Question

Cauchy Integral Formula...

Answer 1

Here's the problem:
Determine the value of
\[(\int\limits)_C[z^3+2z+1/(z−1)^3]dz\]
where C is the contour |z|=2.

Note: That's a line integral being evaluated at C. I did my best to make it look that way. :P

Answer 2

I'm going to keep reposting this until I get the help I need...

Answer 3

I ma looking up how to do it. I don't quite remember.

Answer 4

I have it. Let me type it up.

Answer 5

Okay, First it is important to note that you have a pole of order 3 in your denominator at 1.

Answer 6

so you have something in the form \[f(z)=\psi(x)/(z-z_0)^m\].
To find the Residue of this the formula is:
\[Res_{z = z_0}(f(z))=(\psi(z_0))^{m-1}/(m-1)!\]

Answer 7

so let \[\psi(z)=z^3+2z+1. Therefore, \psi^{(m-1)}(z_0)=\psi''(z_0)=6z\]

Answer 8

Evaluating that you get \[\psi''(1)=6(1)=6.\] From there we know that the contour integral is \[i2\pi*Res=i2\pi*3=i6\pi.\]

Answer 9

Also, I forgot to note, the reason it is only 2pii times the residue is because the contour |z|=2 is simply connected.

Answer 10

Also, I noticed a mistype. \[\psi''(1)=6.\] But notice, you have to divide it by m-1 or in this case 3-1=2. Thats where the i2pi*THREE came from.

Answer 11

Hrm...that's not the solution I got.

Here's what I got:
\[(\int\limits)_C[z^3+2z+1/(z−1)^3]dz\]Considering the form of the denominator in the integrand, I used the Cauchy Integral Extension,\[f^{(n)}(z_0)=n!/2\pi i (\int\limits)_C [f(z)/z-z_0)^{n+1}]dz\]where n=2. Substituting that, I get\[f^{(2)}(z_0)=2!/2\pi i (\int\limits)_C [f(z)/z-z_0)^{3}]dz\]Since \[z_0=1,\] we can rewrite the equation as\[(2\pi i /2)f^{(2)}(1)=(\int\limits)_C [f(z)/(z-1)^3]dz\]since the formula, with f(z)=z³+2z+1, yields the value of the integral. Thus, we find that\[(\int\limits)_C [(z^3+2z+1)/(z-1)^3]dz=\pi i [d^2(x^3+2z+1)/dz^2]|_{z=1}\]\[=\pi i(6z)|_{z=1} =6\pi i\]If the contour C were |z|=1/2, I could instead use the Cauchy-Gourset Theorem because (z³+2z+1)/(z-1)³ is analytic and inside the circle.

Answer 12

We still both got \[6 \pi i\] :D

Answer 13

Haha...yeah we did...sorry about that, I must've missed your final step.

Answer 14

No problem :P Let me know if you need anything else!

Answer 15

OMG...Whats the name of that troll face?

Answer 16

The raisins face xP