1) Find the real and imaginary parts, u(x, y) and v(x, y) of the following functions:
a) f(z) = sin z.
b) f(z) = 2z+3/z + 2.
2) Show that the derivative of f(z) = x2 - y2 + 2ixy exists and is unique by considering Δy = mΔx, that is, Δz goes to zero along a straight line with slope m, thus f(z) is analytic for all z.
3) a. Find out whether the function y-ix/x2+y2 is analytic? Give details to support your results.
b. Using the formal definition of derivative, verify, that d(Inz)/dz =1/z (z ≠ 0) holds. Hint: Expand In(1+Δz/z) in series.
c. Find Cauchy-Riemann conditions in polar coordinates, starting with z = reiθ and f(z) = u(r, θ) + iv(r, θ).
d. Show that u(x, y) = 3x2y - y3 is a harmonic function and find the function f(z) of which u is the real part. Derive v(x, y) and how that v(x, y) is also harmonic.
4) Evaluate the following integrals in the complex plane by direct integration
a) ∫(dz/z2+8i) along the line y =x from 0 to ∞.
b) 0∫1+2i |z|2dz along the indicated paths:
(i) Along the straight line from 0 to 1+2i.
(ii) First from 0 to 2i, then horizontally from 2i to 1+2i.