Question

Consider the complex function $f(z) = \frac{(z - i)(z + i)}{z^2}$. that maps a point $z = x + iy = re^{i\theta}$ of the $z$-plane onto the $w = u + iv$ plane.

          Consider the complex function
$f(z) = \frac{(z - i)(z + i)}{z^2}$.
that maps a point $z = x + iy = re^{i\theta}$ of the $z$-plane onto the $w = u + iv$ plane.

Consider the complex function
f(z) = ((z - i)(z + i))/(z^2).
that maps a point z = x + iy = re^iθ of the z-plane onto the w = u + iv plane.

Added by Milagros W.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Hast Aggarwal

01/08/2022

Step 1

Step 1: Determine the points in the z-plane where the map is conformal. Show more…

Show all steps

Thanks for your feedback!

Please give me a step-by-step solution for each part, I've been struggling with this problem for weeks. 1. Where is this map conformal? Specifically: Determine the points in the z−plane where the map is conformal. 2. Consider now z in the upper half plane only (y > 0). To what curve in the w−plane do the following curves inthe z−plane map onto? a) The semi-circle of radius |z| = r = 1b) The semi-circle of radius |z| = r > 1. What domain in the w−plane does the upper half z−plane map onto?Hint: express u and v in terms of θ and use basic trigonometric relations to obtain a relation that involves onlyu and v (and constants). When doing so 2 cos^2 θ = 1 + cos(2θ) may be useful. Consider the complex function f(z)=(z-i)(z+i) 22