1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 333

Dtanh-^1 z =

1

~ z 2 (z-/= ±1)

the value of the square root to be taken in the formulas of the first four

·rows depending on the chosen branch of the corresponding function.

7. Let z = x + iy and J(z) = Reiw, R = R(x, y), w = w(x, y), f(z) -/= 0
   for z E D f. Suppose that R and w are differentiable in some open
   set A C Dt·

(a) Prove that f is analytic in A iff

at every point of A.

(b) Show that if f' ( z) exists at z E A, then

J'(z) = eiw(Rx + iRwx)

( c) Assuming the existence and continuity of Rxy and Wxy, show that

Rxx + Ryy = R 1 ( Rx^2 + Ry 2) and Wxx +wyy =^0

8. Let z = reilJ and f(z) = Reiw, R = R(r, 8), w = w(r, 8), f(z) -/= 0

for z E Dt. Suppose that Rand w are differentiable in some open set

A C D f that does not contain the point z = 0.

(a) Prove that f is analytic in A iff

RIJ = -Rrwr

(b) Show that if f' ( z) exists at z E A, then

f'(z) = e-i!Jeiw(Rr + iRwr)

(c) Assuming the existence and continuity of RrlJ and Wr!J, show that

r Rrr +^1 -;: RIJIJ + Rr - R 1 ( r Rr^2 +^1 -;: RIJ 2) = 0

and

9. If f(z) = u(x,y) + iv(x,y) is a nonconstant analytic function in a
   region D, prove that the two families of level curves u( x, y) = c1 and
   v(x,y) = c 2 are orthogonal at each point of D, except where f'(z) = 0.
   Find the families of orthogonal curves corresponding to the following
   functions.
   (a) f(z) = z^2
   1
   (c) J(z) = - (z-/= 0)
   z

(b) f(z) = z^3

z-1

(d) f(z) = -

1

(z-/= -1)

z+