# Thread: How To Do Integral Of Complex Conjugate?

1. ## How To Do Integral Of Complex Conjugate?

$\displaystyle \int\limits_{1+i}^{2+i}s\, dx$

Where s is the complex conjugate of x

I have the answer but I don't have the foggiest idea how it is obtained.

2. Originally Posted by soma
$\displaystyle \int\limits_{1+i}^{2+i}s\, dx$

Where s is the complex conjugate of x

I have the answer but I don't have the foggiest idea how it is obtained.

Define $\displaystyle x = e^{i\theta}$.

Then $\displaystyle \overline{x} = e^{-i\theta}$

and $\displaystyle dx = i\,e^{i\theta}$.

Also note that $\displaystyle \theta = \frac{\log{x}}{i}$.

So the integral becomes

$\displaystyle \int{\overline{x}\,dx} = \int{e^{-i\theta}\,i\,e^{i\theta}\,d\theta}$

$\displaystyle = \int{i\,d\theta}$

$\displaystyle = i\theta + C$

$\displaystyle = i\left(\frac{\log{x}}{i}\right) + C$

$\displaystyle = \log{x} + C$.

Now you should be able to substitute the limits of integration.

Remember that $\displaystyle \log{x} = \ln{|x|} + i\,\textrm{arg}\,{x}$.

,

,

,

,

,

,

### integration of complex conjugate

Click on a term to search for related topics.