Thread: Mathematical methods - Conformal mapping Q

Hi, im really stuck on this question, Thankyou for any help in advance.

4) i) For the line segment Lz and the semi-circle Cz
Lz = {x, y : x = ln r, 0 < y < pi}
Cz = {r, θ : r ∈ R+, 0 < θ < pi},
show that the function

f1(z) = ez

maps Lz onto Cz.

ii) Determine the length of the major and minor axis of the ellipse onto which the
function
f2(z) = z +(1/z)

maps the semi-circle Cz.

iii) Construct a conformal map that maps the line segment
L′z = {x, y : x =pi/4, 0 < y < pi} onto an ellipse centered at the origin with major axis length
a = 4 cosh pi/4 and minor axis length b = 4 sinh pi/4.

The attachment is easier to read Q