1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

108 CHAPTER 3 • ANALYTIC ANO HARMONIC FUNCTIONS

Here u.,, Uy, v.,, and Vy are continuous, and u., (x , y) = Vy (x, y) bolds for all
(x, y). But Uy (x , y) = -vx (x, y) iff 6xy = -6xy, which is equivalent to 12xy = 0.

The Cauchy- Riemann equations hold only when x = 0 or y = 0, and according

to Theorem 3.4, f is differentiable only at points that lie on the coordinate axes.
But this means that f is nowhere analytic because any E neighborhood about a
point on either axis contains points that are not on those axes.

When polar coordinates (r, 8) are used to locate points in the plane, we use
Expression (2-2) for a complex function for convenience; that is,

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

f (re;^8 ) = u (rei^9 ) +iv (rei^9 ) = u(rcos8, rsin8) + iv(rcos8, rsin8)

= U(r,8) +iV(r,8),

where U and V are real functions of the real variables r and 8. The polar form

of the Cauchy- Riemann equations and a formula for finding f' (z) in terms of

the partial derivatives of U (r, 8) and V (r, 8) are given in Theorem 3.5, which

we ask you to prove in Exercise 10. This theorem makes use of the validity of

the Cauchy- Riemann equations for u and v , so the relation between them and

the functions U and V- namely, u(x,y) = U(r,8) and v(x, y) = V(r, 8)- is

important.

• EXAMPLE 3.10 Show that if f is the principal square root function given
  by

f (rei^9 ) = f (z) = z~ = r~ cos~+ ir~ sin~.

2 2