discuss the transformation w=ez
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I have a region in the complex plane, defined simply as
0<y<π0<y<π
This is just an infinite horizontal strip from 00to ππ on the zz-plane. Considering the complex map then w=ezw=ez, I am trying to see how this map affects the given region. Starting with z=x+iy,w=u+ivz=x+iy,w=u+iv, I know that
w=ez=exeiy=excosy+iexsiny=u+ivw=ez=exeiy=excosy+iexsiny=u+iv
Thus u=excosy,v=exsinyu=excosy,v=exsiny. Since ex>0ex>0 for all real xx, i focus on the trigonometric components of uu and vv. Since cosy>0cosy>0 for all 0<y<π0<y<π, then u>0u>0. So the map is at least in the right half of the ww-plane. It's here that I start to get lost, so I switch to polar:
w=ez=exeiy=ρeiyw=ez=exeiy=ρeiy
Therefore, ρ=exρ=ex, and θ=yθ=y. Therefore, since ρρ can take all values except, 00, we have our horizontal strip mapped to the upper half of the ww-plane with a singular point at w=0w=0. Is the upper plane, since 0<y=θ<π0<y=θ<π.
Thus, my final intuition is that this horizontal strip is mapped to the upper ww-plane where u∈Ru∈R and v>0v>0. Is this right or am I off?
0<y<π0<y<π
This is just an infinite horizontal strip from 00to ππ on the zz-plane. Considering the complex map then w=ezw=ez, I am trying to see how this map affects the given region. Starting with z=x+iy,w=u+ivz=x+iy,w=u+iv, I know that
w=ez=exeiy=excosy+iexsiny=u+ivw=ez=exeiy=excosy+iexsiny=u+iv
Thus u=excosy,v=exsinyu=excosy,v=exsiny. Since ex>0ex>0 for all real xx, i focus on the trigonometric components of uu and vv. Since cosy>0cosy>0 for all 0<y<π0<y<π, then u>0u>0. So the map is at least in the right half of the ww-plane. It's here that I start to get lost, so I switch to polar:
w=ez=exeiy=ρeiyw=ez=exeiy=ρeiy
Therefore, ρ=exρ=ex, and θ=yθ=y. Therefore, since ρρ can take all values except, 00, we have our horizontal strip mapped to the upper half of the ww-plane with a singular point at w=0w=0. Is the upper plane, since 0<y=θ<π0<y=θ<π.
Thus, my final intuition is that this horizontal strip is mapped to the upper ww-plane where u∈Ru∈R and v>0v>0. Is this right or am I off?
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