3.1 • DIFFERENTIABLE AND ANALYTIC FUNCTIONS 99(a) Show that P' (z) = a1 + 2a2z + .. · + na,.z"- \
(b) Show that, for k = 0, 1, ... , n, ak = pc~,<o>, where p (k ) denotes the
kth derivative of P. (By convention, p(O) (z) = P (z) for all z.)- Let P be a polynomial of degree 2, given by
P(z) =(z-z 1 )(z-z2),
where z1 'I z2. Show that
P' (z) 1 1
P(z) = z-z1 + z-Z2·Note: The quotient 1;:S'/ is known as the logarithmic derivative of P.- Use L'Hopital's rule to find t he following limits.
(a) z-lim i •'-::-~.^1 •
(c) .~~. ::;:.
(dl .i:..!~i r zZ«+• - 2.t:+2.
( ) e 1. im ··-~· 6•
.i:-l+ivJ z
(f) lim •"-512.
.i:--l+iv'3~
Use Equation (3-1) to show that f.~ = ~·
Show that d~ z-" = -nz,,,, where n is a positive integer.
Verify the identity.
:J (z) g (z) h (z) = f' (z) g (z) h(z) + f (z) g' (z) h(z) + J (z)g (z)h' (z).
Show that the function f (z) = JzJ^2 is differentiable only at the point zo = 0. Hint:
To show t hat f is not differentiable at zo 'I 0, choose horizontal and vertical lines
through the point zo and show that ~'; approaches two distinct values as t::i.z --> 0
along those two lines.
Verify
(a) Identity (3-4).
(b) Identity (3-7).
(c) Identity (3- 9).
(d) Identity (3- 10).