Evaluate integrals in the complex plane direct intergration

Question

Assignment:

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.

c) Find Cauchy-Riemann conditions in polar coordinates, starting with z = re^iθ 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⁺² |z|² along the indicated paths:

(i) Along the strategic line from 0 to 1+2i.

(ii) First from 0 to 2i, then horizontally 2i to 1+ 2i.