let f(z) = u + iv, z = x + iy
let u and v be defined as follows:
u(x,y) = (x^3 - y^3) / (x^2 + y^2) if (x,y)≠(0,0), u(0,0)=0
v(x,y) = (x^3 + y^3) / (x^2 + y^2) if (x,y)≠(0,0), v(0,0)=0
solve lim [f(z) - f(c)] / (z-c) as z approaches c. that is, show f'(c) exists
our instructor told us a hint: solve it component-wise
the thing is, I do not know how to apply that in such a complex function please help me.. teach me some sorcery
edit: btw, this limit is defined to be the derivative
Hello Sir Opalg i take it that what you gave me is a hint..
after multiplying a conjugate of the denominator and then some further algebraic manipulations I ended up with these bunch of letters
is this, by any chance, going towards the right direction? i still do not know how to take the limit of this function though
By C-R f'(z) doesn't exist anywhere except possibly at z=0.
If you let x^2 + y^2 = r^2 and note |x|<r and |y|<r, you can show limit of u and v and their derivatives approach 0 as r approaches 0. If you then define u and v and their drivatives = 0 at z=0 they are continuous at z=0 and C-R is satisfied at z=0 so that f'(0)=0.