1. Can someone please prove it explaining the steps. Thank you I will rate.
Solution
We can derive using purely polar coordinates. Start with
z(r,?)f(z)=rei?=u(r,?)+iv(r,?)
We define f?(z) using the limit
f?(z)=limz?0?f?z
where
?f?z=?u+i?v=z(r+?r,?+??)?z(r,?)=(r+?r)ei(?+??)?rei?
Next, we try to first approach from ???0,
?z=(r+?r)ei??rei?=?rei?
Therefore when we take the limit,
f?(z)=lim?r?0?u+i?v?rei?=1ei?(ur+ivr)(1)
On the other hand, approaching from ?r?0 first yields
?z=rei(?+??)?rei?
Using the derivative
ei(?+??)?ei?=dei?d???=iei???
we get
?z=irei???
Therefore
f?(z)=lim???0?u+i?virei???=1irei?(u?+iv?)(2)
Finally, comparing the real and imaginary parts of (1) and (2) gives us what we want:
ur=1rv?,vr=?1ru?