Contour integral around a triangle: why is the result zero?

[elimenohpee]

Homework Statement

Let C be a contour formed by the points O(0,0), A(1,0), B(1,1), with the direction OA->AB->BO. By using the definition of a contour integral, evaluate:

(integral) f(z)dz

Homework Equations

$$\int f[z(t)]z'(t)dt$$

The Attempt at a Solution

I didn't include the work I've done, or even the function upon which I am integrating. I'm looking more for an understanding.

I perform the line integrals about each side of the triangle, and sum each value at the end. But I end up with 0, which I think makes sense since this is a closed region. But should the value equal zero?

 

[Dick]

It might be zero. But since you skipped telling us what the function f(z) is, it's hard to say. It also might not be zero. Is that what you wanted to know? What is f(z)?

 

[elimenohpee]

f(z) = e^pi*z

Its a little too much to try and type all the work I did computing each integral.

 

[Dick]

f(z) = e^pi*z

Its a little too much to try and type all the work I did computing each integral.

That's fine. Since e^(pi*z) doesn't have any poles, then you are ok. The integral should come out to be zero. It's analytic inside the triangle. That's Cauchy's integral theorem. There are other functions that don't satisfy that criterion.

 

[elimenohpee]

That's fine. Since e^(pi*z) doesn't have any poles, then you are ok. The integral should come out to be zero. It's analytic inside the triangle. That's Cauchy's integral theorem.

Thank you, exactly what I figured and actually worked out. Just wanted to verify.

Thanks again!