(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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]

2. Relevant equations

3. 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]

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# Question integrating |f(z)||dz| over a contour C

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