Express f in terms of z

  • #1
Let [tex]f(x+iy) = \frac{x-1-iy}{(x-1)^2+y^2}[/tex]

first of all it asks me to show that f satisfies the Cauchy-Riemann equation which I am able to do by seperating into real and imaginary [itex]u + iv : u(x,y),v(x,y)[/itex] and then partially differentiating wrt x and y and just show that [itex] \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} , \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} [/itex] and then it asks to express f in terms of z i.e f(z) =.....

I have no idea where to begin with this
 

Answers and Replies

  • #2
tiny-tim
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Hi SALAAH_BEDDIAF! :smile:
… express f in terms of z i.e f(z) =.....

Well, the top is obviously ##\bar{z} - 1## …

what do you think the bottom might be? :wink:
 
  • #3
lurflurf
Homework Helper
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write x and y in terms of z and its conjugate, then simplify

$$x=\frac{z+\bar{z}}{2}\\y=\frac{z-\bar{z}}{2 \imath}$$
 
  • #4
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To start, you definitely want to express it in terms of [itex]z[/itex] and [itex]\bar z[/itex].

You can use lurflurf's hint and do it mechanically.

If you want something slightly cleaner...
- Use tiny-tim's hint for the numerator.
- Expand the denominator, and use [itex]x^2+y^2 = |z|^2[/itex] (Pythagoras), which can itself be expressed cleanly as [itex]z\bar z[/itex].
- On what's left (cleaner than before), use lurflurf's hint.
 
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