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Power Series
N. B. Vyas
Department of Mathematics,Atmiya Institute of Tech. and Science,
Rajkot (Guj.)
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 2: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/2.jpg)
Convergence of a Sequence
A sequence {zn} is said to be converge to z0
{as n approaches infinity} if, for each ε > 0 thereexists a positive integer N such that
|z − z0| < ε, whenever n ≥ N
Symbolically we write limn→∞
zn = z0
A sequence which is not convergent is defined to bedivergent.
If limn→∞
zn = z0 we have
(i) |zn| → |z0| as n→∞(ii) the sequence {zn} is bounded
If zn = xn + iyn and z0 = x0 + iy0 then
limn→∞
zn = z0 ⇒ limn→∞
xn = x0 and limn→∞
yn = y0
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 3: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/3.jpg)
Convergence of a Sequence
A sequence {zn} is said to be converge to z0
{as n approaches infinity} if, for each ε > 0 thereexists a positive integer N such that
|z − z0| < ε, whenever n ≥ N
Symbolically we write limn→∞
zn = z0
A sequence which is not convergent is defined to bedivergent.
If limn→∞
zn = z0 we have
(i) |zn| → |z0| as n→∞(ii) the sequence {zn} is bounded
If zn = xn + iyn and z0 = x0 + iy0 then
limn→∞
zn = z0 ⇒ limn→∞
xn = x0 and limn→∞
yn = y0
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 4: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/4.jpg)
Convergence of a Sequence
A sequence {zn} is said to be converge to z0
{as n approaches infinity} if, for each ε > 0 thereexists a positive integer N such that
|z − z0| < ε, whenever n ≥ N
Symbolically we write limn→∞
zn = z0
A sequence which is not convergent is defined to bedivergent.
If limn→∞
zn = z0 we have
(i) |zn| → |z0| as n→∞(ii) the sequence {zn} is bounded
If zn = xn + iyn and z0 = x0 + iy0 then
limn→∞
zn = z0 ⇒ limn→∞
xn = x0 and limn→∞
yn = y0
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 5: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/5.jpg)
Convergence of a Sequence
A sequence {zn} is said to be converge to z0
{as n approaches infinity} if, for each ε > 0 thereexists a positive integer N such that
|z − z0| < ε, whenever n ≥ N
Symbolically we write limn→∞
zn = z0
A sequence which is not convergent is defined to bedivergent.
If limn→∞
zn = z0 we have
(i) |zn| → |z0| as n→∞(ii) the sequence {zn} is bounded
If zn = xn + iyn and z0 = x0 + iy0 then
limn→∞
zn = z0 ⇒ limn→∞
xn = x0 and limn→∞
yn = y0
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 6: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/6.jpg)
Convergence of a Sequence
A sequence {zn} is said to be converge to z0
{as n approaches infinity} if, for each ε > 0 thereexists a positive integer N such that
|z − z0| < ε, whenever n ≥ N
Symbolically we write limn→∞
zn = z0
A sequence which is not convergent is defined to bedivergent.
If limn→∞
zn = z0 we have
(i) |zn| → |z0| as n→∞(ii) the sequence {zn} is bounded
If zn = xn + iyn and z0 = x0 + iy0 then
limn→∞
zn = z0 ⇒ limn→∞
xn = x0 and limn→∞
yn = y0
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 7: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/7.jpg)
Convergence of a Sequence
A sequence {zn} is said to be converge to z0
{as n approaches infinity} if, for each ε > 0 thereexists a positive integer N such that
|z − z0| < ε, whenever n ≥ N
Symbolically we write limn→∞
zn = z0
A sequence which is not convergent is defined to bedivergent.
If limn→∞
zn = z0 we have
(i) |zn| → |z0| as n→∞
(ii) the sequence {zn} is bounded
If zn = xn + iyn and z0 = x0 + iy0 then
limn→∞
zn = z0 ⇒ limn→∞
xn = x0 and limn→∞
yn = y0
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 8: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/8.jpg)
Convergence of a Sequence
A sequence {zn} is said to be converge to z0
{as n approaches infinity} if, for each ε > 0 thereexists a positive integer N such that
|z − z0| < ε, whenever n ≥ N
Symbolically we write limn→∞
zn = z0
A sequence which is not convergent is defined to bedivergent.
If limn→∞
zn = z0 we have
(i) |zn| → |z0| as n→∞(ii) the sequence {zn} is bounded
If zn = xn + iyn and z0 = x0 + iy0 then
limn→∞
zn = z0 ⇒ limn→∞
xn = x0 and limn→∞
yn = y0
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 9: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/9.jpg)
Convergence of a Sequence
A sequence {zn} is said to be converge to z0
{as n approaches infinity} if, for each ε > 0 thereexists a positive integer N such that
|z − z0| < ε, whenever n ≥ N
Symbolically we write limn→∞
zn = z0
A sequence which is not convergent is defined to bedivergent.
If limn→∞
zn = z0 we have
(i) |zn| → |z0| as n→∞(ii) the sequence {zn} is bounded
If zn = xn + iyn and z0 = x0 + iy0 then
limn→∞
zn = z0 ⇒ limn→∞
xn = x0 and limn→∞
yn = y0
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 10: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/10.jpg)
Convergence of a Sequence
A sequence {zn} is said to be converge to z0
{as n approaches infinity} if, for each ε > 0 thereexists a positive integer N such that
|z − z0| < ε, whenever n ≥ N
Symbolically we write limn→∞
zn = z0
A sequence which is not convergent is defined to bedivergent.
If limn→∞
zn = z0 we have
(i) |zn| → |z0| as n→∞(ii) the sequence {zn} is bounded
If zn = xn + iyn and z0 = x0 + iy0 then
limn→∞
zn = z0 ⇒ limn→∞
xn = x0 and limn→∞
yn = y0
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 11: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/11.jpg)
Convergence of a Sequence
The limit of convergent sequence is unique.
limn→∞
zn = z and limn→∞
wn = w then
1 limn→∞
(zn ± wn) = z + w
2 limn→∞
czn = cz
3 limn→∞
znwn = zw
4 limn→∞
znwn
=z
w(w 6= 0)
Given a sequence {an}. Consider a sequence {nk} of positiveintegers 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 andonly if every subsequence converges to p.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 12: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/12.jpg)
Convergence of a Sequence
The limit of convergent sequence is unique.
limn→∞
zn = z and limn→∞
wn = w then
1 limn→∞
(zn ± wn) = z + w
2 limn→∞
czn = cz
3 limn→∞
znwn = zw
4 limn→∞
znwn
=z
w(w 6= 0)
Given a sequence {an}. Consider a sequence {nk} of positiveintegers 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 andonly if every subsequence converges to p.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 13: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/13.jpg)
Convergence of a Sequence
The limit of convergent sequence is unique.
limn→∞
zn = z and limn→∞
wn = w then
1 limn→∞
(zn ± wn) = z + w
2 limn→∞
czn = cz
3 limn→∞
znwn = zw
4 limn→∞
znwn
=z
w(w 6= 0)
Given a sequence {an}. Consider a sequence {nk} of positiveintegers 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 andonly if every subsequence converges to p.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 14: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/14.jpg)
Convergence of a Sequence
The limit of convergent sequence is unique.
limn→∞
zn = z and limn→∞
wn = w then
1 limn→∞
(zn ± wn) = z + w
2 limn→∞
czn = cz
3 limn→∞
znwn = zw
4 limn→∞
znwn
=z
w(w 6= 0)
Given a sequence {an}. Consider a sequence {nk} of positiveintegers 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 andonly if every subsequence converges to p.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 15: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/15.jpg)
Convergence of a Sequence
The limit of convergent sequence is unique.
limn→∞
zn = z and limn→∞
wn = w then
1 limn→∞
(zn ± wn) = z + w
2 limn→∞
czn = cz
3 limn→∞
znwn = zw
4 limn→∞
znwn
=z
w(w 6= 0)
Given a sequence {an}. Consider a sequence {nk} of positiveintegers 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 andonly if every subsequence converges to p.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 16: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/16.jpg)
Convergence of a Sequence
The limit of convergent sequence is unique.
limn→∞
zn = z and limn→∞
wn = w then
1 limn→∞
(zn ± wn) = z + w
2 limn→∞
czn = cz
3 limn→∞
znwn = zw
4 limn→∞
znwn
=z
w(w 6= 0)
Given a sequence {an}. Consider a sequence {nk} of positiveintegers 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 andonly if every subsequence converges to p.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 17: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/17.jpg)
Convergence of a Sequence
The limit of convergent sequence is unique.
limn→∞
zn = z and limn→∞
wn = w then
1 limn→∞
(zn ± wn) = z + w
2 limn→∞
czn = cz
3 limn→∞
znwn = zw
4 limn→∞
znwn
=z
w(w 6= 0)
Given a sequence {an}. Consider a sequence {nk} of positiveintegers 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 andonly if every subsequence converges to p.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 18: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/18.jpg)
Convergence of a Sequence
The limit of convergent sequence is unique.
limn→∞
zn = z and limn→∞
wn = w then
1 limn→∞
(zn ± wn) = z + w
2 limn→∞
czn = cz
3 limn→∞
znwn = zw
4 limn→∞
znwn
=z
w(w 6= 0)
Given a sequence {an}. Consider a sequence {nk} of positiveintegers 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 andonly if every subsequence converges to p.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 19: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/19.jpg)
Convergence of a Sequence
The limit of convergent sequence is unique.
limn→∞
zn = z and limn→∞
wn = w then
1 limn→∞
(zn ± wn) = z + w
2 limn→∞
czn = cz
3 limn→∞
znwn = zw
4 limn→∞
znwn
=z
w(w 6= 0)
Given a sequence {an}. Consider a sequence {nk} of positiveintegers 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 andonly if every subsequence converges to p.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 20: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/20.jpg)
Taylors Series
If f(z) is analytic inside a circle C with centre at z0 then forall z inside C
f(z) = f(z0)+(z−z0)f ′(z0)+(z − z0)2
2!f ′′(z0)+. . .+
(z − z0)n
n!fn(z0)
Case 1: Putting z = z0 + h in above equation, we get
f(z0 + h) = f(z0) + hf ′(z0) +h2
2!f ′′(z0) + . . .+
hn
n!fn(z0)
Case 2: If z0 = 0 then, we get
f(z) = f(0) + zf ′(0) +z2
2!f ′′(0) + . . .+
zn
n!fn(0)
This series is called Maclaurin’s Series.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 21: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/21.jpg)
Taylors Series
If f(z) is analytic inside a circle C with centre at z0 then forall z inside C
f(z) = f(z0)+(z−z0)f ′(z0)+(z − z0)2
2!f ′′(z0)+. . .+
(z − z0)n
n!fn(z0)
Case 1: Putting z = z0 + h in above equation, we get
f(z0 + h) = f(z0) + hf ′(z0) +h2
2!f ′′(z0) + . . .+
hn
n!fn(z0)
Case 2: If z0 = 0 then, we get
f(z) = f(0) + zf ′(0) +z2
2!f ′′(0) + . . .+
zn
n!fn(0)
This series is called Maclaurin’s Series.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 22: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/22.jpg)
Taylors Series
If f(z) is analytic inside a circle C with centre at z0 then forall z inside C
f(z) = f(z0)+(z−z0)f ′(z0)+(z − z0)2
2!f ′′(z0)+. . .+
(z − z0)n
n!fn(z0)
Case 1: Putting z = z0 + h in above equation, we get
f(z0 + h) = f(z0) + hf ′(z0) +h2
2!f ′′(z0) + . . .+
hn
n!fn(z0)
Case 2: If z0 = 0 then, we get
f(z) = f(0) + zf ′(0) +z2
2!f ′′(0) + . . .+
zn
n!fn(0)
This series is called Maclaurin’s Series.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 23: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/23.jpg)
Taylors Series
If f(z) is analytic inside a circle C with centre at z0 then forall z inside C
f(z) = f(z0)+(z−z0)f ′(z0)+(z − z0)2
2!f ′′(z0)+. . .+
(z − z0)n
n!fn(z0)
Case 1: Putting z = z0 + h in above equation, we get
f(z0 + h) = f(z0) + hf ′(z0) +h2
2!f ′′(z0) + . . .+
hn
n!fn(z0)
Case 2: If z0 = 0 then, we get
f(z) = f(0) + zf ′(0) +z2
2!f ′′(0) + . . .+
zn
n!fn(0)
This series is called Maclaurin’s Series.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 24: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/24.jpg)
Taylors Series
If f(z) is analytic inside a circle C with centre at z0 then forall z inside C
f(z) = f(z0)+(z−z0)f ′(z0)+(z − z0)2
2!f ′′(z0)+. . .+
(z − z0)n
n!fn(z0)
Case 1: Putting z = z0 + h in above equation, we get
f(z0 + h) = f(z0) + hf ′(z0) +h2
2!f ′′(z0) + . . .+
hn
n!fn(z0)
Case 2: If z0 = 0 then, we get
f(z) = f(0) + zf ′(0) +z2
2!f ′′(0) + . . .+
zn
n!fn(0)
This series is called Maclaurin’s Series.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 25: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/25.jpg)
Taylors Series
If f(z) is analytic inside a circle C with centre at z0 then forall z inside C
f(z) = f(z0)+(z−z0)f ′(z0)+(z − z0)2
2!f ′′(z0)+. . .+
(z − z0)n
n!fn(z0)
Case 1: Putting z = z0 + h in above equation, we get
f(z0 + h) = f(z0) + hf ′(z0) +h2
2!f ′′(z0) + . . .+
hn
n!fn(z0)
Case 2: If z0 = 0 then, we get
f(z) = f(0) + zf ′(0) +z2
2!f ′′(0) + . . .+
zn
n!fn(0)
This series is called Maclaurin’s Series.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 26: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/26.jpg)
Taylors Series
If f(z) is analytic inside a circle C with centre at z0 then forall z inside C
f(z) = f(z0)+(z−z0)f ′(z0)+(z − z0)2
2!f ′′(z0)+. . .+
(z − z0)n
n!fn(z0)
Case 1: Putting z = z0 + h in above equation, we get
f(z0 + h) = f(z0) + hf ′(z0) +h2
2!f ′′(z0) + . . .+
hn
n!fn(z0)
Case 2: If z0 = 0 then, we get
f(z) = f(0) + zf ′(0) +z2
2!f ′′(0) + . . .+
zn
n!fn(0)
This series is called Maclaurin’s Series.N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 27: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/27.jpg)
Laurent Series
If f(z) is analytic in the ring shaped region R bounded bytwo concentric circles c1 & c2 of radii r1 & r2 (r1 > r2) andwith centre at z0, then for all z in R.
f(z) = a0 +a1(z−z0)+a2(z−z0)2 + . . .+b1
(z − z0)+
b2
(z − z0)2+ . . .
where an =1
2πi
∫Γ
f(ξ)dξ
(ξ − z0)n+1, n = 0, 1, 2, . . .
Γ being any circle lying between c1 & c2 having z0 as itscentre, for all values of n.
∴ f(z) =∞∑n=0
an(z − z0)n +∞∑n=1
bn(z − z0)n
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 28: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/28.jpg)
Laurent Series
If f(z) is analytic in the ring shaped region R bounded bytwo concentric circles c1 & c2 of radii r1 & r2 (r1 > r2) andwith centre at z0, then for all z in R.
f(z) = a0 +a1(z−z0)+a2(z−z0)2 + . . .+b1
(z − z0)+
b2
(z − z0)2+ . . .
where an =1
2πi
∫Γ
f(ξ)dξ
(ξ − z0)n+1, n = 0, 1, 2, . . .
Γ being any circle lying between c1 & c2 having z0 as itscentre, for all values of n.
∴ f(z) =∞∑n=0
an(z − z0)n +∞∑n=1
bn(z − z0)n
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 29: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/29.jpg)
Laurent Series
If f(z) is analytic in the ring shaped region R bounded bytwo concentric circles c1 & c2 of radii r1 & r2 (r1 > r2) andwith centre at z0, then for all z in R.
f(z) = a0 +a1(z−z0)+a2(z−z0)2 + . . .+b1
(z − z0)+
b2
(z − z0)2+ . . .
where an =1
2πi
∫Γ
f(ξ)dξ
(ξ − z0)n+1, n = 0, 1, 2, . . .
Γ being any circle lying between c1 & c2 having z0 as itscentre, for all values of n.
∴ f(z) =∞∑n=0
an(z − z0)n +∞∑n=1
bn(z − z0)n
N.B.V yas − Department of Mathematics, AITS − Rajkot
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Laurent Series
If f(z) is analytic in the ring shaped region R bounded bytwo concentric circles c1 & c2 of radii r1 & r2 (r1 > r2) andwith centre at z0, then for all z in R.
f(z) = a0 +a1(z−z0)+a2(z−z0)2 + . . .+b1
(z − z0)+
b2
(z − z0)2+ . . .
where an =1
2πi
∫Γ
f(ξ)dξ
(ξ − z0)n+1, n = 0, 1, 2, . . .
Γ being any circle lying between c1 & c2 having z0 as itscentre, for all values of n.
∴ f(z) =∞∑n=0
an(z − z0)n +∞∑n=1
bn(z − z0)n
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 31: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/31.jpg)
Laurent Series
If f(z) is analytic in the ring shaped region R bounded bytwo concentric circles c1 & c2 of radii r1 & r2 (r1 > r2) andwith centre at z0, then for all z in R.
f(z) = a0 +a1(z−z0)+a2(z−z0)2 + . . .+b1
(z − z0)+
b2
(z − z0)2+ . . .
where an =1
2πi
∫Γ
f(ξ)dξ
(ξ − z0)n+1, n = 0, 1, 2, . . .
Γ being any circle lying between c1 & c2 having z0 as itscentre, for all values of n.
∴ f(z) =∞∑n=0
an(z − z0)n +∞∑n=1
bn(z − z0)n
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 32: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/32.jpg)
Laurent Series
If f(z) is analytic in the ring shaped region R bounded bytwo concentric circles c1 & c2 of radii r1 & r2 (r1 > r2) andwith centre at z0, then for all z in R.
f(z) = a0 +a1(z−z0)+a2(z−z0)2 + . . .+b1
(z − z0)+
b2
(z − z0)2+ . . .
where an =1
2πi
∫Γ
f(ξ)dξ
(ξ − z0)n+1, n = 0, 1, 2, . . .
Γ being any circle lying between c1 & c2 having z0 as itscentre, for all values of n.
∴ f(z) =∞∑n=0
an(z − z0)n +∞∑n=1
bn(z − z0)n
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 33: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/33.jpg)
Note
If f(z) is analytic at z = z0 then we can expand f(z) bymeans of Taylor’s series at a point z0
Laurent series given an expansion of f(z) at a point z0 evenif f(z) is not analytic there.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 34: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/34.jpg)
Note
If f(z) is analytic at z = z0 then we can expand f(z) bymeans of Taylor’s series at a point z0
Laurent series given an expansion of f(z) at a point z0 evenif f(z) is not analytic there.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 35: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/35.jpg)
Singular Points
A point at which a function f(z) ceases to be analytic iscalled a singular point of f(z)
If the function f(z) is analytic at every point in theneighbourhood of a point z0 except at z0 is called isolatedsingular point or isolated singularity.
Eg. 1 f(z) =1
z⇒ f ′(z) = − 1
z2, it follows that f(z) is analytic at
every point except at z = 0 , hence z = 0 is an isolatedsingularity.
Eg. 2 f(z) =1
z3(z2 + 1)has three isolated singularities at
z = 0, i,−i
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 36: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/36.jpg)
Singular Points
A point at which a function f(z) ceases to be analytic iscalled a singular point of f(z)
If the function f(z) is analytic at every point in theneighbourhood of a point z0 except at z0 is called isolatedsingular point or isolated singularity.
Eg. 1 f(z) =1
z⇒ f ′(z) = − 1
z2, it follows that f(z) is analytic at
every point except at z = 0 , hence z = 0 is an isolatedsingularity.
Eg. 2 f(z) =1
z3(z2 + 1)has three isolated singularities at
z = 0, i,−i
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 37: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/37.jpg)
Singular Points
A point at which a function f(z) ceases to be analytic iscalled a singular point of f(z)
If the function f(z) is analytic at every point in theneighbourhood of a point z0 except at z0 is called isolatedsingular point or isolated singularity.
Eg. 1 f(z) =1
z⇒ f ′(z) = − 1
z2, it follows that f(z) is analytic at
every point except at z = 0 , hence z = 0 is an isolatedsingularity.
Eg. 2 f(z) =1
z3(z2 + 1)has three isolated singularities at
z = 0, i,−i
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 38: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/38.jpg)
Singular Points
A point at which a function f(z) ceases to be analytic iscalled a singular point of f(z)
If the function f(z) is analytic at every point in theneighbourhood of a point z0 except at z0 is called isolatedsingular point or isolated singularity.
Eg. 1 f(z) =1
z⇒ f ′(z) = − 1
z2, it follows that f(z) is analytic at
every point except at z = 0 , hence z = 0 is an isolatedsingularity.
Eg. 2 f(z) =1
z3(z2 + 1)has three isolated singularities at
z = 0, i,−i
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 39: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/39.jpg)
Singular Points
If z = z0 is a isolated singular point, then f(z) can beexpanded in a Laurents series in the form.
f(z) =∞∑n=0
an(z − z0)n +∞∑n=1
bn(z − z0)n
(1)
In (1)∞∑n=0
an(z − z0)n is called the regular part and
∞∑n=1
bn(z − z0)n
is called the principal part of f(z) in the
neighbourhood of z0.
If the principal part of f(z) contains infinite numbers ofterms then z = z0 is called an isolated essential singularity off(z).
N.B.V yas − Department of Mathematics, AITS − Rajkot
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Singular Points
If z = z0 is a isolated singular point, then f(z) can beexpanded in a Laurents series in the form.
f(z) =∞∑n=0
an(z − z0)n +∞∑n=1
bn(z − z0)n
(1)
In (1)∞∑n=0
an(z − z0)n is called the regular part and
∞∑n=1
bn(z − z0)n
is called the principal part of f(z) in the
neighbourhood of z0.
If the principal part of f(z) contains infinite numbers ofterms then z = z0 is called an isolated essential singularity off(z).
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 41: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/41.jpg)
Singular Points
If z = z0 is a isolated singular point, then f(z) can beexpanded in a Laurents series in the form.
f(z) =∞∑n=0
an(z − z0)n +∞∑n=1
bn(z − z0)n
(1)
In (1)∞∑n=0
an(z − z0)n is called the regular part and
∞∑n=1
bn(z − z0)n
is called the principal part of f(z) in the
neighbourhood of z0.
If the principal part of f(z) contains infinite numbers ofterms then z = z0 is called an isolated essential singularity off(z).
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 42: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/42.jpg)
Singular Points
If z = z0 is a isolated singular point, then f(z) can beexpanded in a Laurents series in the form.
f(z) =∞∑n=0
an(z − z0)n +∞∑n=1
bn(z − z0)n
(1)
In (1)∞∑n=0
an(z − z0)n is called the regular part and
∞∑n=1
bn(z − z0)n
is called the principal part of f(z) in the
neighbourhood of z0.
If the principal part of f(z) contains infinite numbers ofterms then z = z0 is called an isolated essential singularity off(z).
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 43: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/43.jpg)
Singular Points
If z = z0 is a isolated singular point, then f(z) can beexpanded in a Laurents series in the form.
f(z) =∞∑n=0
an(z − z0)n +∞∑n=1
bn(z − z0)n
(1)
In (1)∞∑n=0
an(z − z0)n is called the regular part and
∞∑n=1
bn(z − z0)n
is called the principal part of f(z) in the
neighbourhood of z0.
If the principal part of f(z) contains infinite numbers ofterms then z = z0 is called an isolated essential singularity off(z).
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 44: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/44.jpg)
Singular Points
If in equation(1) , the principal part has all the coefficientbn+1, bn+2, . . . as zero after a particular term bn then theLaurents series of f(z) reduces to
f(z) =∞∑n=0
an(z − z0)n +b1
(z − z0)+
b2
(z − z0)2+ . . .+
bn(z − z0)n
i.e. (Regular part) + (Principal part is a polynomial of finite
number of terms in1
z − z0
The the singularity in this case at z = z0 is called a pole oforder n.
If the order of the pole is one, the pole is called simple pole.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 45: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/45.jpg)
Singular Points
If in equation(1) , the principal part has all the coefficientbn+1, bn+2, . . . as zero after a particular term bn then theLaurents series of f(z) reduces to
f(z) =∞∑n=0
an(z − z0)n +b1
(z − z0)+
b2
(z − z0)2+ . . .+
bn(z − z0)n
i.e. (Regular part) + (Principal part is a polynomial of finite
number of terms in1
z − z0
The the singularity in this case at z = z0 is called a pole oforder n.
If the order of the pole is one, the pole is called simple pole.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 46: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/46.jpg)
Singular Points
If in equation(1) , the principal part has all the coefficientbn+1, bn+2, . . . as zero after a particular term bn then theLaurents series of f(z) reduces to
f(z) =∞∑n=0
an(z − z0)n +b1
(z − z0)+
b2
(z − z0)2+ . . .+
bn(z − z0)n
i.e. (Regular part) + (Principal part is a polynomial of finite
number of terms in1
z − z0
The the singularity in this case at z = z0 is called a pole oforder n.
If the order of the pole is one, the pole is called simple pole.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 47: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/47.jpg)
Singular Points
If in equation(1) , the principal part has all the coefficientbn+1, bn+2, . . . as zero after a particular term bn then theLaurents series of f(z) reduces to
f(z) =∞∑n=0
an(z − z0)n +b1
(z − z0)+
b2
(z − z0)2+ . . .+
bn(z − z0)n
i.e. (Regular part) + (Principal part is a polynomial of finite
number of terms in1
z − z0
The the singularity in this case at z = z0 is called a pole oforder n.
If the order of the pole is one, the pole is called simple pole.
N.B.V yas − Department of Mathematics, AITS − Rajkot
![Page 48: Power series](https://reader034.fdocuments.in/reader034/viewer/2022052621/5585948fd8b42abc7b8b4a30/html5/thumbnails/48.jpg)
Singular Points
If in equation(1) , the principal part has all the coefficientbn+1, bn+2, . . . as zero after a particular term bn then theLaurents series of f(z) reduces to
f(z) =∞∑n=0
an(z − z0)n +b1
(z − z0)+
b2
(z − z0)2+ . . .+
bn(z − z0)n
i.e. (Regular part) + (Principal part is a polynomial of finite
number of terms in1
z − z0
The the singularity in this case at z = z0 is called a pole oforder n.
If the order of the pole is one, the pole is called simple pole.
N.B.V yas − Department of Mathematics, AITS − Rajkot