
Canonical name OnTheResidueTheorem1
Date of creation 2013-03-11 19:29:41
Last modified on 2013-03-11 19:29:41
Owner swapnizzle (13346)
Last modified by (0)
Numerical id 1
Author swapnizzle (0)
Entry type Definition