Find the real and imaginary parts of f - show that there

Question

1. Find the 5^th roots of 1 + 2i.

2. Find the real and imaginary parts of f (z) = cos(z).

3. Determine all the values of z ∈ C (if any) at which the given function is differentiable.

(a) f(z) =|z|².

(b) g(z) = xy + 2yi where z = x + iy.

4. Show that

f(z) = Re(z)Im(z) + iIm(z)

is differentiable at only one point in C, and find this point.

5. Consider the function I : C → C defined by

f(z) = e^x= cos(y) + ie^x sin(y),

where z = x + iy. Show that f is differentiable at all points of C and calculate its derivative.

6. Let u(x, y) = 4xy³ - 4x³y.

Show that there exists a real-valued function v(x, y) so that

f(z) = u(x, y) + iv(x, y)

is differentiable at all points z ∈ C.