COMPLEX FUNCTION QUESTION Suppose that f(z) is analytic, with f(0) = 3 - 2i and f(1) =6 - 5i . Find f(1 + i) if Im f'(z) = 6x(2y - 1) ?
My progress so far is given in the answers
- 2 years ago
Using this different analytic function to try to understand how things work.
If f(z) = z^3 = (x + iy)^3 = u + iv, then
u = x^3 – 3xy^2
v = 3x^2y – y^3
These will be conjugate harmonic functions obeying the C R equations
∂u/∂x = 3x^2 – 3y^2 = ∂v/∂y
∂v/∂x = - 6xy = - ∂u/∂y
I want to get to Im f'(z) to make comparisons.
f'(z) = 3z^2 = 3(x^2 – y^2) + 6xy i
Im f'(z) = 6xy
But the problem gives Im f'(z) = 12xy – 6x
Starting with f(z) = 2z^3 would lead to the 12xy part, but not the -6x
Is it somehow valid to let z = x + (y – 1/2)i ……..?? Because then
Im f'(z) = 12xy – 6x
Seems like f(z) = 2z^3 minus something else ??
Can anyone show how to proceed from here ?