How Do You Solve a Complex Contour Integral with a Non-Standard Path?

[dan280291]

Hi I'm really not sure how to start this question. I could do it if it was in terms of z but I'm not sure if trying to change the variable using z = x + iy is correct. If anyone could suggest a method I'd appreciate it.

∫(x³ - iy²)dz

along the path z= $\gamma(t)$ = t + it³, 0≤t≤1

Thanks

 

[pasmith]

Hi I'm really not sure how to start this question. I could do it if it was in terms of z but I'm not sure if trying to change the variable using z = x + iy is correct. If anyone could suggest a method I'd appreciate it.

∫(x³ - iy²)dz

along the path z= $\gamma(t)$ = t + it³, 0≤t≤1

Thanks

I think it's safe to assume that $z = x + iy$ if nothing to the contrary is given.

You have $z = \gamma(t)$ so $dz = \gamma'(t)\,dt$ and $x$ and $y$ are respectively the real and imaginary parts of $\gamma(t)$.