98 CHAPTER 3 • ANALYTIC AND HARMONIC FUNCTIONS
-------~EXERCISES FOR SECTION 3.1
- Find the derivatives of the following functions.
(a) f(z)=5z^3 - 4z^2 +1z-8.
{b) g(z} = (z^2 -iz+9)&.
(c) h (z} =^2 :ti for z f - 2.
(d) F(z) = (z^2 +{1- 3i}z+1) (z^4 +3z^2 +5i).
2. Show that the following functions are differentiable nowhere.(a} f (z) = Re(z}.
(b) f(z) = Im(z).- If f and g are entire functions, which of the following ate neceSSa.tily entire?
(a) [! (z}]^3.
(b) f(z}g(z}.(c) ffit.
(d>IH).
(e) I (z - 1 }.
(f} f(g(z)).- Use Equation (3-1} to verify rule {3· 5).
- Let P (z) = <l-0 + aiz + · · · + a.,z" be a polynomial of degree n 2'. 1.