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1.0. POWER SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL
EQUATIONS
1.1. THE NOTATION
1.1.1 Index or Subscr!" No"#"on
T$e s%&bo' ix
(!ronounced )x sub i *+ s"#nds ,or #n% o, "$e -#'ues
1 / nx x x xL L "$#" "$e -#r#b'e x c#n #ssu&e. T$e 'e""er i n ix s"#nds ,or #n% o,"$e n"eers 0 1 / 2 3 4 #nd s c#''ed "$e subscr!" or ndex.
An% o, "$e 'e""ers j k l m #nd n c#n be used #s subscr!". In so&e 5or6s rs #'so used
#s # subscr!".
N.7.8 9#re $#s "o be exercsed n c$oosn # subscr!". A'5#%s #-od 'e""ers "$#" #re
6no5n s#% "o re!resen" cons"#n"s or re#' or co&!'ex -#r#b'es suc$ #s a b c or x y z
e"c. or 'e""ers "$#" $#-e s!ec#' &e#nns:-#'ues suc$ #s e org.We de,ne "$e Su&"on:S No"#"on b%
1 /
1
.n
i n
i
x x x x x=
= + + + + L
Ex#&!'es
( )
( )
( )( )
( ) ( )
;
3 3 ; ;1 1 / / 2 2 4 413
31 / 21
3
31 / 2 13 3
1 1
1 1 1
.
.
.
.
u#"ons (ODEs+ b% !o5er seres &e"$ods #
ood 6no5'ede o, "$e !ro!er"es o, "$e su&"on( ) no"#"on or "$e #b'"% "o
n!u'#"e "$e s#&e c#nno" be o-er e&!$#s?ed. One $#s "o be con-ers#n" 5"$ -#rous
!ro!er"es or n!u'#"ons o, "$e no"#"on. T$e ,o''o5n s # bre, re-e5 o, "$euse #nd
!ro!er"es o, "$esigmano"#"on.
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Le" us #ree on "$e "er&no'oes "o be used $ere ,or "$e s#6e o, c'#r"% on'%. T$e no"#"on co&!rses o, "5o &!or"#n" re,erences. 7o"$ o, "$ese re,erences #reunds"nus$#b'e exce!" ,or "$e !os"on "$e% occu!% -s@@-s "$e no"#"on "se',. For
ex#&!'e n "$e "er&
( )10
/
2
; nnn
n n x y
=
"$e !#r" ns"ruc"n us "o c#rr% ou" our
su&"ons ,ro& 2n= (n s"e!s o, 1+ "o 1B0n= con"#ns #n n "$#" 5e 5''
c#'' "$e counting index#nd "$#" !#r" n ,ron" o, "$e no"#"on #'so con"#ns #nex!resson n n C "$s n 5e 5'' c#'' "$esummation index. T$s s done on'% "o,#c'"#"e our dscusson "$#" ,o''o5s.
1..1. S$,"n "$e Indces o, 9oun"n #nd Su&"onT$e ndex o, coun"n #s 5e'' #s "$#" o, su&"on '6e "$e du&&% -#r#b'e o,
n"er#"on s # du&&% -#r#b'e.
Ex#&!'e8100 100 100 100 100 100
1 1 1 1 1 1
j jn n k k i i l l r r
n k j i l r
x yx y x y x y x y x y
n k j i l r = = = = = == = = = = 9o&!#re 5"$
( + ( + ( + ( + ( + ( + ( +
b b b b b b b
a a a a a a a
f x dx f z dz f y dy f s ds f r dr f d f d = = = = = = . T$e 5or6 "$#" ,o''o5s de!ends $e#-'% on "$e 6no5'ede o, s$,"n "$e ndces.S$,"n "$e ndces % !resen" so&e !rob'e&s #s " $#s no e>u-#'ence n n"er#"on.
T$e ex#&!'es "$#" ,o''o5 #re n"ended "o exe&!',% so&e &!or"#n" !ro!er"es "$#" 5e
#re on "o use n "$e nex" de-e'o!&en" o, "$epower series methodsn "$e nex"sec"on.
1... Lo5ern "$e 9oun"n or R#sn "$e Su&"on Index #nd ceers#.
9onsder8
[ ]
( )
2 3 4 ;2 3 4 ;
2 / 2 3
/ 2 3
2
2
.
.
No"n "$#" "$e ndces #re du&&% -#r#b'es
n nn n
nm
m
n mn m
n m
n nn n
n n
a x a x a x a x a x a x
a x a x a x a x a x
a x a x
a x a x
=+ + + + +
+ + + + +
++
= =
++= =
= + + + + + +
= + + + + + +
=
=
L L
L L
L L L
Lo5ern "$e coun"n ndex b% 2 Fro& "o
] -es rse "o r#sn "$e
su&"on ndex b% Fro& "o n n
+Z.
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9on-erse'%
nn
n
a x
++
=
2
nn
n
a x
=
( ) .Re-ersn "$e order o, "$e #bo-e re'#"on
R#sn "$e coun"n ndex b% Fro& "o 2 Z -es rse "o 'o5ern "$e
su&"on ndex b% Fro& "on n
+ ]
9onsder #'so "$e ,o''o5n ,n"e ex#&!'e811
; G 10 11; G 10 11
;2 / 3 / 4 / ; / G /
2 / 3 / 4 / ; / G /G
//
211 G
//
; 2
T$#" s
nn
n
mm
m
n nn n
n n
a x a x a x a x a x a x
a x a x a x a x a x
a x
a x a x
=+ + + + +
+ + + + +
++
=
++= =
= + + + +
= + + + +
=
=
Exercse er,% "$e ,o''o5n8
( ) 0
n nn n
n n
i a x a x
++
= =
=
( ) ( ) ( ) ( ) ( ) ( ) ( )
0 0
0
1 2 /n n
n n
n n
ii n n a x x n n a x x
+
= =
+ + = + +
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Counting index up by c
That is if the counting index is raised/lowered by c, then the summation
index is lowered/raised by c as the case may be.Exercses
Wr"e "$e ,o''o5n su&"ons 5"$ "$e ndc#"ed cond"ons8
u#"ons 5"$ cons"#n" coe,,cen"s o, "$e ,or&8@
( ) ( )
5$ere #re cons"#n"s
1
5"$ 0.
d y dya b cy f x
dx dxa b c a
+ + =
L L L
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Prob'e&s #rse n Enneern 5$ere "$e coe,,cen"s #re no" cons"#n"s. A'' "$e
&e"$ods so ,#r s"uded -e n so "o s!e#6 "$#" s ,#' n suc$ c#sesJ Po5er,u' &e"$ods#re re>ured #nd exs" "$#" c#n so'-e second order ordn#r% d,,eren"#' e>u#"ons or
$$er order ODEs 5"$ -#r#b'e coe,,cen"sK5$ere "$e coe,,cen"s #re ,unc"ons o, "$e
nde!enden" -#r#b'eKn "$e !resen" c#sex
. In 5$#" ,o''o5s 5e 5'' concen"r#"e on so&eo, "$e &ore !o5er,u' &e"$ods o, so'u"on o, "$e second order 'ne#r d,,eren"#' e>u#"ons
o, "$e ,or&8@
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
5$ere #nd #re ,unc"ons o, .
5"$ 0x
d y dyP x Q x R x y F x
dx dx
x Q x R x P xP
+ + = L L L
T$e &e"$ods 5e #re #bou" "o de-e'o! #!!'% 5$ere "$e coe,,cen"s #re
!o'%no's or #n% #n#'%"c ,unc"ons. In "$s ,rs" !#r" o, our dscusson 5e #re on "oconcen"r#"e our e,,or"s "o5#rds "$e so'u"on o, "$e e>u#"on8@
( ) ( ) ( ) ( ) 0 /d y dyP x Q x R x ydx dx+ + = L L L
snce "$e &e"$ods o, so'u"on ,or (+ #nd (/+ #re s&'#r mutatis mutandis. Ho5e-er #-er% '#re c'#ss o, !rob'e&s n Enneern or #"$e"c#' P$%scs 'e#ds "o e>u#"on (/+
5"$ !o'%no' coe,,cen"sC ,or ex#&!'e8@
( ) ( )
( ) ( ) ( )
( )
( )
1 T$e 7esse' e>u#"on8 0.
T$e Leendre e>u#"on8 1 1 0.
/ T$e Ar%s e>u#"on8 0.
2 T$e Eu'er e>u#"on 0. 5$ere #nd #re co
d y dyx x x y
dx dxd y dy
x xy ydx dx
d yxy
dxd y dy
x x ydx dx
+ + =
+ + =
=
+ + = ns"#n"s
For "$s re#son 5e 5'' consder "$e ,or& o, c#se (/+ n 5$c$( ) ( ) ( ) #ndP x Q x R x
#re
!o'%no's 5"$ no co&&on ,#c"ors. Ho5e-er "$e &e"$od #!!'es e>u#''% 5e'' "o
e>u#"ons n 5$c$( ) ( ) ( ) #ndP x Q x R x
#re ener#' #n#'%"c ,unc"ons #nd n 5$c$
( ) 0.F x
1./.. THE O9A7ULARY
In "$e ,o''o5n de-e'o!&en" so&e 6e% conce!"s #nd -oc#bu'#r% 5'' be re>ured.I" be$o-es us no5 "o !er,or& # >uc6 #nd s$or" re-e5 o, so&e o, "$e ,und#&en"#' "er&s
#nd conce!"s. T$e s"uden" s re,erred "o e#r'er c#'cu'us '"er#"ure ,or # "$orou$ "re#"&en"o, "$e re'e-#n" "er#'.
1./..1. Ordn#r% #nd Snu'#r Pon"s
Ordinary Point
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I, # !on" 0exs"sx x=
suc$ "$#"( )0 0P x "$en "$e !on" 0 x x= s c#''ed #n
ordinary pointo, (/+. T$en n "$e ne$bour$ood o, 0 x x= e>u#"on (/+ c#n be5r""en n "$e ,or&
( )
( )
( )
( ) ( )
2
Q x R xd y dyy odx dxP x P x+ + = L L L
A !on" 0x x=
s s#d "o be #n ordinary pointo, "$e d,,eren"#' e>u#"on (/+ , bo"$
( +
( +
Q x
P x #nd
( +
( +
R x
P x n "$e s"#nd#rd ,or& (2+ #re #n#'%"c #" 0x x=
. A !on" "$#" s no" #n
ordn#r% !on" s s#d "o be # singular point!"v"#o, "$e e>u#"on.
In "$e dscusson "$#" ,o''o5s 5e 5'' consder "$e so'u"ons o, (/+ n "$e
ne$bour$ood o, "s ordn#r% !on".
$ingular point
I, # !on" 0x x= s suc$ "$#"
( ) 0P x = "$e !on" 0
x x=s c#''ed #singular
pointo, (/+. Snu'#r !on"s #re !o"en"#''% "roub'eso&e. S!ec#' &e"$ods 5'' beex!'ored '#"er on $o5 "o so'-e (/+ ne#r "s snu'#r !on"s. A snu'#r !on" 0
x x=o, "$e
d,,eren"#' e>u#"on (/+ underoes ,ur"$er c'#ss,c#"ons #s ben e"$er regularor
irregular. T$s c'#ss,c#"on de!ends ##n on "$e r#"on#' ,unc"ons
( )
( )
Q x
P x#nd
( )
( )
R x
P x
5$en e>u#"on (/+ s 5r""en n "$e s"#nd#rd ,or& (2+.
Regular or Irregular Singular Point
A snu'#r !on" 0x x=
s s#d "o be # regular singular pointo, "$e d,,eren"#' e>u#"on
(/+ , "$e ,unc"ons0
( +( +( +
Q xx xP x
#nd
0
( +( +( +
R xx xP x
#re bo"$ #n#'%"c #" 0
x x=. A snu'#r
!on" "$#" s no" reu'#r s s#d "o be #n irregular singular point
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NOTE
T$e ,rs" sen"ence n "$e #bo-e box &e#ns "$#" , e"$er one or bo"$ o, "$e ,unc"ons
0
( +( +
( +
Q xx x
P x
#nd
0
( +( +
( +
R xx x
P x
,#'s "o be #n#'%"c #" "$e !on" 0x x="$en "$e !on"
0 x x= s #n rreu'#r snu'#r !on".
1././. 9LASSIFI9ATION OF SIN
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I, 0x x
#!!e#rs at most"o "$e ,rs" !o5er n "$e deno&n#"or o,
( +
( +
Q x
P x #nd at most"o "$e
second !o5er n "$e deno&n#"or o,
( +
( +
R x
P x "$en 0x x=
s # reu'#r snu'#r !on"
ExercsesIn e#c$ o, "$e ,o''o5n !rob'e&s ,nd #'' snu'#r !on"s o, "$e -en e>u#"on #nd
de"er&ne 5$e"$er e#c$ one s reu'#r or rreu'#r.
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
/
1 2 / 3 0 G 0
/ sn 0 2 / 0
3 1 1 0
4 cos sn 0
; 1 0
G 1 / 1
d y dy d y dyx x y x xy ydx dx dx dx
d y dy d y dyx x x y x xy
dx dx dx dxd y dy
x x x x ydx dx
d y dyx x x y
dx dxd y dy
x x x ydx dx
d y dyx x x
dx dx
+ + = + =
+ + = + + =
+ + =
+ + =
+ + + =
+ + ( )
( ) ( )
( ) ( )
0
/sn 1 0
10 /cos 0x
x y
d y dyx x x y
dx dxd y dy
x e x ydx dx
+ =
+ + =
+ + =
( ) ( )
11 2 0 1 sn / 0
d y dy d y dyx xy y x x xy
dx dx dx dx+ + = + + =
1.2. POWER SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL
EQUATIONS NEAR THEIR ORDINARY
POINTS
7e,ore e&b#r6n on "$e !o5er,u' &e"$ods 6no5n #s Po5er Seres So'u"ons o, ordn#r%
d,,eren"#' e>u#"ons 'e" us re&nd ourse'-es o, our !re-ous 6no5'ede "$rou$ #n
ns"ruc"-e ex#&!'e.
Ex#&!'eSo'-e "$e d,,eren"#' e>u#"on8
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( )
0 3
d yy
dx+ = L L L
2) 7% usn "$e &e"$ods #'re#d% 6no5n "o us.3) Ex!ress "$e so'u"on ob"#ned n (+ #s # !o5er seres.
So'u"on
( ) T$e #ux'#r% e>u#"on8 1 0T$e c$#r#c"ers"c roo"sT$e ener#' so'u"on cos sn
i mm iy & x ' x
+ == = +
( ) ( )( )
( )( )
( )( )
( )
2 4 / 3 ;
1
0
A!!'%n "$e #c'#urn seres or "$e T#%'ors seres #bou" 0 "o "$e
ener#' so'u"on n 5e e"
T$#" s
1J 2J 4J /J 3J ;J
1 13
J 1 J
n nn n
n n
ii x
i
x x x x x xy & ' x
x xy & ' a
n n
+
=
=
= + + + + +
= ++
L L
L L L0
=
T$s ex#&!'e de&ons"r#"es "$e ,#c" "$#" # so'u"on o, #n ODE % be ex!ressed n "$e,or& o, # 'ne#r co&bn#"on o, !o5er seres
T$e ,o''o5n s # "$eore& 5e #re on "o use 5"$ou" !roo, n "$e ,o''o5n "re#"&en"
o, !o5er seres so'u"ons o, ODEs. I" u#r#n"ees the existenceo, !o5er seres so'u"onso, ODEs.
THEOREM (EXISTENCE THEOREM)
If 0x x=
is an ordinary point of the differential euation (!)" then the general
solution of (!)" is
0 1
0
( + ( + ( +nnn
y a x x & y x ' y x
=
= = +
#here $ and % are ar&itrary 'onstants and 1( +y x
and ( +y x
are linearly
independent po#er series solutions #hi'h are analyti' at the ordinary point 0x x=
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A so'u"on o, "$e ,or&( )0
0
n
nn
y a x x
=
= s s#d "o be # so'u"on #bou" "$e ordn#r%
!on" 0x x=
. In "$e 5or6 "$#" ,o''o5s 5e s$#'' ,or "$e s#6e o, s&!'c"% concen"r#"e on
so'u"ons #bou" "$e ordn#r% !on" 00x x= =
.0o'u"ons o, "$e ,or&
nn
n$ y a x
= = .
T$ere s #no"$er ,#c" ,ro& A'ebr# "$#" s o, s!ec#' &!or"#nce 5$c$ 5e #reon "o use o,"en. We ##n 5"$ou" !roo, >uo"e "$e ,#c"8
THEOREM (IENTIT PROPERT)
( ) ( )0 00 0
I, "$en 01 / .n n
n n n n
n n
a x x b x x a b n
= =
= = L
( )00
In !#r"cu'#r , 0 "$en 0 01/2 . 0n
n n
n
a x x x a n n
=
= = = L
I''us"r#"-e Ex#&!'eFnd # seres so'u"on n !o5ers o,x o, "$e d,,eren"#' e>u#"on (3+ .e.
( )
0 3
d yy
dx+ = L L L
So'u"on
Tes" ((his must always be performed) henceforth+8
( ) ( ) ( )( )
( )( )
( ) ( )00 0 0
1 0 1
0 1 0
0 s #n ordn#r% !on" o, 3 .T$e #bo-e exs"ence "$eore& u#r#n"ees # !o5er seres so'u"on o, "$e ,or&
0n n
nn n n
n n n
P x Q x R x
P
x
y a x x a x a x
= = =
= = ==
=
= = = No5 'e"
( )0
nn
n
iy a x
=
= L L L
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( )11
nnn
iiy na x
=
= L L L#nd
( ) ( )
M 1 nnn
iiiy n n a x
=
=
L L L
Subs""u"n (+ #nd (+ n"o (3+ 5e e"8y
( )
1 nnn
n n a x
=+
1 4 4 42 4 4 430
0nnn
a x
==
647 48
d y
dx Here 5e $#-e "o e sure #'' enerc "er&s #re ex!ressed n "er&s o,
nx . T$esecond "er& !oses no !rob'e&. Ho5e-er "$e second "er& s no" n "$s re>ured ,or&.
T$s re>ures us "o r#se "$e su&"on ndex b% #nd $ence 'o5er "$e coun"n ndex b%.
( ) ( ) 0 0
1 n nnnn n
n n a x a x
+= =
+ + +
( ) ( ) 0
1 0nnn
nn n a a x
+=
+ + + =
Fro& "$e den""% !ro!er"%
( ) ( )
( ) ( ) ( )
4
1 0 01/ .
01/ 1
n
n
nna
n n a a n
a nn n
+
+
=
+ + + = =
=+ +
L
L L L
E>u#"on (4+ s c#''ed "$e recurrence relation or recursive*recursion formula"Fro& "$e recurrence re'#"on (4+ 5e $#-e8
0 0
1 1 1/
0 02
0 1 J
1/ / 1 /J
1
2 / 2 / J 2J
a a
n a
a a an a
a aan a
= = =
= = = =
= = = =
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/ 1 13
0 024
1/
3 2 3 2 /J 3J
12
4 3 4 3 2J 4J
a a an a
a aan a
= = = =
= = = =
3 1 1;
13
; 4 ; 4 3J ;J
a aan a
= = = =
Subs""u"n "$e #bo-e n"o (+ .e. n"o 0
nn
n
y a x
=
=
0 / 2 3 4 ;
3 ;0 1 / 2 4
0 01 / 20 1
01 13 4 ;
.
J /J 2J
3J 4J ;J
nn
n
y a x
a a x a x a x a x a x a x a xa aa
a a x x x x
aa ax x x
=
== + + + + + + + +
= + + + + +
+ + + +
L
L
( )
( )
( )
( ) ( )
2 4 / 3 ;
0 1
1
0 10 0
1 .J 2J 4J /J 3J ;J
1 1
J 1 J
n nn n
n n
x x x x x xy a a x
x xy a a iv
n n
+
= =
= + + + + +
= +
+
L L
L L L
5$c$ s den"c#' "o "$e so'u"on (3#+ #bo-e 5"$ 0a &=
#nd 1a '=
No"e "$#" no cond"ons $#-e been &!osed on "$e cons"#n"s 0a #nd 1a so "$ese #reser-n #s "$e usu#' #rb"r#r% cons"#n"s.
IS9ELLANEOUS WORED E=APLES
Worded Ex#&!'e One
Fnd # !o5er seres so'u"on o, "$e ,o''o5n e>u#"on #bou" "$e !on" 0x= 8
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( )
1 0
d y dyx x y
dx dx + =
So'u"on
To so'-e ( ) ( )
1 0 1
d y dyx x ydx dx + = L L L .
( ) ( ) ( )( )
( )
T$e "es"8 1
0 1 0
0 s #n ordn#r% !on" o, 1 .
P x x Q x x R x
P
x
= = ==
=
Fro& "$e exs"ence "$eore& #bo-e (1+ $#s # so'u"on o, "$e ,or&
( ) ( )
( )
( )
( ) ( )
0
0 0 0
0
1
1
0
T$#" s "$e so'u"on s
#nd M 1
n nn
n n n
n n n
nn
n
nn
n
nn
n
y a x x a x a x
y a x i
y na x ii
y n na x iii
= = =
=
=
=
= = =
=
=
=
Subs""u"n (+ (+ #nd (+ n"o (1+ 5e e"8
( ) ( )
( ) ( )
1
1 0
1 0
0
s&!',%n "$e ex!ressons b% "#6n "$e coe,,cen"s nsde "$e no"#"on 8
0
s$,"n "$e su&&
1 1
1 1
n n nn n n
n n n
n n n nn n n n
n n n n
x n na x x na x a x
n na x n na x na x a x
= = =
= = = =
=
=
+
+
[ ]
( ) ( ) ( )
[ ]
( ) ( ) ( )
0 1 0
/
#"on ndex n "$e ,rs" "er&
0
n sure "$#" "$e coun"n ndces co&&ence "oe"$er #" n "$s c#se
1 1
1 / 1 1
n n n nn n nn
n n n n
nn
n
n
n n a x n na x na x a x
a a x n n a x n n
+= = = =
+=
=
=
+ + +
+ + + +
0
1 1 0 1
nn
n
n nn n
n n
a x
na x a a xa x a x
=
= =
+ + + =
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( ) ( ) ( ) ( )
( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ){ }
0 /
0 /
0 /
T$#" s
9#''n "$e den""% !ro!er"% n"o !'#% o
4 1 1 0
4 1 1 0
4 1 1 0
nn n nn
n
nnn
n
nn
n
a a a x n n a n na na a x
a a a x n n a n n n a x
a a a x n n a n n x
+=
+=
+=
+ + + + + + =
+ + + + + + = + + + + + + =
( )
( ) ( ) ( ) ( ) ( ) ( )( ) ( )
0
/
r e>u#"n coe,,cen"s #s 5e #re #ccus"o&ed "oE>u#"n coe,,cen"s8
9ons"#n"s8 1
/
0
8 4 0
1 8 1 1 0 1
1
1
0
nn nn n
nn
a a
x a
n nx n n a n n a a a nn n
na a n
n
a a
a
+ +
+
+ =
=
++ + + = = + +
= +
= =
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5/22/2018 Seriees
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1
/ 2 3 4 ; G3 ;0 1 / 2 4 G
2 4 G0 1 2 4 G
0 0 0 2 4 G0 1 0
2 4 G
0
0
.
.
.
3 ;/
13 ;/
nn
n
x
a a x a x a x a x a x a x a x a x
a a x a x a x a x a xa a a
a a x a x x x x
x x xa x
y a x
a
== + + + + + + + + +
= + + + + + += + +
= + +
= L
LL
L
T$ere,ore "$e ener#' so'u"on s
y 12 4 G
0
.13 ;/
xx x x
a x a
= + +L
Worded Ex#&!'e T5o
Fnd # seres so'u"on o,
( ) ( )
2 / 3 0
d y dyx x y
dx dx + + =
n !o5ers o, x .
So'u"on
To so'-e( ) ( )
2 / 3 0
d y dyx x y
dx dx + + = ( )1L L L
In !o5ers o, xK"$#" s ne#r "$e !on" 0.x=
( ) ( ) ( ) ( )( )
( )
Tes"8 2 / 3
0 2 0
0 s #n ordn#r% !on" o, 1
P x x Q x x R x
P
x
= = ==
=
( )
( )
( ) ( )
00
1
1
Assu&e
#nd
%M 1
n
n
nn
n
nn
y a x i
y na x ii
n na x iii
=
=
=
=
L L L
L L L
L L L
Subs""u"n (+ (+ #nd (+ n"o (1+8
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( ) ( ) ( )
( ) ( )
1
1
1
1 1 0
2 1 / 3 0
1 2 1 / 4 3 0
n n nn n n
n n
n n n n nn n n n n
n n n n n
x n na x x na x a x
n na x n na x na x na x a x
= =
= = = = =
+ + =
+ + =
In 5$#" ,o''o5s e,,or"s 5'' be concen"r#"ed on "r#ns,or&n #'' enerc "er&s "o "er&s n
nx 8
( ) ( )( )
( )
0
11 0 0
1
1 2 1
/ 4 3 0
n nn n
n n
n n nn nn
n n n
n
n
n na x n a x
na x a x a x
+= =
+= = =
+
+
+
+ + =
To be #b'e "o ex!ress #'' su&s usn on'% one no"#"on #'' coun"n ndces $#-e "oco&&ence #" "$e s#&e !on"K#nd ob-ous'% end #" "$e s#&e !on"Kso obser-e c#re,u''%$o5 "$s s$,"n o, "$e coun"n ndces s #c$e-ed8
( ) ( )( ) 0
S"#r"n 5"$ 1 2 1n nn nn n
nn na x n a x
+= =
+ +
( ) 11 0 0
1/ 4 3 0n n nn nnn n n
nna x a x a x
+= = =
++ + =
( )
( ) ( )
( )
/
1
1 1
0 1
1
2 1 2 / 2 1
/ 1 /
4 1 4 4 1
3 2 3 0
nn
n
nn
nn
n
nn
n
n
nn
n na x
a a x n n a
a x na x
a a x n a x
a a a x
=
=
=
+=
=
+ +
+ +
+
+ + + =
E#c$ "er& $#s been ex!#nded 5$ere necess#r% on "s o5n se!#r#"e 'ne.
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( ) ( ) ( )
( ) 1
/ 1 1 0 1
/ 4 1 3 0
G 2 / 4 1 3 3
1 2 1
n n nn n n
n n n
n nn n
n n
na x n a x a x
a a x a x a a x a a x
n na x n n a x
+
= = =
+= =
+
+ + + =
+ + + +
+ + +
9o''ec"n "er&s8
( ) ( )
( ) ( ) ( ) ( )
0 1 1 /
1
3 4 G G 1 2
1 / 3 4 1 2 1 0nn n nn
a a a a a a x
n n n a n a n n a x
+ +=
+
+ + + + + + =( ) ( )
( ) ( ) ( ) ( )
0 1 1 /
1
3 4 G 2 / 4
1 / 3 4 1 2 1 0nn n nn
a a a a a a x
n n n a n a n n a x
+ +=
+
+ + + + + + =( ) ( )
( ) ( ) ( ) ( ){ }
0 1 1 /
1
3 4 G 2 / 4
3 4 1 2 1 0nn n nn
a a a a a a x
n n a n a n n a x
+ +=
+
+ + + + + =+ E>u#"n coe,,cen"s usn "$e den""% !ro!er"%8
0 / 0 1
3 /9ons"#n"s8 3 4 G 0
G 2a a a a a a = =
( )1 / / 1 1 18 2 / 4 0 / x a a a a a a = =
1 0 1
1 1 3 // G 2
a a a
=
/ 1 0
1; 32 14
a a a =
( ) ( ) ( ) ( )
( ) ( )
( )
( ) ( )
( ) ( ) ( )
1
1
1
8 3 4 1 2 1 0
4 1 3
2 1 2 1
3 /
2 1
nn n n
nn n
nn n
x n n a n a n n a
nn na a a
n n n n
n na a a n
n n n
+ +
+ +
+ +
+ + + + + =
++ + = + + + +
+ + = + + +
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5/22/2018 Seriees
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. POWER SERIES SOLUTIONS OF ORDINARY
DIFFERENTIAL EQUATIONS NEAR THEIR ORDINARY
POINTS
OTHER THAN 0x =
For "$e c#se 5$ere " 5'' be necess#r% "o ,nd # !o5er seres so'u"on o, #n ODE #bou"
"$e ordn#r% !on" 00x x=
"$e ,o''o5n subs""u"on 5'' be de n "$e e>u#"on8
0z x x= . T$s &!'es 0z= 5$en 0x x= .T$s 'e#ds "o # so'u"on o, "$e ,or&
0
nn
n
y a z
=
= n"o 5$c$ ? 5'' be re!'#ced b% 0
x x.
0
0
Le"z x x
x z x
= = +
( )
.1
#nd .1
T$#" s
dy dy dy dy dyd dzydx dz dx dz dz dx dz
d y dy dy dy d y d yd d d dz dx dx dx dz dz dz dxdx dz dz
d y d y
dx dz
= = = =
= = = = =
=
Su&r%
0
W$en #nd
dy dyz x x
dx dz
d y d y
dx dz
= = =
Worded Ex#&!'e T$ree
So'-e "$e d,,eren"#' e>u#"on
( )
1 0 n !o5ers o, .
d y dyx y x
dx dx+ + =
So'u"on
No5 so'-n #n ODE n !o5ers o, x s e>u-#'en" "o so'-n "$e e>u#"on #bou" "$e!on" x= . So 'e" us ,rs" !er,or& "$e subs""u"onLe" z x x z= = +T$s subs""u"on "r#ns,or&s "$e -en e>u#"on n"o8
( ) ( )
1 0 1
d y dyz y
dx dz + + + = L L L
Tes"
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( ) ( ) ( )( )
( )
1 1 1.
0 0
0 s #n ordn#r% !on" o, 1
P z Q z z R z
P z
z
= = + =
=
( )Le" % nnn
a z i
=
L L L
( )11
nnn
y na z ii
=
= L L L
( ) ( )
#nd M 1 nnn
y n na z iii
=
= L L LSubs""u"n (+ (+ #nd (+ n"o (1+ 5e e"8
( ) ( )
( )( )
( )( ) ( )
( )( )
( )
1
1
1 0
10 1 1 0
10 1 0 0
1 1
1 1 0
1 0
1 1 0
1 1 1
1
n n nn n n
n n n
n n n nn n nnn n n n
n n n nn nn n
n n n n
n nnn
n n
n
n
n
n na z z na z a z
n a z na z na z a z
n a z na z n a z a z
a n a z na z a
n a
= = =
+= = = =
+ += = = =
+= =
+
+
+ +
+ + + =
+ + + + =
+ + + + + =
+ + +
+ +
( ) ( ) ( ) ( ){ }
( ) ( ) ( ) ( ) ( ){ }
1
1 1
1 0 11
1 0 11
0
1 1 0
1 1 1 0
n nnn
n n
nn nn n
n
nnn n
n
z a z
a a a n n a na n a a z
a a a n n a n a n a z
+
= =
+ +=
+ +=
+ =
+ + + + + + + + + =
+ + + + + + + + + =
E>u#"n coe,,cen"s8
( )
( ) ( ) ( ) ( )
( )
( )
0 1 0 1 0
1
1
1
18 0
8 1 1 1 0
0
1 1
n nn n
nn n
nn n
z a a a a a a
z n n a n a n a
n a a a
a a a nn
+ +
+ +
+ +
+ + = = +
+ + + + + + =
+ + + =
= + +
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( ) ( )
( )
/ 1 1 0 1 1 0
/ 0 1
1 1 1 1 1 11
/ / /
14
n a a a a a a a a
a a a
= = + = + + =
=
T$e ener#' so'u"on s
0
/ 20 1 / 2
nn
n
y a z
a a z a z a z a z
==
= + + + + +
L
( ) ( ) /0 1 1 0 0 11 1 4
a a z a a z a a z = + + + +L
( )
( ) ( )( )
( ) ( )
/ /1
/ /
0 1
Re!'#cn b% 5e e"8
.
0 4 4
1
4 4
1
z x
z z z zz
x x x xy a a x
a a
+ +
= + + + +
= +L L
L L
E=ER9ISES
Ques"on OneT$e #bo-e e>u#"on c#n be so'-ed s"r#$" #5#% 5"$ou" $#-n "o c$#ne "$e -#r#b'eJ
In "$s c#se 5e #ssu&e "$e so'u"on o, "$e ,or&
( )
( )
( ) ( )
0
1
1
#nd
M 1
n
nn
n
nn
nn
n
y a x
y na x
y n na x
=
=
=
=
=
=
No5 subs""u"e Ly #nd My n"o "$e -en ODE #nd e>u#"e coe,,cen"s o, '6e !o5ers o,
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( )x .#6e sure %ou ex!ress "$e ,#c"or
( )1x n "$e -en e>u#"on n "er&s o,
( )x .Hence ob"#n "$e so'u"on #nd co&!#re "$e s#&e 5"$ 5$#" 5e $#-e us"
ob"#ned.
You $#-e "o "es" 5$e"$er or no" "$e !on" x= s #n ordn#r% !on" o, "$e ODE.
+,POR(&-(
To so'-e (/+ n !o5ers o,( )0x x e sure #'' "$e !o'%no' coe,,cen"s "$#" s
( ) ( ) ( ) #ndP x Q x R x#re ex!ressed n "er&s o,
( )0x x .
Ques"on "5oSo'-e "$e e>u#"on
/
G 0
d y dyx xy y
dx dx + =
n !o5ers o,x .
Ques"on T$ree
( )
G 0x d y dy
xy ydx dx
+ =
n !o5ers o, (+. ( )/x
(+. ( )/x +
1.4. POWER SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL
EQUATIONS NEAR THEIR SIN
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*RO%ENI+S THEOREM
I, 0x x= s # reu'#r snu'#r !on" o, "$e d,,eren"#'e>u#"on (/+ "$en "$ere exs"s #" 'e#s" one so'u"on
o, "$e ,or& 0 0 0 00 0( + ( + ( + 0c n n cn n
n n
y x x a x x a x x a
+
= == =
5$ere c s # cons"#n" "o be de"er&ned.
Rer68 "$e !$r#se ) at least one solution* n "$e !recedn "$eore& &e#ns "$#" 5e #reu#r#n"eed o, on'% one so'u"on #nd no" #ssured o, "5o seres so'u"ons #s be,oreJNOTE
T$e -#'ue o, c s "o be de"er&ned ,rs" be,ore "$e so'u"on s de"er&ned.
I, c s ,ound "o be # nu&ber "$#" s no" # nonne#"-e n"eer #s " ,re>uen"'% s no""$en
"$e corres!ondn so'u"on ( )0
0
n c
n
n
y a x x
+
=
= s no" # !o5er seres so'u"on
R./&%%
A !o5er seres n !o5ers o, x a s #n n,n"e seres o, "$e ,or& 0( + n
n
x a n
+=
.Suc$ # seres s s#d "o be cen"red #" x a= or #bou" x a=
Ex#&!'e
So'-e "$e ,o''o5n d,,eren"#' e>u#"on
/ 0
d y dyx y
dx dx+ =
n !o5ers o, x .
So'u"on
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( )
( ) ( ) ( )
( ) ( )
To so'-e / 0 1
,ro& "$e e>u#"on / 1 #nd 1TEST
So'-n n !oe5rs o, &e#ns so'-n #bou"
No5 s # snu'#r !on" o, 1 .
0
0 0 0
d y dyx y
dx dxP x x Q x R x
x x
P x
+ =
= = =
=
= =
L L L
( ) ( )
( ) ( )
( )
( )
( ) ( )
( )
0
0
0 0 0
0 0
11'& '& 0 '&//
1'& '& 0/
x x x x
x x x
Q x Q xx x x x
P x P x x
R xx x x
P x x
= = =
= =
7o"$ '&"s exs" "$ere,ore x 0 s # regular singular pointo, (1+.
T$ere,ore 5e #ssu&e
( ) ( )
( )
( )
( )
( )( )( )
0 0 00
00
00 0
1
0
0
bu"
"$#" s
0
M 1
c n
nn
nc
nn
n c c nn n
n n
n cnn
n cn
n
i
ii
y x x a x x x
y x a x x
y a x x a x a
y n c a x
y n c n c a x
=
=
+
= =
+
=
+
=
=
=
= =
= +
= + +
L L L
L L
Subs""u"n (+ #nd (+ n"o (1+ 5e e"8
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( ) ( ) ( )
( ) ( ) ( )
( ) ( )( )
1
0 0
1 1
0 0
10 1
0
10
0
0
1
/ 1
1 / 1
/ 0
0ex!ress #'' enerc "er&s n !o5ers o,
/
n c n cn n
n n
n c n c
n nn n
c n cn
n
c
n cn
n
n cn
n
n c n c a x n c a x
n c n c a x n c a x
c ca x n c n c a x
ca x n c
x a x
a xn c
+ +
= =
+ +
= =
+
+=
+
=
+
=
+ + +
+ + + +
+ + + +
+ + + +
+ =
=+
( )
( )( ) ( ) ( ) ( )( )
( )
( ) ( )
1 10
10
0
1
1
0 0
0
0co''ec"n "er&s
/ 1 1
1
1
0
/
1 / / 1
1
0
0
/ n cnn nn
n cn
n
nn
c
n c
n
n cn
n
n c n c a n c a a x
c
a x
c a
n c n c a a
c ca ca x
x
a x
c x
+
+ +=
+
+=
+
+
=
+
=
+ + + + + +
+ =
+ + + + +
+
=
=
E>u#"n coe,,cen"s8
( ) ( )1 0 0'o5es" !o5er 8 8 / 0 bu" 0cx c c a a =
( ) ( )/ 0c c iii = L L L
( ) ( ) s c#''ed "$e o, 1 . I" s #'5#%s # >u#dr#"c e>u#"on n
0 or
/
iii cindicial e!uation
c c = =
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( ) ( )
( ) ( )
( )
( ) ( )1
1
1
1
1We ob"#ned "5o -#'ues o, no5 5e $#-e "o de"er&ne # se!#r#"e ,or e#c$ -#'ue o, .
W$en 0
01 / 1
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No5 -er,% "$e ,o''o5n8
/ 2/ / 0 1 / 2
0 0
( . . . . +c n nn nn n
y x a x x a x x a a x a x a x a x
= =
= = = + + + + +
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( )
( )
0 0 0 0 / 2/
0
0
/ 2/
0
0
3 1 1 3 G 1 / 3 G 11 1 / 2 3 G 11 12
J3 G 11 /
1 1 1 11
3 1 J 3 G /J 3 G 11 2J 3 G 11 12
J3 G 11 /
n
n
a a a ax a x x x x
ax
n n
a x x x x x
ax
n n
= + + + + +
+ + + +
= + + + +
+ + + +
L LL
LL
NOTE
In "$e !re-ous so'u"ons 5e #'5#%s o" "5o d,,eren" coe,,cen"s 0a
#nd 1a
s#%
,or "$e "5o 'ne#r'% nde!enden" so'u"ons bu" no5 5e $#-e on'% one 0a
J
T$e
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One c#n ob"#n "$e ndc#' e>u#"on be,ore subs""u"n 0
n c
n
n
y a x
+
=
=
n "$e d,,eren"#' e>u#"on (/+. I, 0x= s # reu'#r snu'#r !on" o, e>u#"on (/+ #nd , 5e 'e"
0
( +'&( +x
xQ xpP x
=#nd
0
( +'&( +x
x R x!P x
= "$en "$e
ndc#' e>u#"on o, ( 1+ 0c p c !+ + = (/+ s. . . . (;+
E>u#"on (;+ s # #'5#%s # >u#dr#"c n c #nd s 6no5n #s "$e ndc#' e>u#"on o, (/+
1.4.. Roo"s o, "$e Indc#' E>u#"on
9#se I
T$e "5o roo"s1 #ndc c c c= =
#re ds"nc" #nd do no" d,,er b% #n n"eer. In "$s c#se"$e "5o nde!enden" seres so'u"ons #re ob"#ned #s #bo-e .e. e#c$ one o, "$e "5o roo"s
-es rse n"o # seres so'u"on. T$e ex#&!'e #bo-e ,#''s under "$s c#se.
9#se II
T$e "5o roo"s 1 #ndc c c c= =
#re ds"nc" #nd d,,er b% #n n"eer .e. "$e% #re suc$
"$#" 1c c k k = +
."$s c#se $#s "5o sub c#ses8
Sub@c#se I
One coe,,cen" na
beco&es indeterminate5$en 1c c= # ener#' so'u"on one withtwo linearly independent series solutions#s usu#''% -en b% usn on'% "$s -#'ue
o,c .Sub@c#se II
One coe,,cen" na
beco&es n,n"e 5$en 1c c=
"$e seres s re5r""en b% re!'#cn 0a
b% 1( +k c c
9#se III
T$e "5o roo"s #re e>u#' .e. 1 c c c= =
I''us"r#"-e Ex#&!'es
9#se IIKSub@c#se ITo so'-e "$e d,,eren"#' e>u#"on
( ) ( )
2 0 1d y dyx x x y
dx dx+ + + = L L
n !o5ers o,x .Tes" exercse
0 s # reu'#r snu'#r !on"x =
Assu&e
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5/22/2018 Seriees
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0
0 0
1
0 0
0
( + M ( + ( 1+
c n n c
n n
n n
n c n c
n n
n n
y x a x a x a
y n c a x y n c n c a x
+
= =
+ +
= =
= =
= + = + +
Subs""u"n n"o ( )1 5e $#-e
0 0 0 0
( 1+ ( + 2( + 0n c n c n c n cn n n nn n n n
n c n c a x n c a x a x a x
+ + + + +
= = = =
+ + + + + + =
F'' n "$e #!
( ) ( )
( ) ( ) ( ){ }
10 1
1 2 1
1 2 0
c c
n cn n
n
c c c a x c c a x
n c n c n c a a x
+
+
=
+ + + + + + + + + + + + =
( ) ( ) ( ) ( )
( ) ( ){ }
10 1
1 /
/ 0
c c
n cn n
n
c c a x c c a x
n c n c a a x
+
+
=
+ + + + +
+ + + + + + =
E>u#"n coe,,cen"s8
0 0
1
11
1
8 ( 1+ ( + 0 0 ( +
( 1+ ( + 0 1 ( +
(1+.(0+ 0J
8 ( /+ ( + 0
c
c
x c c a but a From lowest power
c c c or c $,&%%.R ROO(
-owwhen c the smaller root we have
aa becomes +-0.(.R,+-&(.
x c c a(herefore for a general solution
wewill
+
+ + =
+ + = = =
=
=+ + =
. . 1J
usethisvalueof c and discard
asit were theother valuei e c
=
( 1+ ( + 0
/2. . . .( 1+ ( +
n c
n n
n
n
x n c n c a a
from which
a
a nn c n c
+
+ + + + + =
= =+ + + +W$en 8c=
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( )
0 0
1 1
/
1
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9#se IIK Sub@c#se IIHere 5e #re on "o obser-e "$e ,o''o5n8
1. 1 "$e roo"s o, "$e ndc#' e>u#"on #nd d,,er b% #n n"eerc c c c= =
2. 1"$e s''er roo" s#% es ,or so&e n,n"enc c a n=
T$e ,rs" s"e! "o5#rds de"er&nn "$e so'u"on s "o re!'#ce 0a b% ( )0 1b c c
T$e so'u"on no5 # ,unc"on o, #ndc x s ex!ressed n "$e ,or&( )y f x c=
#nd "$en"$e ener#' so'u"on s de"er&ned ,ro&8
1
1( +c
fy &f x c '
c
= +
NOTE8 As n "$e !re-ous c#se c c=
5'' usu#''% -e rse "o # so'u"on "$#" s # sc#'#r&u'"!'e or !#r" o, "$e so'u"on ob"#ned b% usn "$e s''er roo".
Ex#&!'e8
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1 1
0 0 0
0 0
( 1+ ( + ( 1+ ( + ( +
/( + 0
n c n c n c
n n n
n n n
n c n c
n n
n n
n c n c a x n c n c a x n c a x
n c a x a x
+ + +
= = =
+ +
= =
+ + + + + +
+ =
1
0 1
0
( + ( 1+ ( 1+ ( 1+ 0c n cn nn
c c a x n c n c a n c a x
++
=
+ + + + + + = E>u#"n coe,,cen"s8
1
0 0
1
1
8 ( + 0 0
0 ( +
8 ( 1+ ( 1+ ( 1+ 0
1 01/ . . .
1
c
n c
n n
n n
x c c a but a
c or c the smaller root
x n c n c a n c a
n ca a n
n c
++
+
= = =
+ + + + + =
+ + = =
+
W$en 0c= "$e s''er roo"1
1 01/ . . . .
1n n
na a n
n+
+= =
No"e "$#" "$s -#'ue o, c es a
n,n"eJRe-er"n "o our recurson re'#"on
1
1
1n n
n ca a
n c+
+ += + 01/ . . .n=
1 0 0
1 ( 1+( +
1 ( 1+
c c ca a a a
c c c + + + = =
/ 0 2
( +( /+ ( /+ ( 2+
( 1+ ( 1+
c c c ca a a a
c c c c
+ + + += =
0
c n
n
n
y x a x
=
=
/ 2
1 / 2 . . . . .c
ox a a x a x a x a x = + + + + +
/
0 0 0 0
1 ( 1+ ( + ( + ( /+
. . .1 ( 1+ ( 1+
c c c c c c
x a a x a x a xc c c c c
+ + + + +
= + + + +
(No"e8 A'so #" "$s !on" , 5e 'e" 0c= "$e coe,,cen"s beco&e n,n"eJ+.( )0 0 0Le" 0a b c b c= =
/
0
( 1+ ( 1+ ( + ( +( /+. . . .
( 1+ 1 1
c c c c c c cy b x c x x xc c c
+ + + + + = + + + +
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5/22/2018 Seriees
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/
0
( 1+ ( 1+ ( + ( + ( /+. ( + . . . .
( 1+ 1 1
c c c c c c c%et y f x c b x c x x xc c c
+ + + + += = + + + +
T$s &!'es "$#" "$e ener#' so'u"on s -en b%
( )1
c
fy &f x c '
c
= + COT!"E
Here 10c =
"$ere,ore
/ /
0 0
0
/ /
1 1
( 'n + ( 4 . . . .+ 1 3 11 . . .
('n + ( 4 . . .+ 1 3 11 . . .
c
fb x x x b x x x
c
b x x x b x x x
=
= +
= + + + + + + +
No5 ,ro& #bo-e "$e u#"on8 verify+Assu&e
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0
0 0
0.c n n c
n n
y x x x a
+
= =
= = Subs""u"n "$e der-#"-es n"o (1+ #nd s&!',%n8
1 1
0 0 0
( 1+( + ( + 0n c n c n cn n nn n n
n c n c a x n c a x a x
+ + + +
= = =
+ + + + + =
[ ] 1 0
( 1+ ( + ( + 0n c n cn nn n
n c n c n c a x a x
+ +
= =
+ + + + + =
1
0
( + 0n c n cn nn n
n c a x a x
+ +
= =
+ + =
1
0 1
0
( 1+ 0c n c n cn nn n
c a x n c a x a x
+ ++
= =
+ + + + =
[ ]( ) 1 1
0 1 1
( 1+ ( + 1 0c c c n cn nn
c a x c a x c a x n c a a x
+ ++
=
+ + + + + + + + =E>u#"n coe,,cen"s8
1
0 0
1 1
1
1
1
8 0 0 0 ( +
8 ( 1+ 0 ( 1+ 0 0
8 ( + 0 ( + 0 0
8 ( 1+ 0
/2. . . .( 1+
c
c
c
n c
n n
n
n
x c a but a c twice
x c a but c a
x c a but c a
x n c a a
aa n
n c
+
++
+
= =
+ = + =
+ = + =
+ + + =
= =+ +
Fro& 5$c$ one c#n -er,%8
/ 2 ; G . . . . . . 0a a a a= = = = =
0 0 0
/ 4
( /+ ( /+ ( 4+ ( /+ ( 4+ ( +
a a aalso a a a
c c c c c c= = =
+ + + + + +
(er,% "$e #bo-e coe,,cen"s+
( )0
We 'e" c nnn
y f x c x a x
=
= = / 2 3
0 1 / 2 3
4 ; G
4 ; G
(. . . +
c
x a a x a x a x a x a xa x a x a x a x
= + + + + ++ + + + +
/ 4
0 / 4 ( 0 0 0 0 0 0 . . . . +cx a a x a x a x= + + + + + + + + +
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/ 4 0 0 0
0 . . . .
( /+ ( /+ ( 4+ ( /+ ( 4+ ( +
c a a ax a x x xc c c c c c
= + + + + + + + +
/ 4
0 1 . . . .
( /+ ( /+ ( 4+ ( /+ ( 4+ ( +
c x x xa x
c c c c c c
= + +
+ + + + + +
T$ere,ore
/ 4
1 0 ( 0+ 1 . . . .
/ / .4 / .4 .
x x xy f x a
= = + +
11
/ 4
0 1 . .
( /+ ( /+ ( 4+ ( /+ ( 4+ ( +
c
c c
f x x xy a x
c c c c c c c c=
= = + + + + + + + +
/ 4 0
0
0
/ 4 0
0
0
'n 1 . . .
( /+ ( /+ ( 4+ ( /+ ( 4+ ( +
1 . . .( /+ ( /+ ( 4+ ( /+ ( 4+ ( +
c
c
x x xa x x
c c c c c c
x x xa x
c c c c c c c
=
=
= + +
+ + + + + +
+ + + + + + + + +
Ter& P#r"#' der-#"-e 5r" c#'ue o, der-#"-e #" 0c=
1 0 0
1
( /+c + ( )/
/c
+
/
1
( /+ ( 4+c c+ +{ }
1
( /+ ( 4+
'n 'n( /+ 'n ( 4+
1 1 1
/ 4
1 1 .
/( /+ ( 4+
%et zc c
z c c
z
z c c c
z
c c cc c
=+ +
= + + +
= + + +
= + ++ +
/ /
/ .4 / .4 +
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1
( /+ ( 4+ ( +c c c+ + +
/ / /
/ .4 . / .4 . / .4 .
+ +
/ 4
0 .
0
/ 4 4
0 / / / / / /
'n 1 . . ./ / .4 / .4
0 . . . .
/ / .4 / .4 / .4 . / .4 . / .4 .
c
f x x xy a x
c
a x x x x x x
=
= = + + + + + + + +
REAR T$ou$ " 5#s rer6ed #bo-e "$#" se#rc$n
,or # ener#' "er& na n "$e ener#' so'u"on
% #" "&es !ro-e ,ru"'ess bu" 'oo6n
,or # !#""ern s #'5#%s &!er#"-e.
No5 5r"e ou" ne#"'% "$e ener#' so'u"on "$#" s
1 1 .y k y k y= +
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1 1
/ 4
0 1
/ 4
0
/ 4 4
0 / / / / / /
/ 4
0 1 0
1 . . . ./ / .4 / .4 .
'n 1 . . . ./ / .4 / .4 .
. . .
/ / .4 / .4 / .4 . / .4 . / .4 .
( 'n + 1/ / .4
y k y k y
x x xa k
x x xa k x
x x x x x xa k
x xa k a k x
= +
= + +
+ + +
+ + + + +
= + +
/ 4
0 / 3
/ 4
0 1
/ 4
1 / 3
. . . ./ .4 .
1 1 1 1 1 . . .
// / . / .
( 'n + 1 . . . ./ / .4 / .4 .
1 1 1 1 1 . . . // / . / .
x
x x xa k
x x x/ / x
x x x/
+
+ + + + + +
= + + +
+ + + + +
s "$e ener#' so'u"on.
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1.;. APPLI9ATIONS
1.;.1. T$e Leendres E>u#"onKSo'u"on #bou" #n Ordn#r% Pon"T$e d,,eren"#' e>u#"on
( ) ( )
1 1 0 5$ere
d y dyx x k k y k
dx dx + + =
s o, s!ec#' n"eres" n
A!!'ed #"$e"cs P$%scs #nd Enneern !#r"cu'#r'% n bound#r% -#'ue !rob'e&s,or s!$erc#' obec"s. I" s c#''ed "$e%egendre1s e!uation. So c#''ed #,"er "$e Frenc$
#"$e"c#n Adren #re Leendre (1G"$ Se!"e&ber 1;3 10"$ V#nu#r% 1G//+.
We #re no5 e&br#ced "o so'-e( ) ( ) ( )
1 1 0 1
d y dyx x k k y
dx dx + + = L L L
Here "$e !on" 0x= s #n ordn#r% !on" o, "$e e>u#"on. T$e ,o''o5n s # so'u"on o,"$e e>u#"on. Assu&e
1
0 1
M ( 1+n n nn n nn n n
y a x y n a x y n na x
= = =
= = =
Subs""u"n n"o( )1
5e e"8
( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( )
( ) ( ) ( )
1
1 0
1 0
/
1 0 1
1 1 1 0
1 1
1 0
1 / 1 1
1 1 1 1
n n nn n n
n n n
n nn n
n n
n nn
n n
n nn n
n n
n nn n
n n
x n n a x x na x k k a x
n n a x n n a x
na x k k a x
a a x n n a x n n a x
a x na x k k a k k a x k k a x
= = =
= =
= =
+= =
=
+ + =
+ + =
+ + + +
+ + + + + +
0
=
=( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( ){ }
( ) ( )
( ) ( ) ( )
0 / 1 1
0 / 1
0 / 1
1 J /J J 1
1 0
1 J /J
1 0
1 J /J
1 1
nn n
n
nn n
n
n
k k a a a a k k a x
n n a k k n n a x
k k a a a k k a x
n n a k n k n k n a x
k k a a a k k a x
n n a k n k
+=
+=
+
+ + + + +
+ + + + + = + + + + +
+ + + + + + = + + + + +
+ + + + + +
( ){ }
0nnn
n a x
=
=
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E>u#"n coe,,cen"s8
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )( ) ( )
( ) ( )
00 0
/ 1 / 1
18 1 J 0
J
1 8/J 1 0 /J
1 1 0
1
1
nnn
nn
k kx k k a a a a
k kx a k k a a a
x n n a k n a
k n k na a n
n n
+
+
++ + = =
++ + = =
+ + + + + = + +
= + +
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W$en n s # nonne#"-e n"eer 5e ob"#n #n n"$@deree !o'%no' so'u"on.
Le" us ,or "$e "&e ben deno"e "$ese !o'%no' so'u"ons b% ( )n% x 8
( ) ( )
( ) ( ) ( )
( ) ( )
0 0 1 1
/ 0 / 1
2 / 32 0 3 1
31 /
//3 12 11 10
/ / 3
% x a % x a x
% x a x % x a x x
% x a x x % x a x x x
= = = = = + = +
And so on #nd so ,or"$.
I, n "$e #bo-e !o'%no's ( )n% x 5e de,ne 0a #nd 1a n suc$ # 5#% "$#" ( )
1 1n% = "$e
!o'%no's( )n% x "r#ns,or& n"o 5$#" #re 6no5n #s "$e Leendre !o'%no's o,
deree n . T$e% #re deno"ed b% ( )nP x . T$#" s ( )n
P xs "$e Leendre !o'%no' o, deree
n .
Ex#&!'es
( )
( )
( ) ( ) ( ) ( )
0 0 0
Le" 1 11
1 /.1 1 1
1 11 / / 1
%
a a a
% x x P x x
= = = =
= =
( )
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0
1
/ 2 / 2
3 / 4 2 3 4
S&'#r'% (-er,%+81
1 13 / /3 /0 /
3 G
1 14/ ;0 13 /1 /13 103 3
G 14
P x
P x x
P x x x P x x x
P x x x x P x x x x
=
=
= = +
= + = +
1.;.. RODRI
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( ) ( ) ( )
( ) ( )
0
1 J
J J J5$ere
e-en
1 odd
n-
n rn
nr
n rP x x
n r n r n r
nn
- n n
=
=
=
Exercses1. Ex!ress "$e ,o''o5n !o'%no's n "er&s "$e Leendre !o'%no's
3.2 / 1x x x x+ + + +
4. /1 / 2x x x+
5./ 1x +
6. 2 / /x x x x+ + . S$o5 "$#"
7.( + ( 1+ ( +nn nP x P x =
8.
( ) ( )1
1
0nP x dx n
9.
( )1
//
1
2
/3x P x dx
=
1.;./. SOLUTION A7OUT A REu#"on o, order . I"s so'u"ons #re c#''ed 7esse' ,unc"ons o, order
.
1.;./.1.1. So'u"on o, "$e 7esse's e>u#"onRe>ured "o so'-e
( ) ( )
0 1
d y dyx x x y
dx dx+ + = L L L
I" c#n be s$o5n "$#" 0x= s # reu'#r snu'#r !on" o, (1+.
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Assu&e0
0 0
0.c n n cn nn n
y x a x a x a
+
= =
= = Subs""u"n "$s -#'ue n"o (1+8
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
0 0 0 01 / 2
10 1
1
10 1
1 0
1 1 1
1
n c n c n c n cn n n n
n n n n
c c n cn
n
c c n cn
n
n c n c a x n c a x a x a x
c ca x c c a x n c n c a x
ca x c a x n c a x
+ + + + +
= = = =
+ +
=
+ +
=
+ + + + + =
+ + + + +
+ + + + +
1 4 4 4 44 2 4 4 4 4 43 1 4 44 2 4 4 43 1 4 2 4 3 1 442 4 43
1 4 4 4 4 4 4 4 4 4 4 4 2 4 4 4 4 4 4 4 4 4 4 4 3
/
1 0 1
2
0
n cn
n
c c n cn
n
a x
a x a x a x
+
=
+ +=
+
=
1 4 4 4 4 4 4 4 2 4 4 4 4 4 4 4 3 1 4 2 4 3
1 4 4 4 4 44 2 4 4 4 4 4 43
9o''ec"n "er&s #nd c#rr%n ou" so&e #'ebr#c n!u'#"ons8
( ) ( ) ( ){ }
( ) ( ) ( ) ( )
( ) ( ){ }
1
0 1
10 1
1
1 1
1 0
c c n cn n
n
c c
n cn n
n
c a x c a x n c a a x
c c a x c c a x
n c n a a x
+ +
=
+
+
=
+ + + + +
+ + + + +
+ + + + =
E>u#"n coe,,cen"s8
( ) ( )
( ) ( ) ( )
0 0
1
1 1
8 0 bu" 0 .
8 1 1 07u" 1 0 ener#''%. 0.
c
c
x c c a a c c
x c c a c a
+
+ = =
+ + + = + =
( ) ( ) 8 0n c n nx n c n c a a + + + + + =
( ) ( )
1 n na a n
n c n c = + + +
Frs" consder "$e c#se 5$erec =
( )
1
n na a n
n n = +
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( ) ( ) ( )
( )
0 0 0
/ 1 1 /
1 1 1
1 1
1/ bu" 0 0
/ /
n a a a a a
n a a a a
= = = = + + +
= = = =+
No"e (#nd -er,% "$#"+ / 3 ; H0a a a a= = = = =L
( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
2
2 02
4 2 02
4 0 4 04 4
1 1 12
2 2 . 1
1
. 1
1 1 14 4 4 ./. / . 1
1 1
./. 1 / ./J 1 /
n a a
a a
n a a a
a a a a
= = = + + +
=+ +
= = = + + + +
= = + + + + + +
S$o5 "$#"
( ) ( ) ( ) ( )
( )
( ) ( ) ( )
GG
0
1#nd $ence ener#''%
.2J 1 / 2
1
e-en J 1 /
k
k
k
a
a a kk k
=+ + + +
= + + + +
L
No5 !u""n k n= 5e e"( )
( ) ( ) ( ) ( ) 0
1 0
. J 1 /
n
nn
a a nn n
=
+ + + +L
T$ere,ore one o, "$e so'u"ons s
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( )
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
0
2 4 G0 2 4 G
2 40 0 0 0
2 4
0
1 1 1
1 .J 1 ./J 1 /
1
. J 1 /
c nn
n
c
n
n
n
y x a x
x a a x a x a x a x
a a x a x a xx
a xn n
=
=
= + + + + +
+ += + + + + + +
+ + + + + +
L
L
L LL
T$#" s
( )
( ) ( ) ( ) ( ) ( )0
0
1
J 1 /
n
n
nn
y a x xn n
=
=
+ + + + L L LL!ro-ded n s no" # ne#"-e n"eer.
For re#sons "o be ob-ous s$or"'% 'e"( )
0
1
1a
=
+#nd subs""u"e " n"o (+8
0
( 1+.
( 1+ J( 1+( +( /+. . . ( +
n n
nn
xxy
n n
=
=
+ + + + + 2
2
4
4
2 4
2 4
1
( 1+ ( 1+ ( 1+ J( 1+ ( 1+.( +
( 1+. . . .
./J( 1+ ( 1+ ( +( /+ J( 1+ ( 1+ ( +( /+. . .( +
1
( 1+ ( + .J ( /+ /J (
n n
n
x x x
xx
n n
x x x x
= + + + + + + + +
+ + +
+ + + + + + + + + = + + + + +
( 1+. . . . . .
2+ J ( 1+
n n
n
x
n n
+ + +
+ +
xy
= 0n
=
( 1+
J ( 1+
n n
n
x
n n
+ +
0 0
( 1+ ( 1+
J ( 1+ J ( 1+
n nn n
n n
x x x
n n n n
+
= =
= = + + + +
T$e so'u"on 5$c$ s ob"#ned 5$enc v=
s usu#''% deno"ed b%( )2 x
"$#" s n "$s c#se
( ) ( )
( )
0
1
J 1
n n
n
x2 x
n n
+
=
= + +
!ro-ded s no" # ne#"-e n"eer.
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T$s ,unc"on ( )2 x s c#''ed "$e 7esse' ,unc"on o, "$e ,rs" 6nd o, order .
W$en c = 5e e" ,ro& ( )2 x
( ) ( )
( )
0
1
J 1
n n
n
x2 x
n n
=
=
+
"$e 7esse' ,unc"on o, "$e ,rs" 6nd o, order
( ) ( )#nd2 x 2 x #re "5o 'ne#r'% nde!enden" so'u"ons (1+
( + ( +y &2 x '2 x = +s "$e ener#' so'u"on o, "$e 7esse' e>u#"on (1+
Loo6n #" "$e s%&&e"r% 5"$ 5$c$ #!!e#rs n "$e ener#' so'u"on " s e-den" "$#"
"$e e#r'er cons"r#n" "$#" 0 c#n no5 be ',"ed.
cos ( + ( +( + n"
sn
+n general for many practical purposes it is found more
convenient totakethelinear combination2 x 2 x
3 x not an eger
=
In our "re#"&en" o, bo"$ "$e 7esse' e>u#"on #nd "s conco&"#n" 7esse' ,unc"ons 5e #re
on "o s#"s,% ourse'-es 5"$ on'%( )2 x #nd ( )2 x #s ob"#ned #bo-e. T$ose 5$o
s"'' $#-e so&e "$rs" ,or &ore o, )7esse'* #re re,erred "o &ore #d-#nced '"er#"ure ,or"$e s#&e.
We c#n ,or # &o&en" e # s'' c$#ne n "$e s ndex b% !u""n r,or n
ob"#nn8
( ) ( )( )
0
1J 1
r r
r
x2 xr r
+
= = + + #nd
( ) ( )( )
0
1J 1
r r
r
x2 xr r
=
= + No5 , 5e 'e" n= 5e ob"#n
( ) ( )
( )
0
1
J 1
r r n
n
r
x2 x
r r n
+
=
= + +
#nd
( ) ( )
( )
0
1
J 1
r r n
n
r
x2 x
r r n
=
= +
We c#n no5 enu&er#"e so&e o, "$e &os" &!or"#n" !ro!er"es o, "$e 7esse' ,unc"ons.
7e,ore >uo"n #n% resu'"s 'e" us so'-e "$e 7esse's e>u#"on o, ,rs" 6nd #nd order ?ero.
T$#" s "o so'-e "$e e>u#"on
( )
0 1
d y dyx x x y
dx dx
+ + = L L L
9'e#r'% 0x= s # reu'#r snu'#r !on" o, (1+.
Assu&e 5"$ 00a
0 0
c n n c
n n
n n
y x a x a x
+
= =
= = Subs""u"n "$s ex!resson n"o (1+8
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[ ]
0 0 0
1 1
0 1 0 1
1
0 1
( 1+( + ( + 0
( 1+ ( 1+ ( 1+( + ( 1+
( + 0
( 1+ ( + 0
n c n c n c
n n n
n n n
c c n c c c
n
n
n c n c
n n
n n
c c n c
n n
n
n c n c a x n c a x a x
c ca x c c a x n c n c a x ca x c a x
n c a x a x
c a x c a x n c a a x
+ + + +
= = =
+ + +
=
+ +
= =
+ +
=
+ + + + + =
+ + + + + + + +
+ + + =
+ + + + + =
E>u#"n coe,,cen"s8
0 0
1
1 1
0 1
/
1 / 3 ;
0 0
2 4
8 0 0 0 0 ( +
8 ( 1+ 0 ( 1+ 0 0
8 ( + 0 ( +
/ 0( + ( /+
. . . 0
2 4( + ( 2+ ( + (
c
c
nn c
n n n
x c a but a c c twice
x c a but c a
ax n c a a a n
n c
a an a n a
c c
a a a a a
a an a n a
c c c c
+
+
= = =
+ = + =
+ + = = +
= = = = =+ +
= = = = = =
= = = = + + +
0
G
2+ ( 4+
G( + ( 2+ ( 4+ ( G+
c
an a
c c c c
+ +
= =+ + + +
( )0
2 4 G
0 2 4 G
2 4
0
G
( +
. . .
1( + ( + ( 2+ ( + ( 2+ ( 4+
. . .( + ( 2+ ( 4+ ( G+
c n
n
n
c
c
y f x c x a x
x a a x a x a x a x
x x xa x
c c c c c c
x
c c c c
=
= =
= + + + + +
= + +
+ + + + + +
+ + + + +
2 4 G
1 0 ( + ( 0+ 1 . . .
.2 .2 .4 .2 .4 .Gx x x xy x f x a = = + + +
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2 4 G
0 2 4 G
2 4 G
0 2 4 G
0 0 00
1 . . . (.1+ (/..1+ (2./..1+
( 1+1 . . . . . .
(J+ (/J+ (2J+ ( J+
( 1+( + ( +
( J+
n n
n
nn
n
x x x xa
xx x x xa
n
xa a 2 x 4erify
n
=
= + + +
= + + + + +
= =
T$e ex!resson n "$e s>u#re br#c6e" cou'd $#-e been e#s'% deduced ,ro&( +n2 x b%
s&!'% 'e""n 0n= "$us
( ) ( )
( )
0
1
J 1
r r n
n
r
x2 x
r r n
+
=
= + +
5$en 0n= 5e e"
( ) ( )
( )
0
0
1
J 1
r r
r
x2 x
r r
=
=
+
0 0
( 1+ ( 1+
J J ( J+
r nr n
r n
x x
r r n
= =
= =
#s #bo-e.
In on "$rou$ "$e !ro!er"es o, "$e 7esse' ,unc"ons "$e ,unc"ons ( ) ( )#ndn n2 x 2 x
ob"#ned #bo-e 5'' be #ssu&ed.
1.;./.1.. SOE 7ESSEL FUN9TIONS IDENTITIES:PROPERTIEST$e 7esse' ,unc"ons #re re'#"ed b% # -er% '#re #rr#% o, den""es or !ro!er"es.
T$e ,o''o5n #re so&e o, "$e ,und#&en"#' !ro!er"es. We #re on "o !ro-e "$e ,rs" ,e5'e#-n "$e res" "o "$e re#der #s #n exercse.
I.( ) ( )1n nn n
dx 2 x x 2 x
dx =
5$ere n s # !os"-e n"eer.
Proo,8
0
( 1+( +
J( +J
r nrn n
n
r
d d xx 2 x x
dx dx r n r
+
=
= +
0
( 1+
J( +J
r r n
r nr
xd
dx r n r
+
+=
= +
1
0
( 1+ ( +
J( +J
r r n
r nr
r n x
r n r
+ +
=
+= +
1
0
( 1+ ( +
J( +( 1+J
r r n
r nr
r n x
r n r n r
+
+=
+= + +
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1
10
( 1+
J( 1+J
r r n
r nr
x
r n r
+
+ =
= +
1
10
( 1+ .
J( 1+J
r r n n
r nr
x x
r n r
+
+ =
= +
1
0
( 1+
J( 1+J
r nrn
r
xx
r n r
+
=
= +
1
0
( 1+
J( 1 +J
r nrn
r
xx
r n r
+
=
= +
1( +n nx 2 x Q.E.D
II. ( ) ( )1n n
n n
d
x 2 x x 2 xdx
+ = 5$ere n s # !os"-e n"eer
Proo,8
0
( 1+( +
J( +J
r nrn n
n
r
d d xx 2 x x
dx dx r n r
+
=
= +
0
( 1+
J( +J
r r
r nr
xd
dx r n r
+=
= +
1
( 1+1
J J( +J
r r
n r nr
xd
dx n r n r
+=
= + +
1
0
( 1+1
J ( 1+J( 1+J
r r
n r nr
xd
dx n r n r
+ +
+ +=
= + + + +
0
( 1+1
J ( 1+J( 1+J
r r
n r nr
xd
dx n r n r
+
+ +=
= + + +
1
0
( 1+ ( +0
( 1+J( 1+J
r r
r nr
r x
r n r
+
+ +=
+= + + +
1
0
( 1+ ( 1+
( 1+ J( 1+J
r r
n rr
r x
r r n r
+
+ +=
+
= + + + 1
10
( 1+ . .
J( 1 +J
r n n r
n rr
x x x
r n r
+
+ +=
=
+ +
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1
10
1
0
1
( 1+ .
J( 1 +J
( 1+
J( 1 +J
( +
r n rn
n rr
r nrn
r
nn
xx
r n r
xx
r n r
x 2 x
+ +
+ +=
+ +
=
+
=
+ +
= + +
=
T$#" s
1( + ( +n n
n n
dx 2 x x 2 x
dx
+ =
Q.E.D.
III. I, 5e !u" 0n= n "$e #bo-e !ro!er"% 5e ob"#n
[ ]0 1( + ( +
d2 x 2 x
dx=
T$e ,o''o5n s # !roo, o, "$e !ro!er"%8
[ ]
[ ]
[ ]
0 0
1
1
0
0
( 1+( +
( J+
( 1+1
( J+
( 1+
1 ( 1+J
( 1+1
( 1+J
rr
r
rr
r
rr
r
rr
r
d d x2 x
dx dx r
d x
dx r
d x
dx r
d x
dx r
=
=
++
=
+
=
=
= +
= + +
= +
[ ]
1
0
( 1+ ( 1+0 .
( 1+J
r r
rr
r x
r
+
+=
+=
+
[ ]
1
0
( 1+ ( 1+.
( 1+J( 1+J
r r
rr
r x
r r
+
+=
+=
+ +
[ ]
[ ]
1
10
1
1
0
( 1+.( 1+J J
( 1+. ( +
( 1+J J
r r
rr
rr
r
x
r r
x2 x
r r
+
+=
+
=
=
+
= =
+
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T$#" s
[ ]0 1( + ( +d
2 x 2 xdx
=
1.;.2. IS9ELLANEOUS E=APLES
1. Fnd # seres so'u"on ,or "$e e>u#"on
( + 0
d y dyx x y
dxdx+ + =
n !o5ers o, x .HINTS
1
0 1
0
( + ( 1+ ( + ( + 0c n cn nn
c c a x n c n c a n c a x
++
=
+ + + + + + + + =
1
/ 01/. . . .
( 1+n
na n
n n+
= =
+
0
/ 2
( +( 1+( 1+
( + ( /+( +
( + ( 1+ ( +( 1+ ( 1+. . .
( + ( /+ ( 2+ ( + ( /+ ( 2+ ( 3+
c
c c cc x x
c c cy f x c b x
c c c c c c cx x
c c c c c c c
+ + + + + = = +
+ + + + + + + + +
Ter& P#r"#' der-#"-e 5r" c#'ue o, der-#"-e
#" 1c=
1c+( 1+ 1c
c
+ =
1
c
c
+
2
( +
c
c c c
= + +
2
( 1+( +
( + ( /+
c c
c c
+ +
( 1+ ( +
( + ( /+
'n 'n( 1+ 'n ( + 'n ( + 'n ( /+
( 1+( + 1 1 1
1 /( + ( /+
c c%et v
c c
v c c c c
c cv
c c c c cc c
=
+ +
= + + +
= + + ++ +
10
/omplete and write out the general solution5
. Fnd # seres so'u"on o, "$e d,,eren"#' e>u#"on
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(1 + (3 ; + / 0
d y dyx x x y
dxdx + =
n !o5ers o, x .
So'u"on (ex"ens-e+To so'-e "$e e>u#"on
( ) ( ) ( )
1 3 ; / 0 1
d y dyx x x y
dx dx + = L L L
Tes"
( )0 00 s # snu'#r !on"
Px
= =
No5
( ) ( )
( )
( )
( )
( ) ( )
( )
( )
( )
0
0 0
0
0
0
3 ; 3'& '& #nd 1
/'& '& 0
1
x x x
x x x x
x x Q x x x
P x x x
x x R x x
P x x x
= =
= =
7o"$ '&"s exs"C "$ere,ore "$e !on" 0x= s # reu'#r snu'#r !on".
Assu&e
0
0 0
0c n n c
n n
n n
y x a x a x a
+
= =
= =
1
0 0
( + M ( 1+ ( +n c n cn nn n
y n c a x y n c n c a x
+ +
= =
= + = + +
Subs""u"n "$ese der-#"-es n"o (1+ 5e e"
[ ] [ ]
1 1
0 0 0
0 0
1
0 0
( 1+( + ( 1+( + 3( +
;( + / 0
( 1+( + 3( + ( 1+( + ;( + / 0
n c n c n c
n n n
n n n
n c n c
n n
n n
n c n c
n nn n
n c n c a x n c n c a x n c a x
n c a x a x
n c n c n c a x n c n c n c a x
+ + +
= = =
+ +
= =
+ +
= =
+ + + + + +
+ =
+ + + + + + + + + =
(T$e #bo-e s"e! s #cco&!'s$ed b% co''ec"n "oe"$er '6e "er&s8 "er&s 5"$ s#&e
enerc "er&s #nd coun"n ,ro& "$e s#&e ndces+
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1
0 0
1
0 0
( + /( + ( + 3( + / 0
( + ( /+ ( /+( 1+ 0
n c n c
n n
n n
n c n c
n n
n n
n c n c a x n c n c a x
n c n c a x n c n c a x
+ +
= =
+ +
= =
+ + + + + + + =
+ + + + + + + =
(7% s$,"n "$e ndces 8+
1
0 1
0 0
(. /+ ( 1+(. . 3+ (. . /+( 1+ 0c n c n cn n
n n
c c a x n c n c a x n c n c a x
+ +
+
= =
+ + + + + + + + + + =
E>u#"n coe,,cen"1
0 08 ( /+ 0 0cx c c a but a
+ = s8
/0
c or c = =
1
8 ( 1+ ( 3+ ( /+ ( 1+ 0n c
n n
x n c n c a n c n c a+
++ + + + + + + + =
1
/ 01/. . . .
3n n
n ca a n
n c+
+ + = =
+ +Here 5e #re de#'n 5"$ "$e c#se n 5$c$ "$e roo"s o, "$e ndc#' e>u#"on do
no" d,,er b% #n n"eer. T$ere,ore e#c$ -#'ue o, c 5'' -e rse n"o # so'u"on.T#6n "$e -#'ues o, "$e roo"s n "urn8
( )
1
1 0 / 2 0
/0 01/. . . . .
3
/ / / /0 3 ; 11
n n
nc a a n
n
n a a $imilarly a a a a
+
+= = =
+
= = =
/ 2
0 1 / 2( . . . .+c
y x a a x a x a x a x = + + + + +
0 / 2
1 0 0 0 0 0
/ / / /( . . . .+
3 ; 11y x a a x a x a x a x = + + + + +
/ 2
0
/ / / /(1 . . . .+
3 ; 11a x x x x= + + + + +
1 1 /
/ 2
0 1 / 2
/
/ 2 0
/
0
/ 01/ . . . . . . . . . . . 0
1
( . . . .+
( 0. 0. 0. 0. . . . .+
n n
c
nc a n a a a a
n
y x a a x a x a x a x
y x a x x x x
a x
+
= = = = = = = = =+
= + + + + +
= + + + + +
=I" s 5or"$5$'e no"n ##n$ere "$#" one@"er& so'u"ons exs"J
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T$e ener#' so'u"on s
1 1
/
/ 2
0 1 0
/ / / /(1 . . . +
3 ; 11
y k y k y
& x x x x 'x
where & a k and ' a k
= +
= + + + + + +
= =
/. Ex!ress2 / ( + 2 / ; 3 1f x x x x x= + +
n "er&s o, "$e Leendre !o'%no's
So'u"onFro&
2
2
/
/
1 0
1( + (/3 /0 /+
G1
( + (3 / +
1( + (/ 1+
( + ( + 1
P x x x
P x x x
P x x
P x x and P x
= + =
= = =
5e $#-e
2
2
/
/
G 4 /( +
/3 ; /3
/ 1( + ( +
3 3 / /
x P x x
x P x x and x P x
= +
= + = +
Subs""u"n n"o( )f x
5e e"8
2 /
1 0
G 4 / /( + 2 ( + / ( +
/3 ; /3 3 3
1; ( + 3 ( + 1 ( +
/ /
f x P x x P x x
P x P x P x
= + +
+ + +
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2 / 1 0
0 1 0 1 0
2 / 1 0
2
/ 4 12 2 1( + ( + ( + 3 ( + 1 ( + ( +
/3 3 / ; / /
1 ;( + ( + ( + 3 ( + 1 ( +
/3 3 /
/ 4 12 14 G 1 ;( + ( + ( + ( + 3 3 ( + 1
/3 3 / ; 3 ; /3 /
/ 4( +
/3 3
P x P x P x P x P x P x
P x P x P x P x P x
P x P x P x P x P x
P x
= + + + +
+ +
= + + + + + + +
= / 1124 21 13//;
( + ( + ( +1 3 ;/3
P x P x P x+ +
Re-er"n # b" "o "$e 7esse' Func"ons 'e" us #dd one &ore !ro!er"%8
I( ) ( ) ( )1 1
n n n
n2 x 2 x 2 x
x+ + =
2. Ex!ress( ) ( ) ( )2 0 1n "er&s o, #nd2 ax 2 x 2 x
So'u"onFro&
1 1
( + ( + ( +n n n
n2 x 2 x 2 x
x+ + =
5e $#-e
2 / 1 /
./( + ( + ( + ( +2 z 2 z 2 z 2 z
z+= =
1
4 .( + ( + ( +2 z 2 z 2 z
z z
=
1 1 0 1
1 0 1 1 /
1 0/ /
2 4 2 .1 41 ( + ( + 1 ( + ( + ( +
2G2 2 4 4 1 . ( + 1 ( + ( + ( +
2G 2GG 2( + 1 ( +
( +
2 z 2 z 2 z 2 z 2 zz z zz z
2 z 2 z 2 z 2 z z z z zz z z z
2 z 2 zzz z ax
= =
= =
= = 1
G 2( +
( +2 ax
ax ax
. ( ) ( ) ( )1 n
n n2 x 2 x n +
= (Proo, 'e," #s #n exercse+
I (#+.
( )1
sn2 x x
x=
I (b+.
( )1
cos2 x x
x=
(Proo, 'e," #s #n exercse+
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5/22/2018 Seriees
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II( ) ( ) ( )L1 1 n n n2 x 2 x 2 x + = (Proo, 'e," #s #n exercse+
T$ere #re n% &ore suc$ ,or&u'#e so &uc$ so "$#" " be$o-es #n% one o, us "o 'oo6 ,or&ore o, "$ese re'#"ons. #6e sure %ou ,#&'#rse %ourse', 5"$ "$er !roo,s .T$e !roo,s
#re s"#nd#rd bu" re>ure so&e c#re #nd nenu"% "o be #b'e "o ,urns$ # !roo,.
To de&ons"r#"e "$s !on" 'e" us ,urns$ "$e !roo, o, "$e !ro!er"% 5e $#-e used n
exercse (2+ viz8
1 1
( + ( + ( +n n n
n2 x 2 x 2 x
x+ + =
Proo,8 ,ro&
1( + ( +n n
n n
dx 2 x x 2 x
dx
=
5e $#-e
1
1
1
( + ( + ( +
( + ( + ( + . . . ( + ( +
n n n
n n n
n
n n n
x 2 x nx 2 x x 2 x
n2 x 2 x 2 x a by dividing throughout by x
x
+ =
+ =
A'so ,ro&
1( + ( +
n n
n n
dx 2 x x 2 x
dx
+
=
5e $#-e 1
1
1
( + ( + ( +
( + ( + ( + . . . ( + ( +
n n n
n n n
n
n n n
x 2 x nx 2 x x 2 x
n2 x 2 x 2 x b by dividing throughout by x
x
+
+
=
+ =
Addn( ) ( )#nd 5e e"a b
1 1
( + ( + ( +n n n
n2 x 2 x 2 x
x +
= +
Or
1 1
( + ( + ( +n n n
n2 x 2 x 2 x
x +
+ =
III.
( + ( 1+ ( +nn n2 x 2 x =
Proo,8
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0
( 1+8 ( +
J ( 1+
1 0 ( 1+
( + . ( +
( 1+ (1+ . 1 1 .
r nr
n
r
n n
xR./&%% 2 x
r r n
For r n r n for that range of values
thecoefficients of 2 x &%% vanish (he first nonzero terms of 2 x begin
when r n i e when r n when r n $o w
=
= + + + =
+ = + = =
n"
e hane to
split the summation o two parts as will be shown bellow
( +n2 x =
0
( 1+
J ( 1+
r nr
r
x
r r n
=
+
1
0
( 1+
J ( 1+
r nrn
r
x
r r n
=
+
( 1+
J ( 1+
r nr
r n
x
r r n
=
+
0
( 1+
J ( 1+
r nr
r n
x
r r n
=
+
(b% s$,"n ndces+
0
( B+
( +J ( 1+
r nr n
r
x
r n r
++
=
= + +
0
( B+( 1+
( +J ( 1+
r nrn
r
x
r n r
+
=
= + +
0
( B+( 1+ ( 1+ ( + . .
( +J J
r nrn
n
r
x2 x Q . 0
r n r
+
=
= =
+
2. S$o5 "$#"
1
( + sn2 x x
x
=
So'u"on 8
Fro&
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0
1
1
0
1
0
( 1+( +
J ( 1+
( 1+( +
1 J ( 1+
( 1+
1 J ( 1+
r nr
n
r
rr
r
rr
r
x2 x
r n r
x2 x
r r
x x
r r
+
=
+
=
=
= + +
=
+ +
= + +
1 2
1 2
1 1 1. . . . .
/ 3 ; 1J J
1 1 1.
1 1 / 1 1 3 / 1 1 .
x x x
x x x
= +
= +
1 2
2
/ 3
1 . . . .
1 1 /J 3J
1
. . . . ( 8 . 1 /J 3J
. . . . sn . .
/J 3J
x x x
x x x x x
/heck the trickx x
x xx x Q . 0
x x
= +
= + =
= + =
Exercse
S$o5 "$#"8
/ 1 1
1( + ( + ( +2 x 2 x 2 x
x
=
/ /
sn( + cos ( +
x2 x x find 2 x
x x =
3 3
/sn (/ +cos( + ( +
x x x2 x find 2 x
x x x
= +
L
1 1( + ( + ( +n n n2 x 2 x 2 x + =
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( + ( 1+ ( + ( +nn nP x P x usual notation =
[ ]1
1
( +
3P x
=
1.;.3. EQUATIONS REDU9I7LE TO 7ESSELS EQUATIONS
W$'e so'-n #n Enneern !rob'e& one so&e"&es encoun"ers #n ODE "$#" b#,,'es
$&:$er bec#use o, "s s"r#neness. On scru"nsn 5e'' "$e !rob'e& % be ,ound "o be
reducb'e "o # 7esse' e>u#"on. I, suc$ # c#se s dscerned on'% # '""'e scr#"c$ 5or6 sre>ured be,ore one c#n 5r"e do5n # !er"nen" so'u"on. T$s 5'' s#-e %ou # 'o" o, "&eJ
A ,e5 suc$ c#ses #re -en be''o5 #s ''us"r#"ons o, 5$#" #c"u#''% s re>ured o, %ou n
suc$ # s"u#"on.
1.;.3.1. D,,eren"#' E>u#"on Reducb'e "o 7esse's E>u#"on8 9#se ISo'-e "$e e>u#"on
( ) ( )
0
d y dyx x x n y i
dx dx+ + = L L L
So'u"on
9o&!#re( )i
5"$ "$e 7esse' e>u#"on( )1
on !#e //8
In "$e !resen" e>u#"on( )i
5e $#-e n "$e br#c6e"s "$e "er& x 5$ere#s n
( )1"$e
corres!ondn "er& sx ns"e#d.
So 'e" us !u"z x=
. . .
dz
dx
dy dy dz dy dy dyi e
dx dz dx dz dx dz
=
= = =
.
d y d dy d dy d dy dz d y
dx dx dx dz dz dz dxdx dz
= = = =
d y d ydx dz
=Su&rsn 5e $#-e
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.
Put z x
dy dyand
dx dz
d y d ydx dz
= =
=
Subs""u"n n"o( )i
-es (u!on s&!',c#"on+8
( + 0
d y dyz z z n y
dzdz+ + =
T$s s "$e 7esse' e>u#"on o, ,rs" 6nd #nd o, ordern . I"s ener#' so'u"on s
( + ( + ( + ( +n n n ny &2 z '2 z &2 x '2 x = + = +
1.;.3.. D,,eren"#' E>u#"on Reducb'e "o 7esse's E>u#"on8 9#se IISo'-e "$e d,,eren"#' e>u#"on
1 1G 0 . . . . . . ( +d y dy y iix dxdx x
+ + =
So'u"on
u#"on8
1 1G 0
d y dyy
x dxdx x
+ + =
u'"!'%n "$rou$ou" b%x 8
(G 1+ 0
d y dyx x x y
dxdx + + =
9o&!#rn "$s e>u#"on 5"$ "$#" o, reducb'e c#se I #bo-e n#&e'%8
( + 0 . . . . . . ( +
d y dyx x x n y i
dxdx+ + =
5e no"e "$#"
1
G
n
x x
=
"$s &e#ns "$#" "$e ener#' so'u"on s
1 1( + ( +y &2 x '2 x= +
IPORTANT NOTES
1. T$e re'#"on
1 1
( + ( + ( +n n n
n2 x 2 x 2 x
x+ =
ser-es "o ex!ress 7esse' ,unc"ons o, $$er orders n "er&s o, 7esse' ,unc"ons
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o, 'o5er orders # use,u' n!u'#"on. 7u" , 5r""en n "$e ,or&
1 1
( + ( + ( +n n n
n2 x 2 x 2 x
x +=
ser-es "o ex!ress 7esse' ,unc"ons o, '#re ne#"-e orders ,or ex#&!'e n "er&s
o, 7esse' ,unc"ons 5$ose orders #re nu&erc#''% s''er.. A use,u' ,or&u'# n "$e e-#'u#"ons o,
( +n2 x
s
1
( 1+J11
. Jnn
nn
++ + + =
PRO7LES
Prob'e& OneW$#" s &e#n" b% # reu'#r snu'#r !on" o, # d,,eren"#' e>u#"on
Fnd "$e so'u"on o, "$e d,,eren"#' e>u#"on
(/ + 0
d y dyx x y
dxdx + + =
n "$e ,or& o, # seres. u#"on
(1 + 0
d y dyx x y
dxdx+ + + =
$#s # reu'#r snu'#r !on" #" 0x= . Hence s$o5 "$#" // 1 . . . .
/y x x x= + +
s # so'u"on.
Prob'e& T$ree9'#ss,% "$e snu'#r !on"s n "$e ,n"e !'#ne ,or "$e e>u#"on
2 /
( 1+( 1+ 2 ( 1+ ( 1+ 0
d y dyx x x x x x y
dxdx+ + + + =
.
S$o5 "$#" "$e so'u"on o, "$e d,,eren"#' e>u#"on
(1 + 0
dyd yx x y
dxdx+ =
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c#n be ex!ressed n "$e ,or&
0
1 1 1'n 1 . . .
J /
nx x
n
xy &e ' e x
n n
=
= + + + +
5$ere & #nd ' #re #rb"r#r% cons"#n"s.
Prob'e& FourS$o5 "$#" "$e ndc#' e>u#"on o, "$e d,,eren"#' e>u#"on
( + / 0
d y dyx x y
dxdx + =
s( )1 0c c + =
#nd s$o5 ,ur"$er "$#"
1( +( 1+ ( 2+n nc n c n a c n a + + + = + .S$o5 "$#" one so'u"on s
// / 11
( 1+ ( 1+( +
x x x
+
+ + + 5$en s no" #n n"eer.
#nd ob"#n "$e ,rs" "$ree "er&s o, "$e o"$er so'u"on 5$en =
Pro&le, *i/e
A "ubu'#r #s !re@$e#"er 5or6s b% dr#5n coo' #r "$rou$ # c%'ndrc#' $e#"ed "ube.For # !#r"cu'#r "ube "$e o-ernn e>u#"on ,or "$e "e&!er#"ure o, "$e #r s
;300 /300 0
d ( d( (
dxdx x =
5$ere(
s "$e d,,erence be"5een "$e "e&!er#"ure o, "$e 5#'' o, "$e "ube #nd "$e #r #" #ds"#nce x ,ro& "$e n'e". Fnd "$e seres so'u"on ,or (
#s ,#r #s "$e "er& n
3
x . (I" % $e'! !u""nx z= +.
Prob'e& Sx
Pro-e "$#"
/ 1
( + ( + ( +2 x dx c 2 x 2 x
x=
{ } 0 0 11
( + ( + ( +
x2 x dx x 2 x 2 x= +
.0. 9OPLE= ARIA7LES
.1 S$or" Re-e58 9o&!'ex Nu&ber T$eor%
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7e,ore de'-n &uc$ n"o "$e "$eor% o, 9o&!'ex #r#b'es 'e" us o "$rou$ bre,'% so&e
&!or"#n" !on"s n "$e A'ebr# o, co&!'ex nu&bers. We #re no" n"endn "o subs""u"e
,or 5$#" 5#s ex"ens-e'% co-ered n e#r'er &odu'es on "$e subec"
De,n"on
A nu&ber o, "$e ,or& z x i y= + 5$ere x #nd y #re re#' nu&bers #nd i s # nu&ber
suc$ "$#" 1i = s c#''ed # co&!'ex nu&ber
T$e nu&ber i s c#''ed "$e imaginary unit"T$e re#' nu&ber x n z x i y= + s c#''ed
"$e real parto, z#nd s deno"ed b% Re( +z 5$ere#s y s c#''ed "$e i,aginary parto, z
#nd s deno"ed b% I&( +z
De,n"on
T5o co&!'ex nu&bers 1 1 1z x i y= +
#nd z x i y= +
#re s#d "o be e>u#' "$#" s
1 1 1 Re( + Re( + I&( + I&( +z z iff z z and z z= = =
Pro!er"esT$e s"uden" s c#''ed u!on "o ,#&'#r?e onese', 5"$ "$e ,o''o5n !ro!er"es o, "$e
#r"$&e"c o!er#"ons o, co&!'ex nu&bers
Le" 1 1 1z x i y= +
#nd z x i y= +
( ) ( ) ( )1 1 1 i z z x x i y y = +
( ) ( ) ( )1 1 1 1 1ii z z x x y y i x y x y= + +
( ) 1 1
z z ziii
z z z
= 5$ere s "$e co&!'ex conu#"e o,z x iy z= .
( )
( )
1 1 1 1
1 1
#ndiv z z z z z z z z
z zv
z z
+ = + =
=
er,% "$e #bo-e.
A'so
Re( +
z zz
+=
I&( +
z zz
i
=
T$e &odu'us or #bso'u"e -#'ue o, z x i y= + deno"ed b% z s "$e re#' nu&ber -en b% .z x y z z= + =
1 1 z z z z+ + #nd 1 / 1 /. . . . . .n nz z z z z z z z+ + + + + + + +
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T$#" s 1 1
n n
k k
k k
z z= =
T$s s 6no5n #s "$e Sc$5#r"? ne>u#'"%.
T$e Po'#r For& o, # 9o&!'ex Nu&ber
T$e co&!'ex nu&ber z x i y= + n 9#r"es#n coordn#"es c#n be ex!ressed #s(cos sn +z r i = + n !o'#r coordn#"es 5$ere
1&od #r "#n #r
yr z the ulus of z and z the ument of z
x
= = = =
I, < "$en s c#''ed "$e !rnc!#' #ru&en" o, z#nd s deno"ed b% ( +&rg z
.
Note
1 1
#r ( . + #r #rz z z z= +
1
1
#r #r #rz
z zz
=
De o-res T$eore&
W$en1z =
#nd cos snz i = + "$en (cos sn + cos snn nz i n i n = + = +
,ro& 5$c$ one % der-e "$e ,#c"1 1
(cos sn + cos sn 01. . . 1n n
k kz r i z r i k n
n n
+ + = + = + =
5$c$ -es "$e n nth roo"s o, .z
#!S$E%%!& T'E SECO# ()T'
We #'' 6no5 "$#" be,ore "$e #d-en" o, co&!'ex nu&ber "$eor% " 5#s un"$n6#b'e "$#" #n
e>u#"on o, "$e ,or& 1 0x + = cou'd e-er be so'-edJ I" 5#s "$e n"roduc"on o, "$e
nu&ber i 5$c$ s suc$ "$#" 1i = 5$c$ b'e5 u! "$e &%"$8 *negati+e numbers ha+e
no suare roots-No5 5e $#-e "o ds!e' %e" #no"$er &%"$8 *logarithms of negati+e numbers are notdefined-
Le" z#nd w be suc$ "$#"
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( ) ( ) ( ) ( ) ( )
1 'o "$e 'o#r"$& o, # co&!'ex nu&ber $#s #n n,n"e nu&ber o, -#'ues.
T$#" s " s # &u'"@-#'ued ,unc"on.