Complex power series

Let A be a subset of C, and let {fn} = {fn}^∞n=0 be a sequence of functions defined on A (in the previous chapter, sequences were indexed by Z>0; for convenience, in this chapter, they will be indexed by Z≥0). We form the new sequence (known as a series) {S_N}, where S_N(z) =

N n=0

f_n(z), and the formal infinite series ∞ n=0

f_n(z) .

The sequence of complex numbers {S_N(z)}^∞n=0 is also known as the sequence of partial sums associated with the infinite series

∞ n=0

fn(z) at the point z ∈ A. When the indices of summation are clear from the context, we often omit them. For example,

f_n(ζ) usually means the infinite sum

∞ n=0

fn(ζ). Other similar abbreviations are used.

Definition 3.1. i) We say that the infinite series ∞ n=0

f_n(z) con- verges at a pointζ ∈Aif{SN(ζ)}converges. In this case, we write ∞

n=0

f_n(ζ) = lim

n→∞S_N(ζ).

ii) We say that the infinite series ∞ n=0

f_n(z)converges pointwise inA if {SN(ζ)} converges for everyζ ∈A.

iii) We say that the infinite series ∞ n=0

f_n(z) converges absolutely at a point ζ ∈A if the infinite series

∞ n=0

|f_n(ζ)|converges.

iv) We say that the infinite series ∞ n=0

f_n(z) converges uniformly in A if the sequence of partial sums {S_N(z)} converges uniformly in A.

3.1. COMPLEX POWER SERIES 25

v) We say that the infinite series ∞ n=0

f_n(z)converges normally on a set B ⊂ A if there exists a sequence of positive constants {M_n} such that

(1) |f_n(z)| ≤M_n for allz ∈B and all n, and

(2)

M_n<∞.

vi) We say that the infinite series ∞ n=0

f_n(z) diverges at a point of A if it does not converge at that point.

We speak of the pointwise, uniform, absolute, or normal conver- gence of a series as well as of thedivergence of a series.

Remark 3.2. The fact that the sequence {S_N(ζ)}converges if and only if {S_N(ζ)} is Cauchy allows us to rephrase conditions (i–iv) as if and only if statements when useful: (i)

f_n(z) converges at a point ζ ∈Aif and only if{SN(ζ)}is Cauchy; (ii)

fn(z)converges pointwise in A if and only if {S_N(ζ)} is Cauchy for every ζ ∈ A; (iii)

f_n(z) converges absolutely at a point ζ ∈ A if and only if the infinite series |f_n(ζ)| converges; and (iv)

f_n(z) converges uniformly inA if and only if the sequence of partial sums {S_N(z)}converges uniformly inA.

Remark3.3. Many questions on convergence of complex sequences are reduced to the real case by the trivial but important observation that absolute convergence at a point implies convergence at that point.

Remark 3.4. Two other such observations, both trivial but impor- tant, are that if an infinite series

∞ n=0

f_n(z) converges at a point ζ, then

nlim→∞f_n(ζ) = 0 and lim

n→∞

∞ k=n

f_k(ζ) = 0.

Some relationships between some different types of convergence of a series are given in the following result.

Weierstrass M-test. Normal convergence implies uniform and absolute convergence.

Proof. With notation as in the last definition, if N1 < N are positive integers, then

|S_N(z)−S_N1(z)| ≤ N n=N₁+1

M_n for all z ∈A (needed for the uniform convergence argument), and

| |SN(z)| − |SN₁(z)| | ≤ N n=N₁+1

Mn for allz ∈A (needed for the absolute convergence argument).

Given any > 0, we can find a positive integer N₀ such that N >

N1 > N0 implies N n=N₁+1

M_n< .

We shall be mostly interested in series of the form _∞

n=0a_nzⁿ with a_n∈C(these are known aspower series) and theassociated real-valued series _∞

n=0|a_n|rⁿ where r =|z|. We define, for each N ∈Z≥0, S_N^∗ (r) =

N n=0

|a_n|rⁿ for r∈R≥0, and we observe that

S_N^∗₊₁(r)≥ S_N^∗(r) for all N ∈Z≥0 and for all r ∈R≥0 . We refer to S_N^∗(r) as thereal partial sum at r.

An elementary but most important example is provided by the geo- metric series with r ∈R≥0:

1 +r+r²+. . . .

Note thatS_N(1) =N + 1, and S_N(r) = 1−r^N+1

1−r for 0≤r <1, as well as for r >1.

Thus ∞ n=0

rⁿ = 1

1−r if 0≤r <1 and

rⁿ diverges if r≥1.

We now introduce two special cases of divergence of a sequence of real numbers. It will be useful to regard these sequences as convergent sequences with infinite limits.

3.1. COMPLEX POWER SERIES 27

Definition3.5. A sequence of real numbers{bn}converges to+∞ if and only if for all M >0 there exists an N ∈Z>0 such thatbn > M for all n > N. In this case we shall write lim_n→∞b_n = +∞. A similar definition applies to real sequences converging to −∞.

With this notation either:

(a) lim

N→∞S_N^∗(r) exists and is finite: That is,_∞

n=0|an|rⁿconverges.

In this case we write _∞

n=0|a_n|rⁿ<+∞; or

(b) lim

N→∞S_N^∗(r) = +∞: That is,_∞

n=0|a_n|rⁿ diverges (in the pre- vious sense). In this case we write _∞

n=0|a_n|rⁿ= +∞. For real sequences, we have thecomparison test. Let 0≤a_n ≤b_n.

(a) If

a_n= +∞, then

b_n = +∞. (b) If

b_n<+∞, then

a_n <+∞.

Abel’s lemma. Let 0< r < r0. Assume there exists an M ∈R>0

such that

|a_nrⁿ₀| ≤M for all n∈Z>0 . Then the series

a_nzⁿconverges normally for all zwith |z| ≤r. In particular, it converges absolutely and uniformly for all z with |z| ≤r.

Proof. For|z| ≤r we have

|a_nzⁿ|=|a_n| |z|ⁿ ≤ |a_n| |r|ⁿ=|a_n| r

r₀

n

rⁿ₀ ≤M r

r₀

n

. Comparison with the geometric series [or an application of the Weierstrass M-test with M_n = M(_r^r

0)ⁿ] shows the normal conver-

gence.

IfSis any nonempty set of real numbers, then the least upper bound or supremum of S is denoted by supS and the greatest lower bound or infimum is denoted by infS. The possibilities that supS = +∞ or infS=−∞ are allowed.

Definition 3.6. The radius of convergence ρ of the power series a_nzⁿ is given by

ρ= sup{r ≥0;

|a_n|rⁿ <+∞} .

Note that 0 ≤ ρ ≤+∞. As a result of the next theorem it makes sense to define{z ∈C; |z|< ρ}as thedisk of convergenceof the power series

a_nzⁿ.

Theorem 3.7. Let

anzⁿbe a power series with radius of conver- gence ρ.

(a) If 0< r < ρ, then

a_nzⁿ converges normally, absolutely, and uniformly for |z| ≤r.

(b) The series

anzⁿ diverges for |z|> ρ.

Proof. (a) Choose r₀ with r < r₀ < ρsuch that

|a_n|r₀ⁿ<+∞. Thus there exists an M > 0 with |an|rⁿ₀ ≤ M for all n in Z>0. Now apply Abel’s lemma.

(b) We claim that for |z| > ρ, the sequence {|a_n| |z|ⁿ} is not even bounded. Otherwise Abel’s lemma (with r₀ = |z|) would guarantee the existence of an r with ρ < r < |z| and

|a_n|rⁿ < +∞. This

contradicts the definition of ρ.

Corollary 3.8. Let

a_nzⁿ be a power series with radius of con- vergence ρ. Then the function defined by S(z) =

a_nzⁿ is continuous for |z|< ρ.

Proof. It follows immediately from Theorems 2.23 and 3.7.

We now turn to the obvious and important question: How do we compute ρ?

To answer this question, we introduce the concepts of lim sup and lim inf.

Definition 3.9. Let {u_n} be a real sequence. We use ≡ to indi- cate equivalent names [we also use this same notation with a different meaning in other places, such as f ≡ 0 or f ≡ g, to emphasize that these functions are (identically) equal] and define

limn u_n≡lim sup

n

u_n≡ upper limit of {u_n}

≡ limit superior of {u_n}= lim

p→∞sup

n≥p{u_n}= inf

p sup

n≥p{u_n}, and

lim

n

u_n≡lim inf

n u_n≡ lower limit of {u_n}

≡ limit inferior of {u_n}= lim

p→∞inf

n≥p{u_n}= sup

p

ninf≥p{u_n}. Note that every real sequence has a limit superior as well as a limit inferior, which are either real numbers or +∞or −∞.

Properties of limits superior and inferior. Let{un}and {vn} be real sequences. Then:

3.1. COMPLEX POWER SERIES 29

(a) lim inf

n un≤lim sup

n

un. (b) lim inf

n (−u_n) =−lim sup

n

u_n. (c) If r >0, then

lim inf

n (ru_n) =rlim inf

n u_n, and lim sup

n

(ru_n) =rlim sup

n

u_n. (d) If u_n≤v_n for all n, then

lim inf

n u_n≤lim inf

n v_n, and lim sup

n

u_n≤lim sup

n

v_n. (e) lim

n u_n =L if and only if lim inf

n u_n=L= lim sup

n

u_n (L is allowed to be±∞ in this context).

(f) lim inf

n (u_n+v_n)≥lim inf

n u_n+ lim inf

n v_n, and lim sup

n

(u_n+v_n)≤lim sup

n

u_n+ lim sup

n

v_n.

Remark and exercise. Iff is a real-valued function of a complex variable, defined on a set S in C and ifζ is a limit point of S, then it is possible to define lim

z→ζf(z) and lim

z→ζ

f(z).

Example 3.10. Let u_n = sin_nπ

2

, n = 0,1,2, . . . . This is the sequence {0,1,0,−1, . . .}. Hence for all p∈Z_≥0,

a_p = sup

n≥p

u_n= 1 and thus lim sup

n

u_n= lim

p→∞a_p = 1, and b_p = inf

n≥p u_n=−1 and thus lim inf

n u_n= lim

p→∞b_p =−1.

The ratio test will be well known to most readers (see Exercise 3.4). Suppose that v_n>0 for all non-negative integers n.

(a) If lim

n→∞

v_n+1

v_n =L <1, then

v_n converges.

(b) If lim

n→∞

v_n+1 vn

=L >1, then

v_n diverges.

Perhaps less familiar is the root test. Suppose that vn≥0 all n.

(a) If lim

n (v_n)ⁿ¹ =L <1, then

v_n converges.

(b) If lim

n (v_n)ⁿ¹ =L >1, then

v_n diverges.

Proof. (a) Choose > 0 such that 0< L+ <1. There exists a P ∈Z>0 such that

sup

n≥p

vn¹ⁿ

< L+ for all p≥P . Thus

v_n<(L+)ⁿ for all n≥P,

and comparison with the geometric series yields convergence.

(b) Suppose that

v_n<+∞. Then lim

n v_n = 0. Thus there exists a P in Z>0 such that n ≥P implies that v_n <1 for all n ≥P. Hence (v_n)ⁿ¹ <1 for all n≥P and therefore lim

n (v_n)ⁿ¹ ≤1.

Remark 3.11. The root test applies (with the same value of L) whenever the ratio test applies. However, the converse is not true.

To see this take a sequence where the ratios are alternately ¹₂ and ¹₈. Then the root test will apply with L = ¹₄, but the sequence of ratios, obviously, will not converge.

Example 3.12. Consider the sequence n^−s with s a positive real number. Both the ratio and the root test end up with L = 1. The series diverges (converges to +∞) for 0 < s ≤ 1 and converges for s >1.

We now return to the study of the complex power series _∞

n=0a_nzⁿ and to the problem of computing the radius of convergence.

Theorem3.13. Let

a_nzⁿ be a power series. Suppose thata_n = 0 for all n and that the limit lim

n

a_n+1 an

=L exists, with 0≤L≤+∞. Then the radius of convergence ρ of the power series _∞

n=0anzⁿ is

1

L; in other words,

1

ρ = lim

n→∞

an+1

a_n , where 1

+∞ is to be understood as being = 0.

The hypotheses required for this result to hold are strong. As pointed out in Remark 3.11, the ratio test is stronger than the root test. The next result provides a way of computing the radius of con- vergence for any power series.

3.1. COMPLEX POWER SERIES 31

Theorem 3.14 (Hadamard). The radius of convergence ρ of the power series _∞

n=0anzⁿ is given by 1 ρ = lim

n |a_n|ⁿ¹ . Proof. Let L = lim

n |a_n|¹ⁿ. Thus lim

n |a_nrⁿ|¹ⁿ = rL for all r ≥ 0 and we conclude by the root test that the associated series

|a_n|rⁿ converges for 0 ≤r < _L¹ and diverges forr > _L¹. Thus ρ= _L¹.

Lemma 3.15. Let

u_n and

v_n be two absolutely convergent se- ries. Define

wn= n

p=0

upvn−p.

Then

wn is absolutely convergent and

wn= (

un) ( vn).

Proof. Letα_p =

n≥p

|u_n| and β_p =

n≥p

|v_n|. Then limp α_p = 0 = lim

p β_p . Also

N n=0

|w_n| ≤^∞

n=0

|u_n|^∞

n=0

|v_n|=α₀β₀ <+∞,

and therefore, ∞ n=0

|wn| < +∞. Thus we have proven the absolute convergence of the new series.

To show the required equality, choose m and n with m ≥ 2n and consider

m k=0

w_k− _n

k=0

u_k

n k=0

v_k

=L.

We have to show that L →0 as n → ∞ (we already know each of the above series converges). We rewrite

L=

m k=0

k i=0

uivk−i− n

j=0

n i=0

uivj

.

5

4

3

2

1

-1

k

2 4 i

(a) The first sum

5

4

3

2

1

-1 k

2 4 i

(b) The second sum

Figure 3.1. The (i, k) plane

By looking at the diagrams in the (i, k) plane shown in Figure 3.1, we see that

m k=0

k i=0

u_iv_k−i= m

i=0

m k=i

u_iv_k−i

= m

i=0

u_i m

k=i

v_k−i = m

i=0

u_i

m−i

j=0

v_j = m

i=0 m−i

j=0

u_iv_j . Thus we can estimate

L=

n i=0

_m−i

j=0

u_iv_j− n

j=0

u_iv_j

+ m i=n+1

m−i

j=0

u_iv_j

≤

n i=0

m−i

j=n+1

u_iv_j +

m i=n+1

m−i

j=0

u_iv_j

≤ ∞

i=0

∞ j=n+1

|u_i| |v_j|+ ∞ i=n+1

∞ j=0

|u_i| |v_j|= (α₀β_n+1+β₀α_n+1), and the last expression approaches 0 as n goes to∞.

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