3.2 • THE CAUCHY-RIEMANN EQUATIONS 109where the domain is restricted to be { rei^8 : r > O and - 11 < B < 11}, then the
derivative is given by
f
' () z = - (^1) = - r 1 _ , ~ cos - -B i-r. 1 _1 ~ sm. B -
2z! 2 2 2 2'
for every point in t he domain.
Solution We write
U(r,B)=r~cos~ and V(r,B)=r!sin~.
Thus,
Since U, V, Ur, Ue, V,., and Ve are continuous at every point in the domain (note
the strict inequality in - n < B < n), we use Theorem 3.5 and Equation (3-23)
to get
Note that f (z) is discontinuous on the negative real axis and is undefined at the
origin. Using the terminology of Section 2.4, the negative real axis is a branch
cut, and the origin is a branch point for this function.
Two important consequences of the Cauchy-Riemann equations close this
section.