Residue Theorem
[ 1 ] Statement :
- If zo Is An Isolated Singularity In A Region C Then
Integration
∮
f(z) dz = 2πi × (Residue
At z = zo )
- If There Are More Than One Isolated Singularities Then,
∮
f(z) dz = 2πi × {Sum
Of Residue Of f(z)}
[ 2 ] Proof :
- Let Zo Is An Isolated Singularity In A Region Of A Closed Curve
C To Find ∮ f(z) dz Expand The Given Function f(z) According To
Laurent Series At z = zo,Which Is Converges At z = z The
Laurent Series Can Be Written As f(z) = ao + a1(z-zo) + a2(z-
zo)^2 +........ + (b1/z - zo) + (b2/z - zo)^2 + .....
- Now Draw A Small Circle Of Radius ρ & Center zo As Shown
Above Figure By Chauchy's Theorem The Integrals Of The 'a'
Part Of The
Laurent Series Be Analytic To Evaluate The Integrals
Of The 'b' Part Of The Series,We Replsce The Integral Around C
By Integrals Arounds C' With Radius
Ρ And Center zo.
- Calculating The Integrals Of The b1 Terms Of Above Series Is,
- If There Are Sevaral Isolated Singular Point Such That
zo,z1,z2.... Then We Draw A Small Circle About Each As Shown
In Figure So That Function f(z) Is Analytic Inside Region
Between C And Circles. According To Cauchy's Integral Formula,
We Find That The Integral Around Circle ,Is The Sum Of The
Integral Arounds Circle Is 2πi Times The Residue Of f(z) At
Singular Points Inside The Region.
- Thus We Have Generalized Form Of Residue Theorem
∮
f(z) dz = 2πi × {Sum
Of Residus Of Function f(z) Inside C}
[ 3 ] Application Of Residue Theorem :
(1) Evaluating Integrals:
The Residue Theorem Can Be Used To Compute Definite
Integrals That Might Be Challenging Using Traditional Methods.
It's Particularly Useful For Integrals Involving Trigonometric,
Exponential, Or Rational Functions.
(2) Inverse Laplace Transforms:
In Engineering And Physics, The Residue Theorem Can Be
Used To Find Inverse Laplace Transforms, Which Are Crucial
In Solving Differential Equations In The Time Domain Given
Their Laplace-Transformed Counterparts.
(3) Real Integrals:
The Residue Theorem Can Sometimes Be Used To Evaluate
Real Integrals By Considering Complex
Contour Integrals.
(4) Electromagnetic Theory:
In Electromagnetic Theory, The Residue Theorem Is Used To
Solve Problems Related To Electromagnetic Fields, Wave
Propagation, And Scattering.
(5) Quantum Mechanics:
The Residue Theorem Plays A Role In Solving Problems
Related To Quantum Mechanics, Including The Behavior Of
Particles In Potentials And The Calculation Of Transition
Probabilities.
(6) Number Theory:
The Residue Theorem Has Applications In Number Theory,
Especially In Problems Involving Modular Arithmetic And
Congruences.
#ResidueTheorem
#ComplexAnalysis
#ComplexIntegration
#CauchyResidueTheorem
#PolesAndResidues
#ComplexContourIntegration
#ComplexFunctionTheory
#ResidueCalculus
#Singularities
#ContourIntegration
#ComplexIntegrationTechniques
#MeromorphicFunctions
#LaurentSeries
#JordanLemma
#WindingNumber
#ClosedCurveIntegration
#CauchyIntegralFormula
#ComplexFunctionResidues
#ComplexPathIntegration
#ResidueIntegration
#CauchyPrincipalValue
