Differentiation 335and14. Suppose that f = u +iv is analytic in a region G. Prove the following.
(a) ( :x lf(z)I)
2
+ ( :y lf(z)I)2= lf'(z)l
2(b) \7^2 lf(zW = n^2 lf(z)ln-^2 lf'(z)l^2 (n ~ 2)
(c) \7^2 luln = n(n - l)luln-^2 lf'(z)l2 (u f O,n ~ 2)
*(d) \7^2 lnlf(z)I = 0 [f(z) f O]
2 2 4lf'(z)l^2
(e) \7 Log[l + lf(z)l J = [l + lf(z)l 2 ]2
15. Suppose that f = u +iv is analytic in a region G and that 'ljJ = 'l/;(x, y)
has a sufficient number of derivatives, mixed derivatives being equal at
every point of G. Prove the following.
(a) \7'1/; = 'l/Jx + i'l/;y = ('I/Ju+ i'l/;v)f'(z)
(b) 'I/;;+ 'l/J; = ['I/;~+ 'l/J;Jlf'(z)l^2
(c) \7^2 '1/; = ['l/Juu + 'l/Jvv]IJ'(z)l2- (a) Show that if u and v are harmonic conjugates of each other in a
domain D, both u and v must be constant functions.
(b) Suppose that u is a harmonic function in a domain D. Determine
the class of real functions g for which g( u) is also harmonic. - (a) Show that if u(z) = u(x, y) is harmonic in a doma~n D, then u(z) =
u( x, -y) is harmonic in D, where D is the reflection of D on the
x-axis.
(b) Show that if f(z) is analytic in D, then f(z) is analytic in D*.18. If u(x, y) is a real harmonic function in D, show that au/az is an
analytic function in D and that au/az is conjugate analytic in the
same domain.- Show that at the point z = 0 the Cauchy-Riemann equations are
satisfied for the function f(z) = ~' yet f'(O) does not exist.
*20. If u = u(x,·y) is differentiable at a point (x, y) the real directional
derivative of u in the direction a is given by Dau = u., cos a+ Uy sin a.
Suppose that f = u +iv is analytic in a region R.
(a) Show that for (x, y) E R we have ·(b) If s and n are any two perpendicular vectors at z such that arg s -
argn =^1 / 2 7r + 2k7r, deduce that
au av
8n = 8s
and