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Taytor's AND LauRENT'S SERIES
30.1 INTRODUCTION
‘An analytic function within a circle can be expanded by Taylor's series
Ifa function which is not anslytic within a circle is expanded by Laureat’s series,
30.2 CONVERGENCE OF A SERIES OF COMPLEX TERMS
La (4 FA) He Hv) HL, FA) +, HIV) + a
be an infinite series of complex terms:
4), Yy. being real numbers.
(a) If the series Ea, and Xv, converge to the sums 1 and V then series (1) is said to
converge ta the eum I+ iV
(b) IF(1)is a convergent series, then
lim, +4) =0
(©) The series (1) is said to be absolutely convergent if the series
Ie ivy felay ay tay, 4
id
is convergent. Since |
@) Let the series
a are both Tess than lay +4
44 (2) +43 (2) +as(2) +44, (2). Q
converge to the sum (2) and S, (2) be the sum ofits first terms.
‘The series (2) is said to be absolutely convergent in region R, if comesponding to any
positive number e, tere exists positive number WV.
S@)-S,|.N
(6) Weirstras's, M-test holds good for series of complex terms also.
s.
1es (2) is uniformly convergent in a region R if there is a convergent series ZAfy. Such
that Ja, (ois M,
‘A uniformly convergent series can be integrated term by term.
30.3 POWER SERIES
A series in powers of (2 ~ sis ealled power series.
Sac
ag + a2 9) Hey (229) Hg (2 Fg)" + ®
ey 1 sons a€€ KNOWN 26 the evelficient of the series.
™ 30.4
Higher Engineering Mathomatics
Here zs a complex variable and z, is called the centre of the series
(1) is also called the power series ahout the point :,
Zaye ap tayzrane rasta
Here the centre of the series is zero.
REGION OF CONVERGENCE
‘The region of convergence is the set of all points z for which the series converges.
‘There are three distinct possibilities for a convergent series,
(1) The series converges only at the point z = z,
(2) The series converges for all the points in the whole plane.
G) The series converges everywhere inside a circular plane | z~ z, |< R, where R is the
radius of convergence and diverges everywhere outside the eircle/circular ring.
RADIUS OF CONVERGENCE OF POWER SERIES
Consider the power series Ea, 2
By Cauchy theorem on limits’ radius of convergence R is given by
= lim |a, |!" «i i lim
Example 1. Find the radius of convergence of the power series
a -—t
mt Geb
Radius of convergence is given by
1 tim |“) = im =o
RT eef a, | onl
R=
Hence, the radius of convergence ofthe given power series is Ans.
Example 2. Find the radius of convergence of the power series
rods
Solution. Here,
Lo, 1
Was net FRG
Rais of convergence is given by
3
tim |) = tim 2743. = jim —2 =!
wala, | roe 43 909, 2
Ro? "
Hence, the radius of convergence of the given power series is 2 Ans.
Example 3. Find the radius of convergence ofthe power series
fas ye Taylor's and Lauront's Series, 773
Solution. Here,
a 4,2 —a
ee aye!
Racus of convergence 1s given by
1 .
= lim tim
Ra aye
= tim "stim —_1 9
mene D nt on (foe
= R=%
Hence, the radius of convergence of the given power series is =. Ans.
Example 4. Find the radius of convergence of the power series:
LS
yo= dy
Solution. Here, a, =
a = inet
ns =
Now, = ent at ah at (a)
a ne nh ey nl (ey rel
x
= k
Mace, te rats of corvergenceof the piven power sei i Am
EXERCISE 30.1
Fra te rai oF merece filing power sei
nye ns 25 Ans.
ede) ane! a, De ote fn 13
. 2n+3 amt
20 Gns5iin45)*
30.6 METHOD OF EXPANSION OF A FUNCTION
(1) Taylor's series (2) Binomial series (3) Exponential series
30.7 TAYLOR'S THEOREM
Ifa function f (2) is analytic a all points inside a circle C, with its centre at the point a and
‘radius R, then al each point z inside C-
£@)
2
cate LIM ght
flay+ fane-a)>
www. ngineeringFRookePef.com 77 Higher Engineering Mathomatics
Proof. Take any point z inside C. Draw a circle C, with centre a, enclosing the point z, Let w
be a point on circle C,
1 14 ce
Woata-z waa)
1
wae)
Apply Binomial thearem,
5 stay. ea o
woe woe Wwnay Wena Gena)
lea]
as [e-a|e [wal > feel
So the series converges uniformly. Hence the series is integrable.
Multiply (I) by fw).
LO LP 4g gy LO 5 (egy? LO 5 (e-a"
woz waa va) wa)
‘On integrating wet. “W", we get
£0) yf LO sto sce fo), 2
te dem )
[et Seam, ro)
Q)
We know that
j& =2mi fiz) ana [2 LO toe 25 fla)
Lo dw = 281 f(A) and so on,
Substituting these values in (2), we get
Taylor's series as given below
GQ) Proved.
farhasiosipe)s Epos ne yay
Cory 2 = 0, es 8) bees
1012 10)POVELO AZ LO
‘This series is called Muclaurin series,
= faye
www.EngineesingERookePef.com Taylor's and Laurent's Series 78
= fe =
re >
se >
By Taylor's series of f
FO =flM+e-af' tz S a+
4) 50
ee Ee
Example 6. Expand the function
fo
about 2 = 2 in Taylor's series. Obtain its radius of convergence.
Solution. Here, we have,
te)
fO=
= £e
> re
Se
By Taylor's series
Jl) =fQ)+ (= 2) f/Q)+
Alternat
‘We can expand the given function by Binomial expansion.
boot
Radius of convergence
= xl
www.EngineesingERookePef.com Highor Engineering Mathomatics
Example 7. Expand f (2) = cosh z about xi
Solution. Here, we have
Fi =eosn: =
f@=smh: =
Fo@=chz >
fo@smh: 3
By Taylors series
. Gan, (=n
Fa fad) +n f/@ # SS Fre 6 pra s
coh + (ens xis SE” cohen
Ans.
Example 8. Expand J(2)
Solution, Here, we have
JO bie "bem
mial series)
CIC (4)
3 bare) *
Example 9. Show that
logz=(2-I) —
Solution. Let f(2)-log2) => —f(@)=log 1-0
www.EngineesingERookePef.com Taylor's and Laurent's Series ™
1
) =+=1
ro=5
L@=t
7 dy=2
I") =-Bx2xd
By Taylor series
r= 3!
fe0y= fay +f ta). c2maye LUMA , LC)
f(z) = log z=log (142-1)
On substituting the values of f(), (),") elc., we get
(ena)
Loy y2
loge 0+ -D- Lee +2 ey
tox 1D FEW + FEw
2 ep leet
= log? = she + r
log. Wes eve Proved.
Example 10. Expand inthe region
(@ |z|<1 >2 (GP, Bhopal, II Semester, Dee. 2005)
Solution: H @=3) 7-2
(@ Ifz|2 P
Wehave, — f(=
{By Binomial theorer)
ees eet
‘Which isthe required expansion Ans. Highor Engineering Mathomatics
Example 11. Show shat when | 247 <1,
— 1
y-1P
z
Solution. f(2)
Fo
‘Taking common, bigger term out of Land |2+1 [here, 1>|2+1|
So, we take | common.
—1_
-@+br
=e Dorney" Proved.
142+ D+ 3G 4D +44 D+
+0
Find the first four terms ofthe Taylor's series expansion ofthe complex variable
function
10D
about z = 2. Find the region of convergence.
ee
Solution. f(2)= =
IO ye
I centre of a circle is at z= 2, then the distances of the singularities z= 3 and z= 4 from the
centre are 1 and 2
Hence 1° circle 1s drawn with centre ¢ = Z and raaius 1, then Within the eirete |
the given function ns is analyte, hence it can be expanded ina Taylr's sere within the
circle | rele of convergence.
(By Partial fraction method)
-4 5
2-1 @-2)-2 (e-2]< 0)
oy. Sf cet aay -27
[L4G :) IY? + (2 y’ 1 i z 4 3 |
= (+-5)e(s-S)erae(+-3e-ara(- ear
Alternative method. In obtaining the Taylor series we evaluate the coefficients by contour
integration
23
(2) = ==
12) Tya-H 2
ce the differentiation easier le us convert the given fraction into partial fractions
“45
FO Tate y
5 4s
- f=
e :
f Gay 23 ay Taylor's and Laurent's Series 779
3" =a)
Taylor senesis f(2)= flad+(e-0) f+ ES prays Opn
FHL 3,6, MH @-38 (27), G2 177,
@-9E-9 2 soo ay
3 ul 227 35
eat e 2? Fae 2p Be. Ans,
Z1e-teg-2? Zig
Example 13. Find the first three terms of the Taylor series expansion of f(z)
‘about < =~ i. Find the region of convergence.
Solution, (2)
P44
Poles are given by 2 + > g=-4, >
If the centre of a circle is 2=— i, then the distances ofthe singularities = 2 and
the centre are3 and I. Hence if a circle of radius Vis drawn with centre at
[z+ i] 1. the given function f(2) is analytic, Thus the function can he expanded in Taylor series
within the circle |z + é|= 1, which is therefore the circle of convergence. y
1 1 1
fos ame “al
+21
= 2i from,
i, then within the circle
ale am
ere
| {[ titer} "ealeaesoy ‘
EY cope DEDEDE oy y3y
Jew (Jeee|
3
ofroee 9-2 cat ce? LMM)
J
ahi 1 +a ie+)-( +i e+
| CHD ED EHD tat HEED HD fori]
a)
www.EngineesingERookePef.com 780 Highor Engineering Mathomatics
Alternative method
e
IO .
By Taylor expansion /(2)= f(a) +(2-a) f(a) +222" (ays
Putting = iim above, we get
ase sn s+ pci.
= seo. 1.3
Cora ee 3
22
cay 9
Pay
mig), MDB?
(qy=- 2B
f Fay
(On substituting the value of fH. f"(-.f"Ci), we get
1 (21) @4n'/_14),
fo at *) (3)
i 1 esat
= yt ern Lesa?
fe 3 ra Fe iy
Region of convergence is |z+i| <1 ‘Ans.
44
Example 14. For the function S2)="4— find all Taylor series about the
centre zero. (U.P. Il Semester, Dee, 2006)
y
x
Poles are determined by <= 1 = 0 bi
> GDNet DE +=0
> rsohet
By Partial fractions : :
3 Sarat a
f= + 2
= sag}
fine ate aie il
3 a5 1 1 1 ¥
=-Raeay Saver! a(2e+t jase?
Sao Saas ta(-arel}ane'
Siesta? a2) 42%+ [45 pl-ata? 242 2+ at
Guertst celeste defies! hast d4(teed oat eat ed
= 1-4 Ans.
‘Whicit isthe required series.
wa Engine
Fef.com Taylor's and Laurent's Series 781
2241
Example 15. Find Taylor expansion of (=
‘about the point z = 1
Phe
Spy’ sitetlaitcs are piven by 2.(e i
If centre of the circle is at z = 1, then the distance of the
singulanties 2 = O and z = -1 from the centre are 1 and 2.
Hence, ia circle is drawn with centre 2=1 and radius 1, then
wathin the circle |2—1)=1, the given function f(z) is ana:
Itc & therefore, ican be expanded in a Taylor series within
the circle [2—I]=1, which is thus the circle of
2241
z@+)
sya
2) ) (Gy:
sl) ,9 nO»
2a ens
( ) ep 16‘ iy
+UI-@-D+-1-@-D' +.)
2)°8
‘Which is the required expansion, ‘Ans.
Example 16. Expand cos zin a Taylor series about 2 = *
Solution. Here f'
Hence cos =
Which is the required expansion.
Example 17. Expand the function 5
Solution. Putting z~ =f, we have
sin _ sin(w+#) _-sine
Which is the required expansion.
Example 18. Expand the function sin ' z in powers of. (U.P. III Semester, Dec. 2006)
Solution, Let w =sin!z 782 Highor Engineering Mathomatics
a
2
Which is the required expansion, Ans,
:xample 19. Expand the function
Solution. We have, f(z) = tan”! ¢
tan in powers of 2. (U.P. III Sem. 2009 2010)
df)
= w
2
Oz00c > C
(On putting the value of Cin (2), we get
wrtese- 545-54 Ans.
EXERCISE 30.2
Expand the following functions in Taylor's series
as
A. yo abont
el
