Math Is Fun Forum

coffeeking

Member

Registered: 2007-11-18

Posts: 44

Complex Integration

Well, I have the solution to the problem below, just that I don't understand what is going on

Let

be a simple closed positively oriented contour that passes through the point

. Set

Find

, where z approaches

from inside

The solution given is:

Since

is analytic on

, by Cauchy's Integral Formula,

for

inside

Therefore

by continuity of cosine function.

Well, the thing I don't understand is why

? I don't see any link of it to Cauchy Integral Formula, also what

is refering to in this context?

In addition I am totally confuse of why we replace

by

and

by

in Cauchy Integral Formula as I don't understand what is the rationale of doing so and more importantly what we mean by

and integration with respect to

in this context.

I will be more than grateful if any kind soul could explain this to me as I have been struggling to understand many of the concepts even since I taken this course...

PS. Sorry if my question sound stupid or ridiculously easy... I am just someone who are trying his very best to learn...

Thanks in advance.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Complex Integration

What's in a name?  zeta here is just a dummy variable, and nothing more.  You need to integrate with respect to something.  Cauchy's integral formula says (under the proper restrictions):

Again, w is just a dummy variable, it could be any letter (one of my friends likes to integrate with respect to symbols such as a smiley face or a boat).

Cauchy's integral formula tells you a rather startling fact: the value of an analytic function is determined by an integral around a curve.  In order to integrate around a curve, we introduce a variable that takes on the values of Gamma, in my post it's w, in yours its zeta.

Does that clear things up?

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

coffeeking

Member

Registered: 2007-11-18

Posts: 44

Re: Complex Integration

hmm.. Think that should clear things up thanks Ricky. However, there is another thing about Laurent Series that I don't understand

Laurent Series states that:

Let

be analytic in the annulus

. Then

can be expressed there as

Where

is given by:

Well, same confusion here... The first statement with the summation sign are all in

and

however, the second statement that tell us about

are in

... So what

actually means in this context?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Complex Integration

Again, it is just a dummy variable.

It doesn't matter what letter (or thing) you integrate with respect to, but you need to integrate with respect to something.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."