- #1
opticaltempest
- 135
- 0
Homework Statement
I want to compute [tex]\int_{C}^{}{|f(z)||dz|}[/tex] along the contour C given by the curve [tex]y=x^2[/tex] using endpoints (0,0) and (1,1). I am to use [tex]f(z)=e^{i\cdot \texrm{arg}(z)}[/tex]
Homework Equations
The Attempt at a Solution
I know that for all complex numbers z, [tex]|e^{i\cdot \texrm{arg}(z)}}|=1[/tex]. So now I am looking at the integral [tex]\int_{C}^{}{1|dz|}[/tex]
Is the approach I take below correct?
A complex representation of [tex]C[/tex] can be given by [tex]\gamma (t)=t+it^2[/tex] for [tex]0 \leq t \leq 1[/tex]. Then [tex]\gamma^{'}(t)=1+i2t[/tex]. We have
[tex]\int_{C}^{}{|f(z)||dz|}[/tex]
[tex]=\int_{0}^{1}{\gamma^{'}(t)\bigg|\frac{dz}{dt}\bigg||dt|}[/tex]
[tex]=\int_{0}^{1}{(1+i2t)|dt|}[/tex]
[tex]=[t+it^2]_{t=0}^{t=1}[/tex]
[tex]=1+i[/tex]