Thread: 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

well a unit circle in 1 not z= 2

try putting z into polar form and then do a contour integral

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

would it be written then |z-2|>2

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 to differentiate it from the variable .

The unit circle about with radius 1 is and with a going from 0 to .

Then .

However, you should know that is analytic everywhere except where z= 0 which, with , means everywhere inside this contour.

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 at the origin until the center rests at -2i. Now, how does the value of the integral change as the circle falls?