112 CHAPTER 3 • ANALYTIC AND HARMONlC FUNCTIONS
(b) Use the original Cauchy- Riemann equations for u and v and the results
of part (a) to prove that rUr = Ve and rV,. = -Ue, thus verifying
Equation (3-22)
(c) Use part (a) and Equations (3-14) and (3-15) to show that t he rigl1t
sides of Equations (3-23) and (3-24) simplify to f' (zo).11. Determine where the following functions are differentiable and wbere they are
analyt ic. Explain!(a) f(z) =x^3 +3xy^2 +i(y^3 +3x^2 y).
(b) f(z)=8x-x^3 -xy^2 +i(x^2 y+y^3 -8y).
(c) f (z) = x^2 - y^2 + i2 lxvl·- Let f and g be analytic functions in the domain D. If f' (z) = g' (z) for all z in
D, then show that f (z) = g (z) + C, where C is a complex constant. - Explain how the limit definition for the derivative in complex analysis and the
limit definition for the derivative in calculus are different. How are they similar? - Let f be an analytic function in the domain D. Show that if Re[! (z)] = 0 at all
points in D, then f is constant in D. - Let f be a nonconstant analytic function in the domain D. Show that the function
g (z) = f (z) is not analytic in D. - Recall that, for z = x + iy, x = ~ and y = ":;.y;.
(a) Temporarily, think of z and z as durilrny symbols for real variables.
With this perspective, x and y can be viewed as functions of z and z.
Use t he chain rule for a function h of two variables to show that&h = &h &x + &h &y = ~ (&h +i&h)
Oz OXOz &y&z 2 OX &y.(b) Now define the operator t, = ~ ( :. + i:v) that is suggested by
the previous equation. With this construct, show that if f = u +
iv is differentiable at z = (x, y), then, at the point (x, y), ~ =
~ [u, - Vy+ •i (y, + uv)I = 0. Equating real and imaginary parts thus
gives t he complex form of the Cauchy-Riemann equations: ~ = 0.3.3 Harmonic Functions
Let <P (x, y) be a r eal-valued function of the two real variables x and y defined
on a domain D. (Recall that a domain is a connected open set.) The partial
differential equation