Derivative of an complex function at a point $z_0$

Question

Given $f(z) = z {\rm Re}(z) + \bar{z} {\rm Im} (z) + \bar{z}$ is differentiable at point $z_0$. So, $f'(z_0)$ is $ \cdots $ How can I find the real and imaginary part of unknown function?. Give me a hint please.

Answer 1

The function is known. You only don't know what is the point $z_0$. Let's try to find it. We have:

$f(x+iy)=(x+iy)x+(x-iy)y+(x-iy)=(x^2+xy+x)+i(xy-y^2-y)$.

So if we let $u(x,y)=x^2+xy+x$ and $v(x,y)=xy-y^2-y$ then $f(x+iy)=u(x,y)+iv(x,y)$.

Now using Cauchy-Riemann equations you can easily find that this function is only differentiable at the point $1-i$, hence this is $z_0$. Can you finish from here?

Answer 2

Isn't a unknown function. If $z = x + yi$ $$f(z) = (x + yi)x + (x - yi)y + x + yi = \cdots$$ Now, check where $f$ is (complex) differentiable using...