Expansion of Complex function. Taylor's series, Maclaurin's series, Laurent's series

1. Power Series
N. B. Vyas
Department of Mathematics,
Atmiya Institute of Tech. and Science,
Rajkot (Guj.)
N.B.V yas − Department of M athematics, AIT S − Rajkot

2. Convergence of a Sequence
A sequence {zn } is said to be converge to z0
{as n approaches inf inity} if, for each > 0 there
exists a positive integer N such that
N.B.V yas − Department of M athematics, AIT S − Rajkot

3. Convergence of a Sequence
A sequence {zn } is said to be converge to z0
{as n approaches inf inity} if, for each > 0 there
exists a positive integer N such that
|z − z0 | < , whenever n≥N
N.B.V yas − Department of M athematics, AIT S − Rajkot

4. Convergence of a Sequence
A sequence {zn } is said to be converge to z0
{as n approaches inf inity} if, for each > 0 there
exists a positive integer N such that
|z − z0 | < , whenever n≥N
Symbolically we write lim zn = z0
n→∞
N.B.V yas − Department of M athematics, AIT S − Rajkot

5. Convergence of a Sequence
A sequence {zn } is said to be converge to z0
{as n approaches inf inity} if, for each > 0 there
exists a positive integer N such that
|z − z0 | < , whenever n≥N
Symbolically we write lim zn = z0
n→∞
A sequence which is not convergent is defined to be
divergent.
N.B.V yas − Department of M athematics, AIT S − Rajkot

6. Convergence of a Sequence
A sequence {zn } is said to be converge to z0
{as n approaches inf inity} if, for each > 0 there
exists a positive integer N such that
|z − z0 | < , whenever n≥N
Symbolically we write lim zn = z0
n→∞
A sequence which is not convergent is defined to be
divergent.
If lim zn = z0 we have
n→∞
N.B.V yas − Department of M athematics, AIT S − Rajkot

7. Convergence of a Sequence
A sequence {zn } is said to be converge to z0
{as n approaches inf inity} if, for each > 0 there
exists a positive integer N such that
|z − z0 | < , whenever n≥N
Symbolically we write lim zn = z0
n→∞
A sequence which is not convergent is defined to be
divergent.
If lim zn = z0 we have
n→∞
(i) |zn | → |z0 | as n → ∞
N.B.V yas − Department of M athematics, AIT S − Rajkot

8. Convergence of a Sequence
A sequence {zn } is said to be converge to z0
{as n approaches inf inity} if, for each > 0 there
exists a positive integer N such that
|z − z0 | < , whenever n≥N
Symbolically we write lim zn = z0
n→∞
A sequence which is not convergent is defined to be
divergent.
If lim zn = z0 we have
n→∞
(i) |zn | → |z0 | as n → ∞
(ii) the sequence {zn } is bounded
N.B.V yas − Department of M athematics, AIT S − Rajkot

9. Convergence of a Sequence
A sequence {zn } is said to be converge to z0
{as n approaches inf inity} if, for each > 0 there
exists a positive integer N such that
|z − z0 | < , whenever n≥N
Symbolically we write lim zn = z0
n→∞
A sequence which is not convergent is defined to be
divergent.
If lim zn = z0 we have
n→∞
(i) |zn | → |z0 | as n → ∞
(ii) the sequence {zn } is bounded
If zn = xn + iyn and z0 = x0 + iy0 then
N.B.V yas − Department of M athematics, AIT S − Rajkot

10. Convergence of a Sequence
A sequence {zn } is said to be converge to z0
{as n approaches inf inity} if, for each > 0 there
exists a positive integer N such that
|z − z0 | < , whenever n≥N
Symbolically we write lim zn = z0
n→∞
A sequence which is not convergent is defined to be
divergent.
If lim zn = z0 we have
n→∞
(i) |zn | → |z0 | as n → ∞
(ii) the sequence {zn } is bounded
If zn = xn + iyn and z0 = x0 + iy0 then
lim zn = z0 ⇒ lim xn = x0 and lim yn = y0
n→∞ n→∞ n→∞
N.B.V yas − Department of M athematics, AIT S − Rajkot

11. Convergence of a Sequence
The limit of convergent sequence is unique.
N.B.V yas − Department of M athematics, AIT S − Rajkot

12. Convergence of a Sequence
The limit of convergent sequence is unique.
lim zn = z and lim wn = w then
n→∞ n→∞
N.B.V yas − Department of M athematics, AIT S − Rajkot

13. Convergence of a Sequence
The limit of convergent sequence is unique.
lim zn = z and lim wn = w then
n→∞ n→∞
1 lim (zn ± wn ) = z + w
n→∞
N.B.V yas − Department of M athematics, AIT S − Rajkot

14. Convergence of a Sequence
The limit of convergent sequence is unique.
lim zn = z and lim wn = w then
n→∞ n→∞
1 lim (zn ± wn ) = z + w
n→∞
2 lim czn = cz
n→∞
N.B.V yas − Department of M athematics, AIT S − Rajkot

15. Convergence of a Sequence
The limit of convergent sequence is unique.
lim zn = z and lim wn = w then
n→∞ n→∞
1 lim (zn ± wn ) = z + w
n→∞
2 lim czn = cz
n→∞
3 lim zn wn = zw
n→∞
N.B.V yas − Department of M athematics, AIT S − Rajkot

16. Convergence of a Sequence
The limit of convergent sequence is unique.
lim zn = z and lim wn = w then
n→∞ n→∞
1 lim (zn ± wn ) = z + w
n→∞
2 lim czn = cz
n→∞
3 lim zn wn = zw
n→∞
zn z
4 lim = (w = 0)
n→∞ wn w
N.B.V yas − Department of M athematics, AIT S − Rajkot

17. Convergence of a Sequence
The limit of convergent sequence is unique.
lim zn = z and lim wn = w then
n→∞ n→∞
1 lim (zn ± wn ) = z + w
n→∞
2 lim czn = cz
n→∞
3 lim zn wn = zw
n→∞
zn z
4 lim = (w = 0)
n→∞ wn w
Given a sequence {an }. Consider a sequence {nk } of positive
integers such that n1 < n2 < n3 < . . . then the sequence
{ank } is called a subsequence of {an }.
N.B.V yas − Department of M athematics, AIT S − Rajkot

18. Convergence of a Sequence
The limit of convergent sequence is unique.
lim zn = z and lim wn = w then
n→∞ n→∞
1 lim (zn ± wn ) = z + w
n→∞
2 lim czn = cz
n→∞
3 lim zn wn = zw
n→∞
zn z
4 lim = (w = 0)
n→∞ wn w
Given a sequence {an }. Consider a sequence {nk } of positive
integers such that n1 < n2 < n3 < . . . then the sequence
{ank } is called a subsequence of {an }.
If {ank } converges then its limit is called Sub-sequential
limit
N.B.V yas − Department of M athematics, AIT S − Rajkot

19. Convergence of a Sequence
The limit of convergent sequence is unique.
lim zn = z and lim wn = w then
n→∞ n→∞
1 lim (zn ± wn ) = z + w
n→∞
2 lim czn = cz
n→∞
3 lim zn wn = zw
n→∞
zn z
4 lim = (w = 0)
n→∞ wn w
Given a sequence {an }. Consider a sequence {nk } of positive
integers such that n1 < n2 < n3 < . . . then the sequence
{ank } is called a subsequence of {an }.
If {ank } converges then its limit is called Sub-sequential
limit
A sequence {an } of complex numbers converges to p if and
only if every subsequence converges to p.
N.B.V yas − Department of M athematics, AIT S − Rajkot

20. Taylors Series
If f (z) is analytic inside a circle C with centre at z0 then for
all z inside C
N.B.V yas − Department of M athematics, AIT S − Rajkot

21. Taylors Series
If f (z) is analytic inside a circle C with centre at z0 then for
all z inside C
(z − z0 )2 (z − z0 )n n
f (z) = f (z0 )+(z−z0 )f (z0 )+ f (z0 )+. . .+ f (z0 )
2! n!
N.B.V yas − Department of M athematics, AIT S − Rajkot

22. Taylors Series
If f (z) is analytic inside a circle C with centre at z0 then for
all z inside C
(z − z0 )2 (z − z0 )n n
f (z) = f (z0 )+(z−z0 )f (z0 )+ f (z0 )+. . .+ f (z0 )
2! n!
Case 1: Putting z = z0 + h in above equation, we get
N.B.V yas − Department of M athematics, AIT S − Rajkot

23. Taylors Series
If f (z) is analytic inside a circle C with centre at z0 then for
all z inside C
(z − z0 )2 (z − z0 )n n
f (z) = f (z0 )+(z−z0 )f (z0 )+ f (z0 )+. . .+ f (z0 )
2! n!
Case 1: Putting z = z0 + h in above equation, we get
h2 hn
f (z0 + h) = f (z0 ) + hf (z0 ) + f (z0 ) + . . . + f n (z0 )
2! n!
N.B.V yas − Department of M athematics, AIT S − Rajkot

24. Taylors Series
If f (z) is analytic inside a circle C with centre at z0 then for
all z inside C
(z − z0 )2 (z − z0 )n n
f (z) = f (z0 )+(z−z0 )f (z0 )+ f (z0 )+. . .+ f (z0 )
2! n!
Case 1: Putting z = z0 + h in above equation, we get
h2 hn
f (z0 + h) = f (z0 ) + hf (z0 ) + f (z0 ) + . . . + f n (z0 )
2! n!
Case 2: If z0 = 0 then, we get
N.B.V yas − Department of M athematics, AIT S − Rajkot

25. Taylors Series
If f (z) is analytic inside a circle C with centre at z0 then for
all z inside C
(z − z0 )2 (z − z0 )n n
f (z) = f (z0 )+(z−z0 )f (z0 )+ f (z0 )+. . .+ f (z0 )
2! n!
Case 1: Putting z = z0 + h in above equation, we get
h2 hn
f (z0 + h) = f (z0 ) + hf (z0 ) + f (z0 ) + . . . + f n (z0 )
2! n!
Case 2: If z0 = 0 then, we get
z2 zn
f (z) = f (0) + zf (0) + f (0) + . . . + f n (0)
2! n!
N.B.V yas − Department of M athematics, AIT S − Rajkot

26. Taylors Series
If f (z) is analytic inside a circle C with centre at z0 then for
all z inside C
(z − z0 )2 (z − z0 )n n
f (z) = f (z0 )+(z−z0 )f (z0 )+ f (z0 )+. . .+ f (z0 )
2! n!
Case 1: Putting z = z0 + h in above equation, we get
h2 hn
f (z0 + h) = f (z0 ) + hf (z0 ) + f (z0 ) + . . . + f n (z0 )
2! n!
Case 2: If z0 = 0 then, we get
z2 zn
f (z) = f (0) + zf (0) + f (0) + . . . + f n (0)
2! n!
This series is called Maclaurin’s Series.
N.B.V yas − Department of M athematics, AIT S − Rajkot

27. Laurent Series
If f (z) is analytic in the ring shaped region R bounded by
two concentric circles c1 & c2 of radii r1 & r2 (r1 > r2 ) and
with centre at z0 , then for all z in R.
N.B.V yas − Department of M athematics, AIT S − Rajkot

28. Laurent Series
If f (z) is analytic in the ring shaped region R bounded by
two concentric circles c1 & c2 of radii r1 & r2 (r1 > r2 ) and
with centre at z0 , then for all z in R.
b1 b2
f (z) = a0 +a1 (z −z0 )+a2 (z −z0 )2 +. . .+ + +. . .
(z − z0 ) (z − z0 )2
N.B.V yas − Department of M athematics, AIT S − Rajkot

29. Laurent Series
If f (z) is analytic in the ring shaped region R bounded by
two concentric circles c1 & c2 of radii r1 & r2 (r1 > r2 ) and
with centre at z0 , then for all z in R.
b1 b2
f (z) = a0 +a1 (z −z0 )+a2 (z −z0 )2 +. . .+ + +. . .
(z − z0 ) (z − z0 )2
1 f (ξ)dξ
where an = , n = 0, 1, 2, . . .
2πi Γ (ξ − z0 )n+1
N.B.V yas − Department of M athematics, AIT S − Rajkot

30. Laurent Series
If f (z) is analytic in the ring shaped region R bounded by
two concentric circles c1 & c2 of radii r1 & r2 (r1 > r2 ) and
with centre at z0 , then for all z in R.
b1 b2
f (z) = a0 +a1 (z −z0 )+a2 (z −z0 )2 +. . .+ + +. . .
(z − z0 ) (z − z0 )2
1 f (ξ)dξ
where an = , n = 0, 1, 2, . . .
2πi Γ (ξ − z0 )n+1
Γ being any circle lying between c1 & c2 having z0 as its
centre, for all values of n.
N.B.V yas − Department of M athematics, AIT S − Rajkot

31. Laurent Series
If f (z) is analytic in the ring shaped region R bounded by
two concentric circles c1 & c2 of radii r1 & r2 (r1 > r2 ) and
with centre at z0 , then for all z in R.
b1 b2
f (z) = a0 +a1 (z −z0 )+a2 (z −z0 )2 +. . .+ + +. . .
(z − z0 ) (z − z0 )2
1 f (ξ)dξ
where an = , n = 0, 1, 2, . . .
2πi Γ (ξ − z0 )n+1
Γ being any circle lying between c1 & c2 having z0 as its
centre, for all values of n.
∞ ∞
n bn
∴ f (z) = an (z − z0 ) +
n=0 n=1
(z − z0 )n
N.B.V yas − Department of M athematics, AIT S − Rajkot

32. Laurent Series
If f (z) is analytic in the ring shaped region R bounded by
two concentric circles c1 & c2 of radii r1 & r2 (r1 > r2 ) and
with centre at z0 , then for all z in R.
b1 b2
f (z) = a0 +a1 (z −z0 )+a2 (z −z0 )2 +. . .+ + +. . .
(z − z0 ) (z − z0 )2
1 f (ξ)dξ
where an = , n = 0, 1, 2, . . .
2πi Γ (ξ − z0 )n+1
Γ being any circle lying between c1 & c2 having z0 as its
centre, for all values of n.
∞ ∞
n bn
∴ f (z) = an (z − z0 ) +
n=0 n=1
(z − z0 )n
N.B.V yas − Department of M athematics, AIT S − Rajkot

33. Note
If f (z) is analytic at z = z0 then we can expand f (z) by
means of Taylor’s series at a point z0
N.B.V yas − Department of M athematics, AIT S − Rajkot

34. Note
If f (z) is analytic at z = z0 then we can expand f (z) by
means of Taylor’s series at a point z0
Laurent series given an expansion of f (z) at a point z0 even
if f (z) is not analytic there.
N.B.V yas − Department of M athematics, AIT S − Rajkot

35. Singular Points
A point at which a function f (z) ceases to be analytic is
called a singular point of f (z)
N.B.V yas − Department of M athematics, AIT S − Rajkot

36. Singular Points
A point at which a function f (z) ceases to be analytic is
called a singular point of f (z)
If the function f (z) is analytic at every point in the
neighbourhood of a point z0 except at z0 is called isolated
singular point or isolated singularity.
N.B.V yas − Department of M athematics, AIT S − Rajkot

37. Singular Points
A point at which a function f (z) ceases to be analytic is
called a singular point of f (z)
If the function f (z) is analytic at every point in the
neighbourhood of a point z0 except at z0 is called isolated
singular point or isolated singularity.
1 1
Eg. 1 f (z) = ⇒ f (z) = − 2 , it follows that f (z) is analytic at
z z
every point except at z = 0 , hence z = 0 is an isolated
singularity.
N.B.V yas − Department of M athematics, AIT S − Rajkot

38. Singular Points
A point at which a function f (z) ceases to be analytic is
called a singular point of f (z)
If the function f (z) is analytic at every point in the
neighbourhood of a point z0 except at z0 is called isolated
singular point or isolated singularity.
1 1
Eg. 1 f (z) = ⇒ f (z) = − 2 , it follows that f (z) is analytic at
z z
every point except at z = 0 , hence z = 0 is an isolated
singularity.
1
Eg. 2 f (z) = 3 2 has three isolated singularities at
z (z + 1)
z = 0, i, −i
N.B.V yas − Department of M athematics, AIT S − Rajkot

39. Singular Points
If z = z0 is a isolated singular point, then f (z) can be
expanded in a Laurents series in the form.
N.B.V yas − Department of M athematics, AIT S − Rajkot

40. Singular Points
If z = z0 is a isolated singular point, then f (z) can be
expanded in a Laurents series in the form.
∞ ∞
n bn
f (z) = an (z − z0 ) +
n=0 n=1
(z − z0 )n
N.B.V yas − Department of M athematics, AIT S − Rajkot

41. Singular Points
If z = z0 is a isolated singular point, then f (z) can be
expanded in a Laurents series in the form.
∞ ∞
n bn
f (z) = an (z − z0 ) + (1)
n=0 n=1
(z − z0 )n
∞
In (1) an (z − z0 )n is called the regular part and
n=0
∞
bn
is called the principal part of f (z) in the
n=1
(z − z0 )n
neighbourhood of z0 .
N.B.V yas − Department of M athematics, AIT S − Rajkot

42. Singular Points
If z = z0 is a isolated singular point, then f (z) can be
expanded in a Laurents series in the form.
∞ ∞
n bn
f (z) = an (z − z0 ) + (1)
n=0 n=1
(z − z0 )n
∞
In (1) an (z − z0 )n is called the regular part and
n=0
∞
bn
is called the principal part of f (z) in the
n=1
(z − z0 )n
neighbourhood of z0 .
If the principal part of f (z) contains infinite numbers of
terms then z = z0 is called an isolated essential singularity of
f (z).
N.B.V yas − Department of M athematics, AIT S − Rajkot

43. Singular Points
If z = z0 is a isolated singular point, then f (z) can be
expanded in a Laurents series in the form.
∞ ∞
n bn
f (z) = an (z − z0 ) + (1)
n=0 n=1
(z − z0 )n
∞
In (1) an (z − z0 )n is called the regular part and
n=0
∞
bn
is called the principal part of f (z) in the
n=1
(z − z0 )n
neighbourhood of z0 .
If the principal part of f (z) contains infinite numbers of
terms then z = z0 is called an isolated essential singularity of
f (z).
N.B.V yas − Department of M athematics, AIT S − Rajkot

44. Singular Points
If in equation(1) , the principal part has all the coefficient
bn+1 , bn+2 , . . . as zero after a particular term bn then the
Laurents series of f (z) reduces to
N.B.V yas − Department of M athematics, AIT S − Rajkot

45. Singular Points
If in equation(1) , the principal part has all the coefficient
bn+1 , bn+2 , . . . as zero after a particular term bn then the
Laurents series of f (z) reduces to
∞
b1 b2 bn
f (z) = an (z − z0 )n + + 2
+ ... +
n=0
(z − z0 ) (z − z0 ) (z − z0 )n
N.B.V yas − Department of M athematics, AIT S − Rajkot

46. Singular Points
If in equation(1) , the principal part has all the coefficient
bn+1 , bn+2 , . . . as zero after a particular term bn then the
Laurents series of f (z) reduces to
∞
b1 b2 bn
f (z) = an (z − z0 )n + + 2
+ ... +
n=0
(z − z0 ) (z − z0 ) (z − z0 )n
i.e. (Regular part) + (Principal part is a polynomial of finite
1
number of terms in
z − z0
N.B.V yas − Department of M athematics, AIT S − Rajkot

47. Singular Points
If in equation(1) , the principal part has all the coefficient
bn+1 , bn+2 , . . . as zero after a particular term bn then the
Laurents series of f (z) reduces to
∞
b1 b2 bn
f (z) = an (z − z0 )n + + 2
+ ... +
n=0
(z − z0 ) (z − z0 ) (z − z0 )n
i.e. (Regular part) + (Principal part is a polynomial of finite
1
number of terms in
z − z0
The the singularity in this case at z = z0 is called a pole of
order n.
N.B.V yas − Department of M athematics, AIT S − Rajkot

48. Singular Points
If in equation(1) , the principal part has all the coefficient
bn+1 , bn+2 , . . . as zero after a particular term bn then the
Laurents series of f (z) reduces to
∞
b1 b2 bn
f (z) = an (z − z0 )n + + 2
+ ... +
n=0
(z − z0 ) (z − z0 ) (z − z0 )n
i.e. (Regular part) + (Principal part is a polynomial of finite
1
number of terms in
z − z0
The the singularity in this case at z = z0 is called a pole of
order n.
If the order of the pole is one, the pole is called simple pole.
N.B.V yas − Department of M athematics, AIT S − Rajkot