dr. nirav vyas power series.pdf

8/20/2019 Dr. Nirav Vyas Power Series.pdf

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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 M athematics, AIT S − Rajkot


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Convergence of a Sequence

A sequence {z n} is said to be converge to z 0{as n approaches infinity} if, for each > 0 there

exists a positive integer N such that



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|z − z 0| < , whenever n ≥ N



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Symbolically we write limn→∞ z n = z 0



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A sequence which is not convergent is defined to bedivergent.



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If limn→∞

z n = z 0 we have


C S


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If limn→∞

z n = z 0 we have

(i) |z n| → |z 0| as n →∞


C S


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If limn→∞

z n = z 0 we have

(i) |z n| → |z 0| as n →∞(ii) the sequence {z n} is bounded


C S


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If limn→∞

z n = z 0 we have


If z n = xn + iyn and z 0 = x0 + iy0 then


C S


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If limn→∞

z n = z 0 we have


If z n = xn + iyn and z 0 = x0 + iy0 then

limn→∞

z n = z 0 ⇒ limn→∞

xn = x0 and limn→∞

yn = y0




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The limit of convergent sequence is unique.




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limn→∞

z n = z and limn→∞

wn = w then



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limn→∞


wn = w then

1 limn→∞

(zn ± wn) = z + w

2 limn→∞

czn = cz




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limn→∞


wn = w then

1 limn→∞

(zn ± wn) = z + w

2 limn→∞

czn = cz

3 limn→∞

znwn = zw



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limn→∞


wn = w then

1 limn→∞

(zn ± wn) = z + w

2 limn→∞

czn = cz

3 limn→∞

znwn = zw

4 limn→∞

zn

wn=

z

w (w = 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}.




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limn→∞


wn = w then

1 limn→∞

(zn ± wn) = z + w

2 limn→∞

czn = cz

3 limn→∞

znwn = zw

4 limn→∞

zn

wn=

z

w (w = 0)


If {ank} converges then its limit is called Sub-sequentiallimit




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limn→∞


wn = w then

1 limn→∞

(zn ± wn) = z + w

2 limn→∞

czn = cz

3 limn→∞

znwn = zw

4 limn→∞

zn

wn=

z

w (w = 0)


If {ank} converges then its limit is called Sub-sequentiallimit

A sequence {an} of complex numbers converges to p if andonly if every subsequence converges to p.


Taylors Series


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Taylors Series

If f (z ) is analytic inside a circle C with centre at z 0 then forall z inside C


Taylors Series


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Taylors Series


f (z ) = f (z 0)+(z −z 0)f (z 0)+(z − z

0)22!

f (z 0)+. . .+(z − z 0)n

n! f n(z 0)


Taylors Series


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f (z ) = f (z 0)+(z −z 0)f (z 0)+(z − z

0)22!

f (z 0)+. . .+(z − z 0)n

n! f n(z 0)

Case 1: Putting z = z 0 + h in above equation, we get



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Taylors Series


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f (z ) = f (z 0)+(z −z 0)f (z 0)+(z − z

0)2

2! f (z 0)+. . .+(z − z

0)n

n! f n(z 0)


f (z 0 + h) = f (z 0) + hf (z 0) +

h2

2!f (z 0) + . . . +

hn

n!f n(z 0)

Case 2: If z 0 = 0 then, we get


Taylors Series


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f (z ) = f (z 0)+(z −z 0)f (z 0)+(z − z

0)2

2! f (z 0)+. . .+(z − z

0)n

n! f n(z 0)


f (z 0 + h) = f (z 0) + hf (z 0) +

h2

2!f (z 0) + . . . +

hn

n!f n(z 0)


f (z ) = f (0) + zf (0) + z 2

2!f (0) + . . . +

z n

n!f n(0)


Taylors Series


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f (z ) = f (z 0)+(z −z 0)f (z 0)+(z − z

0)2

2! f (z 0)+. . .+(z − z

0)n

n! f n(z 0)


f (z 0 + h) = f (z 0) + hf (z 0) +

h2

2!f (z 0) + . . . +

hn

n!f n(z 0)


f (z ) = f (0) + zf (0) + z 2

2!f (0) + . . . +

z n

n!f n(0)

This series is called Maclaurin’s Series.N.B.V yas−Department of M athematics, AIT S − Rajkot

Laurent Series


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If f (z ) is analytic in the ring shaped region R bounded by

two concentric circles c1 & c2 of radii r1 & r2 (r1 > r2) andwith centre at z 0, then for all z in R.


Laurent Series


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f (z ) = a0+a1(z −z 0)+a2(z −z 0)2+ . . .+

b1

(z − z 0)+

b2

(z − z 0)2 + . . .


Laurent Series


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f (z ) = a0+a1(z −z 0)+a2(z −z 0)2+ . . .+

b1

(z − z 0)+

b2

(z − z 0)2 + . . .

where an = 1

2πi

Γ

f (ξ )dξ

(ξ − z 0)n+1, n = 0, 1, 2, . . .


Laurent Series


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f (z ) = a0+a1(z −z 0)+a2(z −z 0)2+ . . .+

b1

(z − z 0)+

b2

(z − z 0)2 + . . .

where an = 1

2πi

Γ

f (ξ )dξ

(ξ − z 0)n+1, n = 0, 1, 2, . . .

Γ being any circle lying between c1 & c2 having z 0 as itscentre, for all values of n.

N.B.V yas−

Department of M athematics, AIT S−

Rajkot

Laurent Series


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f (z ) = a0+a1(z −z 0)+a2(z −z 0)2+ . . .+

b1

(z − z 0)+

b2

(z − z 0)2 + . . .

where an = 1

2πi

Γ

f (ξ )dξ

(ξ − z 0)n+1, n = 0, 1, 2, . . .


∴ f (z ) =∞n=0

an(z − z 0)n +

∞n=1

bn

(z − z 0)n

N.B.V yas−


Rajkot

Laurent Series


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f (z ) = a0+a1(z −z 0)+a2(z −z 0)2+ . . .+

b1

(z − z 0)+

b2

(z − z 0)2 + . . .

where an = 1

2πi

Γ

f (ξ )dξ

(ξ − z 0)n+1, n = 0, 1, 2, . . .


∴ f (z ) =∞n=0

an(z − z 0)n +

∞n=1

bn

(z − z 0)n

N.B.V yas−


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Note


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If f (z ) is analytic at z = z 0 then we can expand f (z ) bymeans of Taylor’s series at a point z 0

N.B.V yas−


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Singular Points


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A point at which a function f (z ) ceases to be analytic iscalled a singular point of f (z )

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Singular Points


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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 theneighbourhood of a point z 0 except at z 0 is called isolatedsingular point or isolated singularity.

Eg. 1 f (z ) = 1z ⇒ f (z ) = − 1

z 2 , 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

z 3(z 2 + 1) has three isolated singularities atz = 0, i,−i

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Singular Points


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If z = z 0 is a isolated singular point, then f (z ) can beexpanded in a Laurents series in the form.

N.B.V yas−


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Singular Points


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f (z ) =∞n=0

an(z − z 0)n +

∞n=1

bn

(z − z 0)n

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Singular Points


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f (z ) =∞n=0

an(z − z 0)n +

∞n=1

bn

(z − z 0)n (1)

In (1)

∞n=0

an(z − z 0)n

is called the regular part and

∞n=1

bn

(z − z 0)n is called the principal part of f (z ) in the

neighbourhood of z 0.If the principal part of f (z ) contains infinite numbers of terms then z = z 0 is called an isolated essential singularity of f (z ).

N.B.V yas−


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Singular Points


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f (z ) =∞n=0

an(z − z 0)n +

∞n=1

bn

(z − z 0)n (1)

In (1)

∞n=0

an(z − z 0)n

is called the regular part and

∞n=1

bn

(z − z 0)n is called the principal part of f (z ) in the

neighbourhood of z 0.If the principal part of f (z ) contains infinite numbers of terms then z = z 0 is called an isolated essential singularity of f (z ).

N.B.V yas−


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Singular Points


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If in equation(1) , the principal part has all the coefficientbn+1, bn+2, . . . as zero after a particular term bn then the

Laurents series of f (z ) reduces to

f (z ) =∞

n=0

an(z − z 0)n +

b1

(z − z 0)

+ b2

(z − z 0)2

+ . . . + bn

(z − z 0)n

i.e. (Regular part) + (Principal part is a polynomial of finite

number of terms in 1

z − z 0

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Singular Points


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f (z ) =∞

n=0

an(z − z 0)n +

b1

(z − z 0)

+ b2

(z − z 0)2

+ . . . + bn

(z − z 0)n



z − z 0The the singularity in this case at z = z 0 is called a pole of order n.

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Singular Points


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f (z ) =∞

n=0

an(z − z 0)n +

b1

(z − z 0)

+ b2

(z − z 0)2

+ . . . + bn

(z − z 0)n



z − z 0The the singularity in this case at z = z 0 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−


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dr. nirav vyas power series.pdf

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