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


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)
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