Mathematical Methods in the Physical Sciences · Problem 9.12

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · jacobian-conformal-mapping

Problem

w = f(z) = u(x, y) + iv(x, y)

f(z)

analytic, defines a transformation from the variables

x

y

to the variables

u

v

, show that the Jacobian of the transformation (see Chapter 10, Section 13) is

\partial(u,v)/\partial(x,y) = |f'(z)|^2

. Hint: To simplify the determinant, use the Cauchy-Riemann equations and the equations (Section 2) used in obtaining them.