Power series

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

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







zn = z0


If limn→∞

zn = z0 we have



limn→∞



yn = y0







zn = z0


If limn→∞

zn = z0 we have



limn→∞



yn = y0







zn = z0


If limn→∞

zn = z0 we have



limn→∞



yn = y0







zn = z0


If limn→∞

zn = z0 we have



limn→∞



yn = y0







zn = z0


If limn→∞

zn = z0 we have

(i) |zn| → |z0| as n→∞

(ii) the sequence {zn} is bounded


limn→∞



yn = y0







zn = z0


If limn→∞

zn = z0 we have



limn→∞



yn = y0







zn = z0


If limn→∞

zn = z0 we have



limn→∞



yn = y0







zn = z0


If limn→∞

zn = z0 we have



limn→∞



yn = y0



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.




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)



limit





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)



limit





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)



limit





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)



limit





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)



limit





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)



limit





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)



limit





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)



limit



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.


Taylors Series


f(z) = f(z0)+(z−z0)f ′(z0)+(z − z0)2

2!f ′′(z0)+. . .+

(z − z0)n

n!fn(z0)


f(z0 + h) = f(z0) + hf ′(z0) +h2

2!f ′′(z0) + . . .+

hn

n!fn(z0)


f(z) = f(0) + zf ′(0) +z2

2!f ′′(0) + . . .+

zn

n!fn(0)



Taylors Series


f(z) = f(z0)+(z−z0)f ′(z0)+(z − z0)2

2!f ′′(z0)+. . .+

(z − z0)n

n!fn(z0)


f(z0 + h) = f(z0) + hf ′(z0) +h2

2!f ′′(z0) + . . .+

hn

n!fn(z0)


f(z) = f(0) + zf ′(0) +z2

2!f ′′(0) + . . .+

zn

n!fn(0)



Taylors Series


f(z) = f(z0)+(z−z0)f ′(z0)+(z − z0)2

2!f ′′(z0)+. . .+

(z − z0)n

n!fn(z0)


f(z0 + h) = f(z0) + hf ′(z0) +h2

2!f ′′(z0) + . . .+

hn

n!fn(z0)


f(z) = f(0) + zf ′(0) +z2

2!f ′′(0) + . . .+

zn

n!fn(0)



Taylors Series


f(z) = f(z0)+(z−z0)f ′(z0)+(z − z0)2

2!f ′′(z0)+. . .+

(z − z0)n

n!fn(z0)


f(z0 + h) = f(z0) + hf ′(z0) +h2

2!f ′′(z0) + . . .+

hn

n!fn(z0)


f(z) = f(0) + zf ′(0) +z2

2!f ′′(0) + . . .+

zn

n!fn(0)



Taylors Series


f(z) = f(z0)+(z−z0)f ′(z0)+(z − z0)2

2!f ′′(z0)+. . .+

(z − z0)n

n!fn(z0)


f(z0 + h) = f(z0) + hf ′(z0) +h2

2!f ′′(z0) + . . .+

hn

n!fn(z0)


f(z) = f(0) + zf ′(0) +z2

2!f ′′(0) + . . .+

zn

n!fn(0)



Taylors Series


f(z) = f(z0)+(z−z0)f ′(z0)+(z − z0)2

2!f ′′(z0)+. . .+

(z − z0)n

n!fn(z0)


f(z0 + h) = f(z0) + hf ′(z0) +h2

2!f ′′(z0) + . . .+

hn

n!fn(z0)


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

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


Laurent Series


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, . . .


∴ f(z) =∞∑n=0

an(z − z0)n +∞∑n=1

bn(z − z0)n


Laurent Series


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, . . .


∴ f(z) =∞∑n=0

an(z − z0)n +∞∑n=1

bn(z − z0)n


Laurent Series


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, . . .


∴ f(z) =∞∑n=0

an(z − z0)n +∞∑n=1

bn(z − z0)n


Laurent Series


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, . . .


∴ f(z) =∞∑n=0

an(z − z0)n +∞∑n=1

bn(z − z0)n


Laurent Series


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, . . .


∴ f(z) =∞∑n=0

an(z − z0)n +∞∑n=1

bn(z − z0)n


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.


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.


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


Singular Points



Eg. 1 f(z) =1

z⇒ f ′(z) = − 1



Eg. 2 f(z) =1


z = 0, i,−i


Singular Points



Eg. 1 f(z) =1

z⇒ f ′(z) = − 1



Eg. 2 f(z) =1


z = 0, i,−i


Singular Points



Eg. 1 f(z) =1

z⇒ f ′(z) = − 1



Eg. 2 f(z) =1


z = 0, i,−i


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).


Singular Points


f(z) =∞∑n=0

an(z − z0)n +∞∑n=1

bn(z − z0)n

(1)

In (1)∞∑n=0


∞∑n=1

bn(z − z0)n





Singular Points


f(z) =∞∑n=0

an(z − z0)n +∞∑n=1

bn(z − z0)n

(1)

In (1)∞∑n=0


∞∑n=1

bn(z − z0)n





Singular Points


f(z) =∞∑n=0

an(z − z0)n +∞∑n=1

bn(z − z0)n

(1)

In (1)∞∑n=0


∞∑n=1

bn(z − z0)n





Singular Points


f(z) =∞∑n=0

an(z − z0)n +∞∑n=1

bn(z − z0)n

(1)

In (1)∞∑n=0


∞∑n=1

bn(z − z0)n





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.


Singular Points


f(z) =∞∑n=0

an(z − z0)n +b1

(z − z0)+

b2

(z − z0)2+ . . .+

bn(z − z0)n


number of terms in1

z − z0




Singular Points


f(z) =∞∑n=0

an(z − z0)n +b1

(z − z0)+

b2

(z − z0)2+ . . .+

bn(z − z0)n


number of terms in1

z − z0




Singular Points


f(z) =∞∑n=0

an(z − z0)n +b1

(z − z0)+

b2

(z − z0)2+ . . .+

bn(z − z0)n


number of terms in1

z − z0




Singular Points


f(z) =∞∑n=0

an(z − z0)n +b1

(z − z0)+

b2

(z − z0)2+ . . .+

bn(z − z0)n


number of terms in1

z − z0




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Power series