1550251515-Classical_Complex_Analysis__Gonzalez_

334 Chapter^6

10. Show that each of the following functions is harmonic in some domain,
    then determine the corresponding harmonic conjugate and f(z) = u+iv
    as a function of z.
    (a) u = x^4 - 6x^2 y^2 + y^4
    (b) v = 4x^3 y - 4xy^3 + 2xy ·
    x
    (c) u -- x2 + y2

( d) v = Arctan '#..

x

(e) u = e"'(xcosy - ysiny)

(f) v = e-"'(ycosy - xsiny)

(g) u = sin x cosh y

11. (a) Given u = aox^3 + a 1 x^2 y + a 2 xy^2 + aay^3 , find conditions on the
    coefficients so that u be harmonic in <C. Compute the harmonic

conjugate v and the corresponding analytic function f = u +iv.

(b) Determine a, b, c, d so that

w = x^2 +.axy + by^2 + i(cx^2 + dxy + y^2 )

be analytic in <C. Express w in terms of z.

12. Suppose that 1/J = 1/J(x, y) has continuous partial derivatives up to the
    second order in a region G. Show that

V^2 ¢ = EJ2¢ + EJ2¢ = ( ! + i !) ( 0¢ _ i o'ljJ) = 4 o

2

¢

ox2 oy^2 ox oy ox oy oz oz

13. (a) Let f E 'D(A), A open. Show that

fz = fz, f z = fz, d] = df

(b) If w = f ( z) is monogenic at z, show that

~~ = f1(z)

( c) Show that the Cauchy-Riemann equations are equivalent to the

equation Uz = ivz or, alternatively, to the equation Uz = -ivz.

( d) Let f = u +iv, where u and v have continuous partial derivatives

of the first two orders in some open set A. Show that

fzz =^1 / 4 (V^2 u + iV^2 v)

and deduce that· f zz = 0 in A if both u and v are harmonic in A

( v not necessarily a harmonic conjugate of u ).

*(e) Suppose that f E 'D(A) and g E D(B), A, B open, B ::J f(A), and

let ( = f(z), w = g((), so that w = (gof)(z) = g(f(z)). Prove that

ow og o( og o(

oz = o( oz + o( oz