principal value mapping

**chloe561** · December 14th 2009, 03:51 PM

Sketch an image under $w = \text{Log} (z)$ of the half disk $|z| < 1, \text{Re} (z) > 0$ .

To this point, I do not have much experience with mappings under the principal value mapping of the logarithm. I know that

$\text{Log} (z) := \ln | z | + i \cdot \text{Arg} (z)$ .

However, I do not see what this image would look like.

**shawsend** · December 14th 2009, 04:12 PM

If you plot $\ln |r|$ in the x-direction and the principal argument $\text{Arg}(z)$ in the y-direction, then for the right half unit circle, won't the argument go from $-\pi/2$ to $\pi/2$ ? So that's a range in the y direction from $\pi/2$ to $-\pi/2$ . Now, the x-direction: when r is very small, then $\ln |r|$ is very big and when r=1, then $\ln |r|=0$ . Isn't that then a block from zero to infinity along the x-axis with the upper boundary at $\pi/2$ and lower one at $-\pi/2$ ? And since $|z|<1$ , then it's that block except for the left side at x=0.

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