1550251515-Classical_Complex_Analysis__Gonzalez_

332 Chapter6

(c) w = eaz (d) w = sinz

(e) w = cosz (f) w = tanz

(g) w = sinhz (h) w = coshz

(i) w = tanhz (j) w =^1 / 5 ez(sin^2 z - sin2z + 2)

3. Show that the functions w = Rez, w = Imz, w = lzl, and w = Argz
   do not have a derivative anywhere in the complex plane.


4. Find the points where each of the following functions is monogenic and
   compute J'(z) at those points.
   (a) w=(z+i)^2 +3z (b) w=(x-y)^2
   (c)w=z+: (d)w=sin.Z
   z+z


5. Let z = rei^6 and f(z) = u(r,B) + iv(r,B) for z ED. Suppose that u
   and v are differentiable in some open set A C D that does not contain
   the point z = 0.

(a) Prove that f is analytic in A iff

and

at every point of A (Cauchy-Riemann equations in polar coordi-

nates).

(b) Show that if f(z) exists at z EA, then f'(z) = e-i^6 (ur + ivr)

( c) Assuming the existence and continuity of Ure and Vre in A, prove

that both u and v satisfy in A the Laplace equation in polar form,

namely,

2 82 ,,P 8,,P 82 ,,P

r 8r 2 + r or + 882 =^0

6. Apply the formula in problem 5(b) to show that if w = log z, then
   w' = 1/ z for z + lzl # 0, no matter what particular branch of log z is
   chosen (so as to make logz single-valued).
   Apply the preceding result to the appropriate formula in Section 5.25
   to obtain the following formulas for the derivatives of the inverse
   circular and hyperbolic functions:

Darcsinz =

1

vlf=Z2 (z # ±1)

*

·~ (z#±i)

* 1 +z^2

-1

vlf=Z2 (z # ±1)

Darccosz ==

1

*JZZ"=l (z # ±1)

1

Darctanz = -

1 2

(z # ±i)

+z