Mathematical Methods in the Physical Sciences · Problem 10.8

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · conformal-mapping-complex-velocity

Problem

Observe that the magnitude of the velocity in Problem 7 can be obtained from

V = V_0|dw/dz|

. Show that this result holds in general as follows. Let

w = f(z)

be an analytic mapping function such that the lines

v = \text{const.}

map into the streamlines of the flow you want to consider in the

z

plane. Then

V_0 w = V_0(u + iv) = \Phi(x, y) + i\Psi(x, y).

Show that

V_0 \frac{dw}{dz} = \frac{\partial \Phi}{\partial x} - i\frac{\partial \Phi}{\partial y} = V_x - iV_y

(this expression is called the complex velocity). Hence show that

V = V_0|dw/dz|