Thread: Complex Line Integration

1. Complex Line Integration

I'm stuck on a question about complex line integration; I've managed to do some simpler questions, but integrating $\int \exp(it) dt$ is giving me problems. Here's the full question:

Calculate $\int_\gamma |z| dz$ when $\gamma = \{ z : z = \exp(it), -\pi \leq t \leq 0 \}$

So my first step is
$\int_{-\pi}^0 |\exp(it)| * \gamma ' (t) dt$

With $\gamma ' (t) = i \exp(it)$ and $|\exp(it)| = 1$
that becomes
$i \int_{-\pi}^0 \exp(it) dt$

I put that into Wolframm's online integrator and it said that's
$\sin(t) - i \cos(t)$
but I don't get how it works that out, or if it's even correct. TIA

2. Originally Posted by InfernoZeus
I'm stuck on a question about complex line integration; I've managed to do some simpler questions, but integrating $\int \exp(it) dt$ is giving me problems. Here's the full question:

Calculate $\int_\gamma |z| dz$ when $\gamma = \{ z : z = \exp(it), -\pi \leq t \leq 0 \}$

So my first step is
$\int_{-\pi}^0 |\exp(it)| * \gamma ' (t) dt$

With $\gamma ' (t) = i \exp(it)$ and $|\exp(it)| = 1$
that becomes
$i \int_{-\pi}^0 \exp(it) dt$

I put that into Wolframm's online integrator and it said that's
$\sin(t) - i \cos(t)$
but I don't get how it works that out, or if it's even correct. TIA
$\displaystyle \int e^{it} \, dt = \frac{1}{i} e^{it}$ (I have omitted the arbitrary constant since you will use this result to evaluate a definite integral).

3. Originally Posted by mr fantastic
$\displaystyle \int e^{it} \, dt = \frac{1}{i} e^{it}$ (I have omitted the arbitrary constant since you will use this result to evaluate a definite integral).
Cool, I'd already worked that out, but assumed that it was wrong because of what Wolframm told me Thanks for the help

4. What wolfram told you and what you derived are both correct since they are the same by euler's formula!

5. Originally Posted by Vlasev
What wolfram told you and what you derived are both correct since they are the same by euler's formula!
Duh! I should have been able to work that out Especially as I should know Euler's relations for my upcoming exam...

6. Specifically, $e^{it}= cos(t)+ i sin(t)$ so that $i \int_{-\pi}^0 e^{it}dt= i\int_{-\pi}^0 cos(t)dt- \int_{-\pi}^0 sin(t)dt$