finding the points where a complex function is differentiable (Need guidance) Recently I have encountered the topic on complex differentiation and i had these two questions
$f(z) = (z+5)/(z-5i) + (z-5)^10$
and
$f(x+iy) = (6x+y^2) + i(5xy+y)$
What I had to do was to determine every point that the functions can be differentiated and also provide them with the formula for the derivative.
I have been reading through lectures notes and websites but I still don't understand the theory behind how to actually determine where the points are and differentiate them. can anyone provide any hints and guides on how to do so?
 A: The first function is given in terms of $z$. It has a reasonable algebraic form: it's a rational function. You may remember from calculus that rational functions are differentiable at every point of their domain. The same is true in complex analysis, and for the same reasons: 


*

*the linear functions $f(z)=az+b$ are differentiable, with $f'(z)=a$ (check using the definition)

*the sum of differentiable functions is differentiable

*the product of differentiable functions is differentiable (and there is a product rule for derivative)

*the quotient of  differentiable functions is differentiable, as long as denominator is not $0$.


Since a rational function is constructed by the above means, it is differentiable at every point where the denominator is not $0$. Of course, when the denominator is $0$, it's not even defined, so definitely not differentiable. This should give you enough to deal with the first $f$.

The second $f$ is given in terms of $x$ and $y$, not in terms of $z$. In terms of $x$ and $y$, it is a polynomial, which implies that it has continuous partial derivatives with respect to $x$ and $y$. There is a theorem that says that if such  a function satisfies the Cauchy-Riemann equations, then it's complex differentiable. Here $u= 6x+y^2$ and $v=5xy+y$. So, you should find the derivatives $u_x,u_y,v_x,v_y$, put them into the Cauchy-Riemann equations, and see if they term into equalities. It looks like   $u_x=6$ and $v_y=5x+1$; these are equal only when $x=1$. Also, $u_y=2y$ and $v_x=5y$; when does $u_y=-v_x$ hold? and you have the only point on the plane where $f$ is complex differentiable.
