Cauchy's Theorem
[1] Cauchy's Theorem
- Let C Be Any Simle Closed Curve With A Continuously Turning
Tangent Except Possibly Of A Finite Number Of Point I.E. We
Allowe A Finite Numbers Of Corners But Otherwise The Curve Is
Smooth.
- If F(Z) Is Analytics On And Insid Curve C Then
∮
f(z) dz = 0
- Here The Line Integration Shown In Above Equation Is Called
Counter Integral.
[2] Proof :
- let
z = x + iy
dz = dx + idy
- function f(z) = u(x,y) + iν(x,y)
f(z) = u +
iν
- ∮f(z)dz = ∮ (u + iν) + (dx + idy)
= ∮
(udx + iudy + ivdx - vdy) (i*i = -1)
= ∮
(udx - vdy ) + i∮(udy + vdx)......(1)
- Now ,Using Stoke's Theorem,Which Special Case Of Green
Function
- The Stoke's Theorem Is,
∮(pdx
+ Qdy) = ∫∫ [∂p/dy
- ∂Q/∂x]dxdy
- Here P,Q & Their Partial Derivative ∂p/dy & ∂Q/∂X Must Be
Continuous C Is A Simple Close Curve In R Using This Stoke's
Theorem In Eq.(1) We Can Write
∮f(z)dz = ∫∫ [∂u/dy -(-∂ν/∂x)]dxdy + i ∫∫ [∂u/dx - ∂ν/
∂y]dxdy.....(2)
- According To Cauchy Reimann's Condition,
∂u/dx = ∂ν/∂y &
∂ν/∂x = - ∂u/dy
- If We Use Above Cauchy Reimann Condition We Get,
∮
f(z) dz = 0
[3] Cauchy's Theorem Use :
- Calculation Of Complex Integrals:
Cauchy's Theorem Allows You To Calculate The Value Of Certain
Complex Integrals Using The Properties Of Holomorphic
Functions.
- Residue Calculus:
Cauchy's Residue Theorem Is A Powerful Consequence Of
Cauchy's Theorem. It Provides A Method For Evaluating Certain
Real Integrals Using Complex Analysis Techniques.
- Solving Differential Equations:
Cauchy's Theorem, Along With Cauchy's Integral Formula, Can
Be Used To Solve Certain Types Of Partial
Differential Equations.
- Inverse Laplace Transforms:
In The Field Of Signal Processing And Control Theory, Cauchy's
Residue Theorem Can Be Used To Compute Inverse Laplace
Transforms And Determine The Behavior Of Systems
Represented In The Laplace Domain.
- Conformal Mapping:
Cauchy's Theorem Is Also Crucial In The Study Of Conformal
Mappings, Which Are Transformations That Preserve Angles
Locally.
- Harmonic Functions:
Cauchy's Theorem Has Implications For Harmonic Functions,
Which Are Solutions To Laplace's Equation.
- Complex Analysis Research:
Cauchy's Theorem Is A Foundational Result In Complex
Analysis, And Many Further Results And Theorems Are Built
Upon It.
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