7.2 • TAYLOR SERIES REPRESENTATIONS 265- Find f <^3 > (0) for
00
(a) f (z) = L: (3 + ( - 1)")" z".
n=O
(b) g (z) = E <^1 +;>" z".
f\=l
00 n
(c) h(z) = .. ~o (v'iH)".00- Suppose that f ( z) = L: c,.zn is an entire function.
n=O
(a) Find a series representation for f (z), using powers of z.
(b) Show that f (z) is an entire function.
(c) Does f (z) = f (z)? Why or why not?
00- Let f (z) = L: c,.z" = l+z+2z^2 +3z^3 +5z^4 +8z^5 +13z^6 + · ·.,where the coefficients
n=O
c,, are t he F ibonacci numbers defined by Co = 1, c 1 = 1, and c,.. = Cn- 1 + Cn-2, for
n ~ 2.
(a) Show that f (z) = l-•'- • 2 , for all z E DR (0) for some number R.
(b) Find the value of R in part (a) for which t he series representation is
valid. Hint: Find the singularities off (z) and use Corollary 7.3.
Complete the details in the verification of Lemma 7.1.
We used Lemma 7.1 in establishing Identity (7-6). However, Lemma 7. 1 is valid
provided z :f Zo and z :f er. Explain why these conditions are indeed the case in
Identity (7-6).
12. Prove by mathematical induction that f (n) (z) = c/~:,!/! 2 in Example 7.3.
- Establish the validity of Identities (7-15) and (7-16).
1 4. Use the Maclaurin series and the Cauchy product in Identity (7-17) to verify that
sin 2z = 2 cos z sin z up to terms involving z^5.
Compute the Taylor series for the principal logarithm f (z) = Log z expanded
about the center zo = -1 + i.
The Fresnel integrals G (z) and S (z) are defined by
G (z) = 1• cos (e) d{ and S (z) = 1• sin (e) d{.
We define F (z) by F (z) = C (z) + iS (z).(a) Verify the identity F (z) = J; exp (i{2) cl{.
(b) Integrate the power series for exp ( i{^2 } and obt ain the power series for
F(z).