- #1

Mayhem

- 322

- 213

- Homework Statement
- Integrate ##\int_{\Gamma} z^7 dz## along the line ##5i## to ##3 + i##

- Relevant Equations
- ##z = x + iy##

First I parameterize ##z## by ##z(t) = 5i + (3 + i - 5i)t## such that ##z(0) = 5i## and ##z(1) = 3 + i##, which means that ##0 \leq t \leq 0## traces the entire line on the complex plane. By distributing ##t##, we achieve a parameterized expression of the form ##z(t) = x(t) + iy(t)##

$$z(t) = 3t + i(5 - 4t)$$

Then to rewrite the contour integral in terms of ##t##, we determine ##dz/dt##

$$\frac{dz}{dt} = 3 - 4i$$

Yielding

$$\int_{\Gamma} z^7 dz = \int_{0}^{1} z^7 \frac{dz}{dt} dt = (3-4i)\int_{0}^{1} (3t + i(4-5t))^7 dt$$

Not sure where to go from here. The naive approach would be to expand the binominal and factor out all constants (including imaginary units ##i##), but that is tedious. Is there a trick? If this was a calc I problem, I'd just do a simple u-substitution, but not sure if the same logic holds when ##i## is involved.

$$z(t) = 3t + i(5 - 4t)$$

Then to rewrite the contour integral in terms of ##t##, we determine ##dz/dt##

$$\frac{dz}{dt} = 3 - 4i$$

Yielding

$$\int_{\Gamma} z^7 dz = \int_{0}^{1} z^7 \frac{dz}{dt} dt = (3-4i)\int_{0}^{1} (3t + i(4-5t))^7 dt$$

Not sure where to go from here. The naive approach would be to expand the binominal and factor out all constants (including imaginary units ##i##), but that is tedious. Is there a trick? If this was a calc I problem, I'd just do a simple u-substitution, but not sure if the same logic holds when ##i## is involved.