Question

af af az az df Calculate 07 o2, and r (if it exists) for fx + iy)xy

Answer 1

f(x + iy) = x^3 + xy^2 - i(x^2 y + y^3) Put x = z + z bar/2 rightwardsdoublearrow x + iy = z y = z - z bar/2i x^3 = (z + z bar/2)^3 = z^3 + z bar^3 + 3z^2 z bar + 3z z bar^2/8 y^2 = (z - z bar/2i)^2 = (z - z bar)^2/4i^2 = z^2 + z bar^2 - 2 z z bar/-4 y^3 = (z - z bar)^3/8i^3 = z^3 - z bar^3 - 3z^2 z bar + 3 z z bar^2/-8i x^2 = (z + z bar)^2/4 = z^2 + z bar^2 + 2 z z bar/4 f(z) = z^3 + z bar^3 + 3z^2 z bar + 3 z z bar^2/2 + (z + z bar)/2 (z^2 + z bar^2 -2 z z bar)/-4 - i[(z^2 + z bar^2 + 2 z z bar)/4 times (z - z bar)/2i - (z^3 - z bar^3 - 3 z^2 z bar + 3/8i]