Regarding a Complex Integral Along a Semi-Circle

Question

I'm interested in the integral of $f(z)=|z|$ along

• the line between $-i$ and $i$
• the line between $i$ and $-i$
• the semi-circle starting at $-i$ and ending at $i$

and I would need someone to verify whether my calculations are correct:

• For the first I have $\gamma_1(t)=i(2t-1)$ for $t\in [0,1]$ and thus$\int_{\gamma_1}|2t-1|\cdot 2idt=i$.
• For the second I have $\gamma_2(t)=-\gamma_1(t)$ and thus $\int_{\gamma_2}f(z)dz=-i$
• For the third I have $\gamma_3(t)=e^{it}$ for $t\in[3\pi/2,5\pi/2]$ and thus $\int_{\gamma_3}f(z)dz=\int_{3\pi/2}^{5\pi/2}ie^{it}dt=2i$

Does this look fine?

Answer 1

1. There is a problem with the notation that you are using. Instead of $\int_{\gamma_1}\lvert2t-1\rvert\cdot2i\,\mathrm dt$, you should write $\int_0^1\lvert2t-1\rvert\cdot2i\,\mathrm dt$. But the answer is correct.
2. Same problem here.
3. It's just fine.