Evaluating Complex Integrals Using Cauchy's Integral Formula

In summary, the conversation was about using Cauchy's integral formula to evaluate an integral over two different curves: the unit circle and the circle mod(Z)=2. The formula requires the point z0 to be contained inside of the curve being integrated around, which was discussed in relation to the two curves. The use of a different point, z-Pi/2, was also mentioned as a possible solution for the unit circle case.
  • #1
AlBell
11
0

Homework Statement



Use Cauchy's integral formula to evaluate
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when
a) C is the unit circle
b) c is the circle mod(Z)=2

Homework Equations



I know the integral formula is
14mtco.png



The Attempt at a Solution


for the unit circle I was attempting F(z)=sin(z) and Z0=∏/2, which would give a solution of 2∏i, however if this is the correct method I am unsure how to modify it for a larger unit circle as I thought the final result was independent of radius
 
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  • #2
The integral formula requires the point z0 to be contained inside of the curve gamma that you are integrating around. Draw some pictures and you should see the difference between the two curves they are asking you to integrate on
 
  • #3
Ah so the unit circle wouldn't actually contain the point pi/2 whereas the circle mod(z)=2 would?
 
  • #4
That's right. So in the unit circle case you need to figure out something else that let's you calculate the integral
 
  • #5
Can I then use the integral theorem that says it will equal 0?
 
  • #6
That will work
 
  • #7
I'm a bit confused again, sorry!
I thought that the z-Pi/2 on the denominator of the integral means we just shift the origin of the circle to a new position?
 

1. What is the Cauchy Integral Formula?

The Cauchy Integral Formula is a fundamental theorem in complex analysis that relates the values of a holomorphic function inside a closed contour to its values on the contour itself. It states that if f(z) is a holomorphic function inside a closed contour C, then the value of f(z) at any point inside the contour is equal to the average value of f(z) around the contour.

2. How is the Cauchy Integral Formula used in complex analysis?

The Cauchy Integral Formula is used to evaluate complex integrals, which are difficult to compute using traditional methods. It allows for the calculation of integrals by simply knowing the values of the function on the contour, making it a powerful tool in solving problems in complex analysis.

3. What are the conditions for the Cauchy Integral Formula to hold?

The Cauchy Integral Formula holds for any holomorphic function f(z) inside a closed contour C. Additionally, the contour C must be continuously differentiable and the function f(z) must be continuous on and inside the contour.

4. How is the Cauchy Integral Formula related to Cauchy's Theorem?

Cauchy's Theorem states that the integral of a holomorphic function around a closed contour is equal to zero. The Cauchy Integral Formula is a direct consequence of this theorem, as it provides a way to evaluate the integral by using the values of the function on the contour.

5. What are some applications of the Cauchy Integral Formula?

The Cauchy Integral Formula has various applications in complex analysis and other fields, such as physics and engineering. It is used to solve problems involving complex integrals, to evaluate real integrals by using contour integration, and to prove other theorems in complex analysis. It is also used in the development of other important mathematical concepts, such as Laurent series and the residue theorem.

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