Line integral of complex function

[randybryan]

I have to evaluate this line integral in the complex plane by direct integration, not using Cauchy's integral theorems, although if I see if a theorem applies, I can use it to check.

$$\int (z^2 - z) dz$$

between i + 1 and 0

a) along the line y=x

b) along the broken line x=0 from 0 to 1, and then y=i, from 0 to i

I really have no idea how to tackle this

 

[tiny-tim]

hi randybryan!

use a parameter

 

[randybryan]

Can you be slightly more specific? As in, which parameter should I use?

 

[tiny-tim]

any parameter will do

you choose ​

 

[randybryan]

please just lend a guy a hand here. I have no idea what I'm doing :(

 

[tiny-tim]

choose distance (along the line)

 

[randybryan]

can you not just write down the example? Until the parameter is put into equation form, I have no idea how to use them. It's been a while since I've done parametric integration and all I remember is making x and y functions of the same variable. I'm assuming I do something similar here?

 

[randybryan]

the problem is that the book doesn't give an answer, so I don't know what I'm working towards. The thing I'm having a real trouble getting my head around is the conversion of variables. If z= x + iy, then dz = dx + idy. If I try multiplying everything out by expanding the z^2 and z in terms of x and y, I get a very complicated integral in both dx and dy and then I don't know how to change the upper and lower limits.

 

[tiny-tim]

your parameter could be x or y or x+y or (x+y)/2 or (x+y)/√2 or x² - 7y or …

(basically it can be anything except a function of x-y, since that wouldn't change along the line y=x ! )

you choose ​