How to Express Function f in Terms of Complex Variable z?

[SALAAH_BEDDIAF]

Let $$f(x+iy) = \frac{x-1-iy}{(x-1)^2+y^2}$$

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 $u + iv : u(x,y),v(x,y)$ and then partially differentiating wrt x and y and just show that $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} , \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$ and then it asks to express f in terms of z i.e f(z) =...

I have no idea where to begin with this

 

[tiny-tim]

Hi SALAAH_BEDDIAF!

… 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? ​

 

[lurflurf]

write x and y in terms of z and its conjugate, then simplify

$$x=\frac{z+\bar{z}}{2}\\y=\frac{z-\bar{z}}{2 \imath}$$

 

[economicsnerd]

To start, you definitely want to express it in terms of $z$ and $\bar z$.

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 $x^2+y^2 = |z|^2$ (Pythagoras), which can itself be expressed cleanly as $z\bar z$.
- On what's left (cleaner than before), use lurflurf's hint.

 

[blue_raver22]

question, as it is not specified what z represents. If z is a complex number, then f(z) can be expressed as f(z) = \frac{z-1}{(z-1)^2+1}. However, if z represents a real variable, then f(z) cannot be expressed in terms of z as it is a function of complex variables. It is important to have more context and information in order to accurately express f in terms of z.