1550251515-Classical_Complex_Analysis__Gonzalez_

338 Chapter^6

is a harmonic conjugate function in R.

34. ~et f(z) = u(r, B) + iv(r, B) E '.D(A), A open, z = rei^9 -/:-0. Show that
    in polar coordinates the complex differential operators have the forms


35. Let f(zJ = u(r,B)+iv(r,B), where z = reie p,ndu and v have continuous
    partial ·derivatives of the first two orders in a regidn R that does not
    contain the origin. Suppose that

(1)

when~ >. = >.(r) is differentiable for r > O.

(a) Show that both u and v satisfy in R a partial differential equation

of the form

(2)

(b) Show that u = (lnr)cosO satisfies (2) for>.= rlnr, find the most

general conjugate function v satisfying (1), and determine f =

u +iv.

( c) F'ind the general form of a complex function with components sat-

isfying (1) and the special equation (2) for which 'I/Jee = 0 and

>.(r) -/:- 0.

(B. A. Case [23])

36. Let f = u +iv be of class c<^1 )(R), R a region. Prove that in order

that either f or J be analytic in R it is necessary and sufficient that

all three surfaces

( = u(x,y), ( = v(x, y), ( = (u2 + v2)1/2

have the same area over an arbitrary subregion R 1 of R. Investigate

under what conditions on 'ljJ the third surface can be replaced by ( =

'lj;(u, v).

(A. W. Goodman [53])

6.9 THE COMPLEX DIRECTIONAL DERIVATIVE

Let 1: z = z( t), a :=::; t :=::; /3, be a regular arc, z a fixed point on 7, and
z + ~z = z(t + ~t) E 1 (Fig. 6.4). Let w = f(z) be a function defined
at least on 1*.