L 0 IIJ
3. Historical remarks on term-wise differentiation of series. For
EULER it went without saying that term-wise differentiation of power series and function series resulted in the derivative of the limit function. ABEL was the first to point out, in the 1826 letter to HOLMBOE already cited in 3.1.4, that the theorem on interchange of differentiation and summation is
not generally valid for convergent series of differentiable functions. ABEL, who had just found his way in Berlin from the mathematics of the Euler period to the critical logical rigor of the Gauss period, writes with all the enthusiasm of a neophyte (loc. cit., p. 258): "La theorie des series infinies en general est jusqu'a present tres mal fondee. On applique aux series infinies toutes les operations, comme si elles etaient finies; mais cela est-il bien permis? Je crois que non. Ou est-il demontre qu'on obtient la differentielle d'une serie infinie en prenant la differentielle de chaque terme?
Mien n'est plus facile que de donner des exemples ou cela n'est pas juste;
par exemple
(1) 2 = sin x - sin 2x + 3 sin 3x - etc.
En differentiant on obtient
(2)
1
2 = cos x - cos 2x + cos 3x - etc.,
resultat tout faux, car cette serie est divergente. (Until now the theory of infinite series in general has been very badly grounded. One applies all the operations to infinite series as if they were finite; but is that permissible?
I think not. Where is it demonstrated that one obtains the differential of an infinite series by taking the differential of each term? Nothing is easier than to give instances where this is not so; for example (1) holds but in differentiating it one obtains (2), a result which is wholly false because the series there is divergent.)"
The correct function-theoretic generalization of the theorem on term- wise differentiation of power series is the famous theorem of WEIERSTRASS on the term-wise differentiation of compactly convergent series of holomor- phic functions. We will prove this theorem in 8.4.2 by means of the Cauchy estimates.
4. Examples of holomorphic functions. 1) From the geometric series
1111
= Eo z",
which converges throughout the open unit disc E, after k-fold differentiation we get1
(1 -
Z+-,
(k) z"_k z E E.2) The exponential function exp z = is holomorphic throughout C:
exp'(z)=expz, zEC.
This important differential equation can be made the starting point for the theory of the exponential function (cf. 5.1.1).
3) The cosine function and the sine function
cos z = ((21v)! z2v sin z = LJ (2v +)1)i z2v+1 z E C are holomorphic in C:
cos'(z) = -sin z ,
sin'(z)=cosz,zEC.
These equations also follow immediately if one is willing to use the fact al- ready established that exp' = exp, together with the Euler representations
cos z =1 [exp(iz)+ exp(-iz)] , sin z =1 [exp(iz) - exp(-iz)].
2 2i
4) The logarithmic series A (z) = z - z + 3 - is holomorphic in E:
A'(z)=1-z+z2-...=1+z zEE.
5) The arctangent series a(z) = z - 3 + s - is holomorphic in E with derivative a'(z) = Imo. The designation "arctangent" will be justified in 5.2.5 and 5.5.2.
6) The binomial series b°(z) _ E (')zv , a E C, is holomorphic in E:
bv(z) = ab°_I(z) =
1 + z(z) , z E E.
To see the first of these equalities, use the fact that v(") = a(0-1) to getV V-1 b°(z)' = 00 'i'v(v)z-1
= a ,V-1
I(- ') z
1 = aThe second equality is now a consequence of the multiplication formula 2.3.
11
The exponential series, the logarithmic series and the binomial series are connected by the important equation
(*) b°(z) = exp(aa(z)) , z E E,
a special case of which is
1 + z = exp A(z) for all z E E.
Proof. The function f (z) := b,(z) exp(-aA(z)) is holomorphic in E and by the chain and product rules for differentiation
f'(z) = [bi(z) - ab°(z).\'(z)] exp(-a.A(z)) = 0,
since by 4) and 6) b', = o bA. Consequently, according to 1.3.3 f is constant in E. That constant is f (0) = 1, so (*) follows if we anticipate 5.1.1, according to which exp(-w) = (exp(w))-I for all w.
Exercises
Exercise 1. Show that the hypergeometric function F(a, b, c, z) introduced in Exercise 1 of §2 satisfies the differential equation
z(1 - z)F"(a, b, c, z) + {c - (1 + a + b)z}F'(a, b, c, z) - abF(a, b, c, z) = 0 for all z E E.
Exercise 2. Show that for c E V, d E C \ {c}, k E N
1 1
(v) (z -
d)V-k(c - z)k+l = (c - d)k+l k c - d z E Bjc_dj(d).
Exercise 3. (Partial fraction decomposition of rational functions). Let f (z) := a= be a rational function, in which the degree of the polynomial p is less than that of q and q has the factorization q(z) = c(z - c1)"1 (z -
C2)1'2 .. (z - c,)"'^ with c E C" and distinct complex numbers c2.
a) Show that f (z) has a representation of the form
I
a12 al", a21f(z) =
z - cl+ (z-cl)2 ...+
(z-cl)"' + z-c2
+...aml anz arnv,,,
+z
- c.m + (z-Cm)2 +... } (z-CM).'- for certain complex numbers ajk.b) Show that the particular coefficients akv,,, 1 < k < m, from a) are determined by
_
p(Ck)akvk C(Ck - Cl)"' ... (Ck - Ck_1)"k-1 (Ck - Ck+1)"k+1 ... (Ck - Cm)v-
Hint to a): Induction on n := degree q = ill + v2 + + vm.
Exercises 2 and 3, in connection with the Fundamental Theorem of Al- gebra (cf. 9.1) insure that any rational function can be developed in a power series about each point in its domain of definition.
Exercise 4. Develop each of the given rational functions into power series about each of the given points and specify the radii of convergence of these series:
a)
x - iz - x + i
about 0 and about 2.b) z4z3z3
4z8z2 + about 0 and about i.
Hint to b): First use long division to get a proper fraction.
Exercise 5. The sequence defined recursively by co := cl := 1, c,+ :_
c,i_1 + Cn_2 for n > 2 is called the Fibonacci numbers. Determine these numbers explicitly. Hint: Consider the power series development about 0 of the rational function-3+;_
§4 Structure of the algebra of convergent power series
The set A of all (formal) power series centered at 0 is (with the Cauchy multiplication) a commutative C-algebra with 1. We will denote by A the set of all convergent power series centered at 0. Then
A is a
C-subalgebra of A which is characterized by(1) A = {f =
a"z" E A : 3 positive real s, M such that MVv E N}.
This latter is clear by virtue of the convergence lemma 1.1.
In what follows the structure of the ring A will be exhaustively described;
in doing so we will consistently use the language of modem algebra. The tools are the order function v : A - N U {oo} and theorem 2 on units, which is not completely trivial for A. Since these tools are trivially also available in A, all propositions of this section hold, mutatis mutandis, for the ring of formal power series as well. These results are, moreover, valid with any complete valued field k taking over the role of C.
1. The order function. For every power series f = E a,,z" the order v(f) of f is defined by
v(f) - S
min {v E N : a , , 0 } , in case f 360,l 00 in case f = 0.
For example, v(zn) = n.
Rules of Computation for the order function. The mapping v : A -- N U {oo} is a non-archimedean valuation of A; that is, for all f, g E A
1) v(fg) = v(f) + v(g) (product rule) 2) v(f + g) > min{v(f ), v(g)} (sum rule).
The reader can easily carry out the proof, if he recalls the conventions n + oo = n, min{n, oo} = n for n E R U {oo}. Because the range N U {oo}
of v is "discrete" in R U {oo}, the valuation v is also called discrete.
The product rule immediately implies that:
The algebra A and so also its subalgebra A is an integral domain, that is, it contains no non-zero zero-divisors: from f g = 0 follows either f = 0
org=0.
The sum rule can be sharpened: in general, for any non-archimedean valuation, v(f + g) = min{v(f ), v(g)} whenever v(f) i4 v(g).
2. The theorem on units. An element e of a commutative ring R with 1 is called a unit in R if there exists an e E R with ee = 1. The units in R form a multiplicative group, the so-called group of units of R. To characterize the units of A we need
Lemma on units. Every convergent power series e = 1 - b1z - b2z2 - b3z3 - is a unit in A.
Proof. We will have ee = 1 fore := 1 + k1z + k2z2 + k3z3 + E A if we define
(*) k1 :=b1 , for all n>2.
It remains then to show that e E A. Because e E A there exists a real s > 0 such that IbnI < sn for all n > 1. From this and induction we get
I kn l < 211(2s)n , n = 1, 2,... .
Indeed, this is clear for n = 1 and the passage from n -1 to n proceeds via (*) as follows:
n-1 n-1
Iknl <- Ib.Ilkn-.I + Ibnl <- 2 s"(2s)n + sn = 2(2s)n.
Therefore for the positive number t := (2s)-1 we have Iknltn < 1/2 for all n > 1 and this means (recall the defining equation (1) in the introduction to this section) that e E A.
The preceding proof was given by HURWITZ in [12], pp. 28, 29; it prob- ably goes back to WEIERSTRASS. In 7.4.1 we will be able to give a, wholly different, "two-line proof"; for the polynomial ring C[z] there is however no analog of this lemma on units.
Theorem on units. An element f E A is a unit of A if and only if
f(0) 0 0.Proof. a) The condition is obviously necessary: if f E A and f j = 1, then f (0)j(0) = 1 and so f (0) 34 0.
b) Let f = E a"z" E A with ao = f (0) 54 0. For the series e:= ao l f = 1 + ao lalz + as Ia2Z2 + E A the preceding lemma furnishes an e E A with ee = 1. It follows that f (ao le) = 1.
The theorem on units can also be formulated thus:
f EAis aunit of Aav(f)=0.
The lemma and the theorem on units both naturally hold as well for formal power series; in this case the proof of the lemma just reduces to the first two lines of the above proof.
3. Normal form of a convergent power series. Every f E A\{0} has the form
(1)
f = ez"
for some unite of A and some n E N.The representation (1) is unique and n = v(f).
Proof. a) Let n = v(f), so that f has the form f = a" z" + a"+1 zn+1 +...
with an # 0. Then f = ez", where e := an + an+1 z + is, by the theorem on units, a unit of A.
b) Let f = ezI be another representation of f with m E N and e a unit of A. By the theorem on units, v(e) = v(e) = 0. Therefore from ez" = ez,
and the product rule for the order function it follows that n = v(e) + v(z") = v(ez") = V(ezm) = m and then e = e also follows.
We call (1) the normal form of f.
An element p of an integral domain R is called a prime element if it is not a unit of R and if whenever f, g E R and p divides the product f g, then p divides one of the factors f or g. An integral domain R each of whose
non-zero elements is a product of finitely many primes is called a unique factorization domain (in Bourbaki, simply factorial).
From the normal form (1) we see immediately that
Corollary. The ring A is factorial and, up to multiplication by units, the element z is the only prime in A.
In contrast to A, its subring C[z], which is itself a unique factorization domain, has the continuum-many prime elements z - c, c E C.
In the foregoing the prime z played a distinguished role. But the theorem and its corollary remain valid if instead of z any other fixed element r E A with v(r) = 1 is considered. Every such r is a prime element of A, on an equal footing with z, and is known in the classical terminology as a uniformizer of A.
Every integral domain R possesses a quotient field Q(R). It is immediate from the corollary that:
The quotient field Q(A) consists of all series of the form E">n a"z", n E Z, where Eo a"z" is a convergent power series.
Series of this kind are called "Laurent series with finite principal part"
(cf. 12.1.3). The reader is encouraged to supply a proof for the above statement about Q(A), as well as the following easy exercise:
The function v : A --' N U {oo} can be extended in exactly one way to a non-archimedean valuation v : Q(A) -. Z U {oo}. This valuation is given
by
v(f)=n
iff =>a"z"
with an i0.">n