We already know about real sequence and series. Here we will study some important series of complex numbers.
Sequence:
Definition 1.1. A function of a positive integral variable, designated by f(n) or un, where n = 1, 2, 3, . . . , is called a sequence.
Thus, a sequence is a set of numbersu1,u2, u3, ... in a definite order of arrangement and formed according to a definite rule. Each number in the sequence is called a term and un, is called the nth term. The sequence is denoted by {un}.
Example 1.1. We give some examples.
• The sequence i, i2, i3, ..., i100 is a finite sequence.
• The sequence 1 +i, (1 +i)2, (1 +i)3, ... is an infinite sequence.
Limit of a Sequence:
Definition 1.2. A number l is called the limit of an infinite sequence u1, u2, u3, ... if for any positive number we can find a positive number N depending on such that
|un−l|< f or all n > N.
In such case, we write limn→∞un = l. If the limit of a sequence exists, the sequence is called convergent; otherwise it is called divergent.
A sequence can converge to only one limit, i.e., if a limit exists it is unique.
Series:
Let u1, u2,u3, ... be a given sequence. Form a new sequenceS1, S2, S3, ... defined by S1 =u1
S2 =u1+u2 S3 =u1+u2+u3
...,
Sn =u1+u2+u3+...+un
where Sn is called the nth partial sum.
The sequence is symbolized by
u1+u2+u3+...=
∞
X
n=1
un.
which is called an infinite series.
If limn→∞Sn=S exists, the series is called convergent and S is its sum; otherwise the series is called divergent.
A necessary condition that a series converges is limn→∞un = 0, however, this is not sufficient.
Sequences of functions:
Letu1(z),u2(z),u3(z), ... denoted briefly by un(z) be a sequence of functions of z defined and single-valued in some region of the z plane.
Definition 1.3. We call U(z) the limit of un(z) as n → ∞, and write limn→∞un(z) = U(z) if given any positive number , we can find a number N [depending in general on both and z] such that
|un−U(z)|< f or all n > N.
In such a case, we say that the sequenceconverges or is convergent to U(z).
If a sequence converges for all values of z (points) in a region R, we call R theregion of convergence of the sequence. A sequence that is not convergent at some value (point) z is called divergent at z.
Series of functions:
From the sequence of functions un(z) let us form a new sequence Sn(z) defined by S1(z) = u1(z)
S2(z) =u1(z) +u2(z) S3(z) =u1(z) +u2(z) +u3(z)
...
Sn(z) = u1(z) +u2(z) +u3(z) +...+un(z) where Sn(z) is called the nth partial sum of un(z).
The sequence is symbolized by
u1(z) +u2(z) +u3(z) +...=
∞
X
n=1
un(z) (1.1)
which is called an infinite series.
If limn→∞Sn(z) = S(z) exists, the series is called convergent and S(z) is its sum;
otherwise the series is called divergent.
If a series converges for all values of z (points) in a region R, we call R the region of convergence of the series.
2 Absolute and Uniform Convergence
Absolute Convergence:
Definition 2.1. A series P∞
n=1un(z) is called absolutely convergent if the series of absolute values, i.e., P∞
n=1|un(z)| converges.
If P∞
n=1un(z) converges but P∞
n=1|un(z)| does not converge, we call P∞
n=1un(z) condi- tionally convergent.
Uniform Convergence of Sequences and Series:
In the definition of limit of a sequence of functions, it was pointed out that the number N depends in general on and the particular value of z. It may happen, however, that we can find a number N such that
|un−U(z)|< f or all n > N.
where the same number N holds for all z in a region R [i.e., N depends only on and not on the particular value of z (point) in the region]. In such a case, we say that un(z) converges uniformly, or is uniformly convergent, to U(z) for all z in R.
Similarly, if the sequence of partial sums{Sn(z)}converges uniformly to S(z) in a region, we say that the infinite series (1.1) converges uniformly, or is uniformly convergent, to S(z) in the region.
We call
Rn(z) =un+1(z) +un+2(z) +....=S(z)−Sn(z) the remainder of the infinite series (1.1) after n terms.
Then, we can equivalently say that the series is uniformly convergent to S(z) in R if,
given any >0 we can find a number N such that for all z in R,
|Rn(z)|=|S(z)−Sn(z)|< , ∀ n > N.
3 Power Series
A series having the form
a0(z) +a1(z−a) +a2(z−a)2+...=
∞
X
n=0
an(z−a)n (3.1) is called a power series in (z−a).
Clearly the power series (1.2) converges for z =a, and this may indeed be the only point for which it converges.
In general, however, the series converges for other points as well.
In such cases, we can show that there exists a positive number R such that (1.2) converges for |z−a| < R and diverges for |z−a| > R , while for |z−a| =R, it may or may not converge.
Geometrically, if Γ is a circle of radius R with center at z = a, then the series (1.2) converges at all points inside Γ and diverges at all points outside Γ, while it may or may not converge on the circle Γ.
R is called the radius of convergence of (1.2) and the corresponding circle is called the circle of convergence.
Theorems on Power series:
Theorem 3.1. A power series converges uniformly and absolutely in any region that lies entirely inside its circle of convergence.
Theorem 3.2. 1. A power series can be differentiated term by term in any region that lies entirely inside its circle of convergence.
2. A power series can be integrated term by term along any curve C that lies entirely inside its circle of convergence.
3. The sum of a power series is continuous in any region that lies entirely inside its circle of convergence.
Theorem 3.3. (Abel’s Theorem:) Let Σanzn have radius of convergence R and suppose that z0 is a point on the circle of convergence such that Σanz0n converges. Then,
z→zlim0Σanzn= Σanz0n from within the circle of convergence.
Theorem 3.4. SupposeΣanzn converges to zero for all z such that|z|< R where R >0.
Then an = 0. Equivalently, if
Σanzn = Σbnzn for all z such that |z|< R where R >0. Then an=bn.
4 Taylor Series
4.1 Taylor’s Theorem
Let f(z) be analytic inside and on a simple closed curve C. Let a and (a + h) be two points inside C. Then
f(a+h) =f(a) +hf0(a) + h2
2!f00(a) +...+hn
n!f(n)(a) +... (4.1) or writing z =a+h, h=z−a,
f(z) = f(a) + (z−a)f0(a) + (z−a)2f00(a)
2! +...+ (z−a)nf(n)(a)
n! +... (4.2) This is called Taylors theorem and the series (2.1) or (2.2) is called a Taylor series or expansion for f(a + h) or f(z). The region of convergence of the series (2.2) is given by
|z−a|< R where R is the radius of convergence.
If a = 0 in (2.1) or (2.2), the resulting series is often called a Maclaurin series.
Some Special Series:
The following list shows some special series together with their regions of convergence.
1. ez = 1 +z+z2!2 + z3!3 +...+ zn!n +... |z|<∞
2. sinz =z− z3!3 + z5!5 +...+ (−1)n−1(2n−1)!z2n−1 +... |z|<∞ 3. cosz = 1− z2!2 + z4!4 +...+ (−1)n−1(2n−2)!z2n−2 +... |z|<∞
4. ln(1 +z) = z− z22 +z33 +...+ (−1)n−1zn!n +... |z|<1 5. tan−1z =z− z33 + z55 +...+ (−1)n−1(2n−1)z2n−1 +... |z|<1 6. (1 +z)p = 1 +pz− p(p−1)2! z2+...+p(p−1)...(p−n+1)
n! +... |z|<1.
4.2 Some Problems:
Example 4.1. Expand f(z) = ln(1 +z) in a Taylor series about the point z= 0.
Solution: Here
f(z) = ln(1 +z), f(0) = 0 f0(z) = 1
1 +z, f0(0) = 1 f00(z) = − 1
(1 +z)2, f00(0) =−1 f000(z) =− 2
(1 +z)3, f000(0) = 2!
...
f(n+1)(z) = − (−1)nn!
(1 +z)n+1, f(n+1)(0) = (−1)nn!.
Then
f(z) = ln(1 +z) =f(0) +zf0(0) +z2f00(0)
2! +...+znf(n)(0)
n! +...
=z− z2 2 +z3
3 − z4
4 +...
and converges for |z|<1.
Another method:
If |z|<1,
1
1 +z = 1−z+z2−z3+...
Then integrating from 0 to z yields
ln(1 +z) = z−z2 2 + z3
3 − z4
4 +...
Example 4.2. Expand f(z) = sinz in a Taylor series about the point z = π4. Solution: Here
f(z) = sinz, f(π
4) = 1
√2 f0(z) = cosz, f0(π
4) = 1
√2 f00(z) =−sinz, f00(π
4) = − 1
√2 f000(z) =−cosz, f000(π
4) =− 1
√2 ...
Then
f(z) = ln(1 +z) =f(0) +zf0(0) +z2f00(0)
2! +...+znf(n)(0)
n! +...
= 1
√2+ (z−π 4) 1
√2 − 1
√2
(z−π4)2
2! − 1
√2
(z− π4)3
3! +...
= 1
√2[1 + (z−π
4)− (z− π4)2
2! − (z− π4)3
3! +...]
and converges for |z|<∞.
Exercise:
Expand each of the following functions in a Taylor series about the indicated point and determine the region of convergence in each case.
1. e−z;z = 0 2. ze2z;z =−1 3. cosz;z= π2 4. (zsin2+4)z ;z = 0 5. (ezz+1);z = 0.
• Prove that
tan−1z=z− z3 3 +z5
5 − z7
7 +..., |z|<1.
• Prove that
sinz2 =z2− z6 3! + z10
5! − z14
7 +..., |z|<∞.
