Ian H
Lv 7
Ian H asked in Science & MathematicsMathematics · 2 years ago

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) ?

Update:

My progress so far is given in the answers

1 Answer

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  • Ian H
    Lv 7
    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 ?

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