- #1

Rectifier

Gold Member

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**The problem**

I am trying to calculate the integral $$ \int_{\gamma} \frac{z}{z^2+4} \ dz $$

Where ## \gamma ## is the line segment from ## z=2+2i ## to ## z=-2-2i ##.

**The attempt**

I would like to solve this problem using

__substitution__and a

__primitive function to__## \frac{1}{u} ##.

*I am not interested in alternative ways of solving this problem right now, thank you for your understanding.*

## \int_{\gamma} \frac{z}{z^2+4} \ dz = [u=z^2+4 \, \ \ du = 2z dz] = \\

= \frac{1}{2} \int_{\gamma} \frac{1}{u} \ du = \frac{1}{2} \left[ log(u) \right]_{\gamma} ##

## log(u) = ln|u| + iarg(u) ##

**I am not used to line segments that pass through the origin. I don't now which branch to choose for the argument of the complex logarithm since I was thought that when you make the cuts for the branches you cut out 0 and go to infinity on one of the axes. (**

*The trouble is:**Is that perhaps wrong?*)

Example:

**Principal branch**for angles ##-\pi<Arg<\pi##

How do I proceed?