Complex Analysis: Theory and Applications · Problem 10.8.3

Applications of Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · joukowski-circle-through-pm1

Problem

Let

C_0

be a circle that passes through the points

1

and

-1

and has center

c_0 = ia

. (a) Find the equation of the circle

C_0

. (b) Show that the image of the circle

C_0

under

w = \dfrac{z-1}{z+1}

is a line

L_0

that passes through the origin. (c) Show that the line

L_0

is inclined at the angle

\alpha_0 = \dfrac{\pi}{2} - \operatorname{Arctan}(a)