1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 351

3. Find the directional derivative of f(z) = (z + i)^2 + 2z at z = 1 + i in
   the direction of the arc given by z = t^2 + it, 0 ~ t ~ 2.


4. (a) Let f(z) be monogenic at z, and h = lhlei^8 , where() is kept fixed
   while h ---+ 0. Prove that

lim lf(z + h)l2 - lf(z)l2 = J'(z)f(z) + f(z)f'(z)e-2ie

h--+0 h

(b) Let w = Argz (z '!-0). Show that

(

dw) = ~ (~ _ ~e-2i8)

dz 8 2i z z

5. Let f, g E '.D(A).

(a) If f and g have the same areolar derivative on the open set A, show

that f and g differ by an analytic function in A.

(b) If a and b are constants, verify that

(af + bg)~(z) = af~(z) + bge(z)

(c) Prove: (fg) 0 (z) = f(z)g8(z) + f9(z)g(z).

6. Suppose that f = u + iv and that u and v have continuous partial
   derivatives of the first two orders in some open set A. Let w = lf(z)l^2 =

f(z)f(z), z E A.

(a) Show that

and deduce that if f is analytic in A, then

\7^2 w = 4lf'(z}l2

and that if f is conjugate analytic in A, then

'Y'^2 w = 4lf:zl^2 = 4lf9(z)l^2 = 4l](z)l^2

(b) If u and v are harmonic in A (not necessarily harmonic conjugates),

then

\7^2 w = 4 {lfzl^2 + lf:l^2 }

Deduce that w = ff is subharmonic whenever u and v are

harmonic.

( c) Prove also that if u and v are harmonic in A (not necessarily

harmonic conjugates), then

\7^2 \7^2 w = 16 {lfzzl

2

+ lfzzl

2

}

7. Construct the Kasner circle for the following functions at the given
   points.