# contour integrals

• Feb 28th 2010, 02:29 PM
stumped765
contour integrals
Compute integral of:
Integral over C of 1/z dz
where C is the unit circle centered at some point z.
|z.| >2
• Feb 28th 2010, 07:07 PM
wowsomeme
well a unit circle in 1 not z= 2

try putting z into polar form and then do a contour integral
• Mar 1st 2010, 12:05 PM
stumped765
the center of the circle is at a point z, |z|>2 its just saying the unit circle is shifted from the origin..
• Mar 1st 2010, 12:08 PM
wowsomeme
would it be written then |z-2|>2
• Mar 1st 2010, 12:59 PM
HallsofIvy
Quote:

Originally Posted by stumped765
the center of the circle is at a point z, |z|>2 its just saying the unit circle is shifted from the origin..

Then use something like $z_0$ to differentiate it from the variable $z$.

The unit circle about $z_0$ with radius 1 is $|z-z_0|= 1$ and $z= z_0+ e^{i\theta}$ with $\theta$a going from 0 to $2\pi$.

Then $dz= ie^{i\theta}d\theta$ $\oint \frac{1}{z}dz= \int_0^{2\pi}\frac{ie^{i\theta}}{z_0+ e^{i\theta}}d\theta$.

However, you should know that $\frac{1}{z}$ is analytic everywhere except where z= 0 which, with $|z- z_0|> 2$, means everywhere inside this contour.
• Mar 1st 2010, 02:19 PM
shawsend
Quote:

Originally Posted by stumped765
Compute integral of:
Integral over C of 1/z dz
where C is the unit circle centered at some point z.
|z.| >2

Hi. Here's something fun to check your knowledge about contour integrals: pin the center of the unit circle at the point 2i. Now, drop it straight through the singular point of $1/z$ at the origin until the center rests at -2i. Now, how does the value of the integral $\mathop\oint\limits_{C} \frac{1}{z}dz$ change as the circle falls?