Cauchy's Integral Formula

 Cauchy's Integral Formula

Cauchy's Integral Formula

 

[ 1 ] Statement :

 

- if f(z) Is Analytic On And Inside A Single Closed Curve C The

  Value Of Fuction f(z) At Point z=a Inside C Is Given By The

  Following Counter Integral Along C.

  f(a) = 1/2πi  ∮ (f(z)/z-a)dz

 

[ 2 ] Proof :

 

- Let 'a' Be A Fixed Point Inside The Simple Closed Curve C And Consider.

  φ(z) = f(z)/z-a

  Wher f(z) Is Analytic On And Inside Curve C.

- Let C' Be A Small Circle (Inside C) With A Center At 'A' And

  Radius With ρ.

- According To Cauchy's Theorem Its Contour Integral Along C Is

  Anti Clockwise Along Cut AB & Along C' Is Clockwise Along Cut

  Cd Totally Become Zero In Liting Case.Cut Is Extremly Small

  Therefor Integration Along AB & CD Being Equal And Opposite.

  ∮ φ(z)dz + ∮ φ(z)dz = 0

 (Along C         (Along C'   

  Counter          Counter

  Clockwise)     Clockwise)

 

 ∮ φ(z)dz = - ∮ φ(z)dz = ∮ φ(z)dz

(Along C               (Along C'                (Along C' 

Counter                 Counter                  Counter

 Clockwise)           Clockwise)             Clockwise)

  

- Here Integration Has Been Taken In Such A Way That Area Of

  The Region Always Remains On The Left  Of Integration.

   ∮ φ(z)dz =  ∮ φ(z)dz

  c                             c'

                    =  ∮ (f(z)/z-a)dz

                      c'

- Now Take z = a + ρe^(iθ)

         z-a = ρe^(iθ)

          dz = iρe^(iθ)dθ

- At A Point z → a The Function f(z) → f(a)

- Which Is Known As Cauchy's Integral Formula.

#CauchysIntegralFormula

#ComplexAnalysis

#ResidueTheorem

#ContourIntegration

#ComplexFunctions

#HolomorphicFunctions

#CauchyResidueTheorem

#CauchyIntegralTheorem

#AnalyticFunctions

#ComplexIntegration