Multiple choice

For the function of a complex variable W = ln Z (where, W = u + jv and Z = x + jy), the u = constant lines get mapped in Z-plane as

  1. set of radial straight lines

  2. set of concentric circles

  3. set of confocal hyperbolas

  4. set of confocal ellipses

Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\text{We have} W = In \hspace{0.1cm}z \\ u + jv = In(x + jy) \\ or \hspace{1cm} e^{u+jv} = x + jy \\ or \hspace{1cm} e^u e^jv = x + jy \\ e^u (cos v + jsinv) = x + jy \\ Now \hspace{0.5cm} x = e^u cos v \hspace{0.2cm} y = e^u sin v \\ Thus \hspace{0.2cm} x^2 + y^2 = e^{2u} \hspace{2cm} \text{Equation of circle}$