Complex Analysis: Cauchy's Theorem

Click For Summary
SUMMARY

Cauchy's Theorem states that if functions u and v satisfy the Cauchy-Riemann equations within and on a simple closed contour C, then the integral of the function f(z) equals zero. In the example provided, with f(z) = 1/(z+20) and a closed contour defined as a circle around the origin with a radius of 1, the integral is indeed zero because the singularity at z = -20 lies outside the contour. This confirms that f is differentiable within and on the boundary defined by |z| = 1.

PREREQUISITES
  • Understanding of complex functions and their properties
  • Familiarity with the Cauchy-Riemann equations
  • Knowledge of contour integration in complex analysis
  • Basic concepts of singularities in complex functions
NEXT STEPS
  • Study the implications of Cauchy's Integral Theorem in complex analysis
  • Explore examples of contour integrals involving singularities
  • Learn about the application of residue theorem in evaluating integrals
  • Investigate the relationship between analytic functions and their integrals
USEFUL FOR

Students preparing for exams in complex analysis, mathematicians focusing on theoretical aspects of calculus, and educators teaching advanced mathematics concepts.

jokusan13
Messages
1
Reaction score
0
As I am studying for an exam I am trying to wrap my head around the concepts I learned. I want to make sure I fully understand the concepts before the exam in 1.5 weeks.

Cauchy's Theorem
If u and v satisfy the Cauchy-Riemann equations inside and on the simple closed contour C, then the integral of f(z)=0

Now for example, if we have f(z)=1/(z+20) and our closed contour is a circle around the origin with radius=1. If I am understanding this correctly, we can say that the integral is equal to 0 since the 'bad point' of z=-20 is outside of the circle correct meaning that f is differentiable in and on |z|=1.

Does this sound correct?
 
Physics news on Phys.org
Yes that sounds correct.
 

Similar threads

Graduate Cauchy’s integral theorem and residue theorem, what’s the difference
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Find f(z) given f(x, y) = u(x, y) + iv(x,y)
  • · Replies 8 ·
Replies
8
Views
1K
Graduate Question about proof of Great Picard Theorem
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Why are analyticity and convergence related in complex analysis?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Existence of a limit implies that a function can be harmonic extended
  • · Replies 1 ·
Replies
1
Views
3K
Graduate A question about a complex integral
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Apparent counterexample to Cauchy-Goursat theorem (Complex Analysis)
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad What is the value of the integral for higher order poles in the real axis?
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Contour integration over Riemann surface
  • · Replies 10 ·
Replies
10
Views
4K
Undergrad When and How Does an Integral Have a Cauchy Principal Value?
  • · Replies 3 ·
Replies
3
Views
4K