- #1
mattmns
- 1,128
- 6
This issue of infinity (undefined?) keeps coming up in the following problems.
For example, the following question:
Computer the image of the sector [itex]0 \leq r \leq 1, 0 \leq \theta \leq \pi[/itex], under the map ln(z).
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So I first graphed this thing in the x,y (z-plane) and obviously we get a half circle with radius 1, above/including x-axis.
Then I looked at the following points. A=0, B=1, C=i, D=-1. Then if we map them under [itex]w= ln(z)[/itex] we get,
A' = w(A)= ln(A) = ln(0) = ?
B' = ln(1) = 0
C' = ln(i) = [tex]i \frac{\pi}{2}[/tex]
D' = ln(-1) = [tex]i \pi[/tex]
If we then look at each segment, and map them under w, well I know what to do with everything but the parts that involve, or go though A. But the whole infinity, or undefined, issue bugs me.
Any ideas? Thanks.
For example, the following question:
Computer the image of the sector [itex]0 \leq r \leq 1, 0 \leq \theta \leq \pi[/itex], under the map ln(z).
-------------
So I first graphed this thing in the x,y (z-plane) and obviously we get a half circle with radius 1, above/including x-axis.
Then I looked at the following points. A=0, B=1, C=i, D=-1. Then if we map them under [itex]w= ln(z)[/itex] we get,
A' = w(A)= ln(A) = ln(0) = ?
B' = ln(1) = 0
C' = ln(i) = [tex]i \frac{\pi}{2}[/tex]
D' = ln(-1) = [tex]i \pi[/tex]
If we then look at each segment, and map them under w, well I know what to do with everything but the parts that involve, or go though A. But the whole infinity, or undefined, issue bugs me.
Any ideas? Thanks.
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