determine all points where this function is differentiable and where applicable give the formula for the derivative ... f(x+iy)=(x-y^2)+i(2xy-y)

Question

anonymous · Answer

Should we differentiate this?!is it applicable?!

anonymous · Answer

what is $i$ here ?  $\sqrt{-1}$ ?

anonymous · Answer

yes.

anonymous · Answer

Sorry, but I think this question is beyond the capabilities of most of the people on here. Here's something that I hope can help you at least a bit. http://www.wolframalpha.com/input/?i=derivative+f%28x%2Biy%29%3D%28x-y^2%29%2Bi%282xy-y%29 Not sure why it has put it in the form it has but you should be able to rearrange the equation to get f'(x+iy).

anonymous · Answer

btw ... thanks ... :)

anonymous · Answer

are u familiar with Cauchy-Riemann equations for functions like $$f(z)=u(x,y)+i \ v(x,y)$$

anonymous · Answer

yes ... i know it

anonymous · Answer

well u have  $u(x,y)=x-y^2$  and  $v(x,y)=2xy-y$

Cauchy-Riemann equations states that for Derivability of $f(z)$ at point 

$z_0=x_0+i  y_0$ u must have  $u_x=v_y$  and  $u_y=-v_x$

anonymous · Answer

yes...

anonymous · Answer

so u just set up the equations for our particular case and see what happens and what are the possible values for $x$ and $y$ for Derivability

anonymous · Answer

yes ... so we will get -2y=-2y and 2x-1=1 ~> x=1 ...

anonymous · Answer

now should we differentiate it?!or it is not applicable?!

anonymous · Answer

well Cauchy-Riemann equations are necessary for Derivability but not sufficient

lol i cant explain my english is bad

@Neemo

anonymous · Answer

:D ... thanks .

anonymous · Answer

Cauchy-Riemann equations are necessary  and sufficient conditions for complex differentiation once we assume that its real and imaginary parts are differentiable real functions of two variables.

anonymous · Answer

A function f(z) = u(x, y) + i v(x, y), where z = x + i y, has a
complex derivative f′(z) if and only if its real and imaginary parts are continuously differentiable and satisfy the Cauchy–Riemann equations : 
(mentionned by @mukushla)

anonymous · Answer

yeah...

anonymous · Answer

@parto 

note that  $u$  and  $v$   are continuously differentiable and according to what u got satisfy the Cauchy–Riemann equations for $x=1$

anonymous · Answer

|dw:1344440942150:dw|

Mapping of complex functions am I right?