Math Is Fun Forum

coffeeking

Member

Registered: 2007-11-18

Posts: 44

Indented Contour

Hi, below is a question on indented contour where I have the solution but don't understand what's going on.

The question goes like this :

Find

and the solution goes something like:

By defining indented contour on a rectangle contour

around the point

and

Well my question is why we take only half the residue on

and

? Isn't it by Cauchy's Residue Theorem that for a closed positively oriented contour

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Indented Contour

A lot of times, you take f(z) to be different than f(x) (ignoring the abuse of notation).  Make sure that isn't happening here.  Otherwise, you seem to be correct.

---

"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..."

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Indented Contour

By defining indented contour on a rectangle contour

Does this mean that your contour is defined so your singularities lie outside of it?  If that's the case, then you have no residues.

---

"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: Indented Contour

The contour

for this question goes something like:

                            __<__
                           /         \
                          | 2(pi)i  |
-R+2pi(i) -----<---    .       -----<-------  R+2pi(i)
              |                                        |
              |                                        |
              |                                        |
              v                                       ^
              |                                        |
              |                                        |
        -R   |______>   .     ____>_____| R
                          |    0    |
                           \__>_/

Where the contour around

and and

is a semi-circular arc.