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# Please help with this contour integral?

We have to solve the integral for C of (x^2 - iy^3)dz and C is the straight line from z=1 to z=i .

I know I have to parametrize C, but I'm not sure how to do that.

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- kbLv 78 years agoFavorite Answer
C: z(t) = 1 + (i - 1)t for t in [0, 1].

==> z(t) = (1 - t) + it.

Alternately, we can write this as x = 1 - t, and y = it.

So, the integral equals

∫(t = 0 to 1) [(1 - t)^2 - i(it)^3] * (d/dt)(1 + (i - 1)t) dt

= ∫(t = 0 to 1) [(1 - t)^2 - t^3] * (i - 1) dt

= [-(1 - t)^3/3 - t^4/4] * (i - 1) {for t = 0 to 1}

= (1/12) (i - 1).

I hope this helps!

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