1. Formal Power Series and Polynomials
Let Z+ = N ∪ {0} be the set of all nonnegative integers. Let us denote the space of functions (or space of sequences of complex numbers) from Z+ to C by V. For f, g ∈ V, and a ∈ C, we define their sum f + g and the scalar multiplication af by
(f + g)(n) = f (n) + g(n), (af )(n) = af (n)
for n ≥ 0. Then V is a complex vector space. Moreover, we define the multiplication f ∗ g by
(f ∗ g)(n) = X
i+j=n
f (i)g(j).
Two functions f, g ∈ V are equal if and only if f (n) = g(n) for all n ≥ 0. We can verify that (V, +, ∗) forms a commutative ring. We denote this ring by C[[z]]. Elements of C[[z]]
are called formal power series. We denote C[z] the subset of C[[z]] consisting of sequences f : Z+→ C such that there exists N > 0 so that f(n) = 0 for all n > N. Elements of C[z]
are called polynomials over C. We see that C[z] forms a subring of C[[z]].
Let z ∈ V be the function from Z+ to C defined by z(n) = δ1n for n ≥ 0. (In other words, z can be identified with the sequence (0, 1, 0, · · · ).) Let z0∈ V be the function z0(n) = δn0 for n ≥ 0 and define zk inductively by zk = zk−1∗ z for k ≥ 1. In fact, we can verify that z0 is the identity of the ring C[[z]]. We also denote z0 by 1. For each f ∈ C[z], there exists N > 0 so that f (n) = 0 for n > N. Then we have
f = a0+ a1z1+ · · · + aNzN,
where ai = f (i) for i ≥ 0. In fact, f can be identified with the sequence f = (a0, · · · , aN, 0, · · · ).
This also shows that {zn : n ≥ 0} generates C[z]. It is easy to verify that {zn : n ≥ 0} is linearly independent. Thus {zn : n ≥ 0} forms a basis for C[z]. For elements of C[[z]], we identity f with {an}∞n=0 where f (n) = an for n ≥ 0. Formally, we write f =P∞
n=0anzn. In this way, the sum of f = P∞
n=0anzn and g = P∞
n=0bnzn and the multiplication f ∗ g are given by
f + g =
∞
X
n=0
(an+ bn)zn, f ∗ g =
∞
X
n=0
(
n
X
k=0
akbn−k)zn.
Let S be a subset of C. A polynomial f = a0+ a1z + · · · + anzn∈ C[z] defines a function fS : S → C by
α ∈ S 7→ a0+ a1α + · · · + anαn.
Such a function fS is called a polynomial function on S. Notice that the notion of polynomi- als is different from that of polynomial functions. For example, the polynomial f (z) = 1 + z and g(z) = 1 + z2 are two different polynomials. Take S = {0, 1}. Then f and g defines the same polynomial functions on S, i.e. fS = gS on S. For simplicity, we shall use f for fS when we can distinguish the difference between polynomials and polynomial functions.
Similarly, given a formal power series f =P∞
n=0anzn, can we find a subset S of C such that f defines a function fS from S to C? Let us take S = {0}. Then fS : S → C is the function 0 7→ a0. Hence every the formal power series f ∈ C[[z]] defines a function from {0}
to C. Can we find a larger S 6= {0} so that f defines a function from S into C? We cannot always do this, for example, consider f =P∞
n=0n!znthen α = 0 is the only complex number so thatP∞
n=0anαn is convergent. Hence we need to study the convergence of infinite series P∞
n=0anαn for α ∈ C.
Theorem 1.1. (Theorem and Definition) Given any formal power series f =P∞ n=0anzn in C[[z]], there exists 0 ≤ R ≤ ∞ such that
1
2
(1) if |α| < R, thenP∞
n=0anαnis convergent, (2) if |α| > R, thenP∞
n=0anαnis divergent.
The number R constructed above is called the radius of convergence. The open disk {α ∈ C : |α| < R} is called the disk of convergence.
