!"#$%&' (

!"#$%&' %()*"!+,-")$*

*./!,)0.&

!"# $% $&!'()*%+ #%*,") )*!()-%# .%# /-)&,0%# 0% 1(/'#,"+ 2%3),"45,)%# %)

6,/7%*8 7!#-%# #9* 0%# !''*,:(/!)(,"# ',.;",/(!.%#< =%# /-)&,0%# (")%*',.!4

)(>%# 0% );'% ?!9## "% #%*,") '!# !7,*0-%# 0% /@/% A9% .%# /-)&,0%# 7!#-%# #9*

0%# )(*!8%# !.-!),(*%# 09 );'% B,")%45!*., 0%# !9)*%# /-)&,0%# 0C(")-8*!)(,"<

!" #$%$&'()*$+

=% /-)&,0%# 0C(")-8*!)(," "9/-*(A9% D ,9 !"#$"%!$& "9/-*(A9% D ,") ',9*

79) 0C->!.9%* .! >!.%9* "9/-*(A9% I 0% .C(")-8*!.% 0C9"% E,"$)(," f $,")("9% #9*9" 0,/!("% [a, b] (a < b)<

I =

∫ b

a

f(x)dx = I + ε FG<HI

ε *%'*-#%")% .% )%*/% 0C%**%9* !##,$(- J I+ %**%9* A9% .C," #,9&!()%*! /("(/(#%*<=C!''*,:(/!)(," I %#) $!.$9.-% J '!*)(* 0% >!.%9*# '*(#%# '!* f %" 9" %"#%/7.%0% n ',(")# {xi} ∈ [a, b]< =%# /-)&,0%# 0-*(>-%# 0% .! /-)&,0% 0%# )*!'KL%#F1(/'#,"+ 2%3),"45,)%# %) 6,/7%*8I *%A9(K*%") A9% .%# xi #,(%") !'()*$+,-

+&'% $,."$%(/ 0!"# .C(")%*>!..% 0C(")-8*!)(,"< 1( $% "C%#) '!# .% $!#+ 9"% /-)&,0%

(")%*',.!)(>% 0,() @)*% 9)(.(#-%<

!"#$%&! '"()$*#+*! =! '*-$(#(," 0C9"% /-)&,0% "9/-*(A9% $*,M) 8-"-4

*!.%/%") .,*#A9% .% ",/7*% n 0% ',(") !98/%")%< 2-!"/,("#+ !98/%")%* n (/4'.(A9% 9" ",/7*% !$$*9 0C->!.9!)(,"# 0% f !("#( A9% 09 ",/7*% 0C,'-*!)(,"#+$% A9( ! ',9* $,"#-A9%"$% 0% E!(*% $*,M)*% .C%**%9* 0C!**,"0(< N" $,/'*,/(# %#)

0,"$ "-$%##!(*%<

!"!" #$%&'()**& +,-$%./')%-0$ $0$ 10'$.

=% ",/7*% 0%# ',(")# 0C->!.9!)(," -)!") O"(+ I %#) 0,"$ "-$%##!(*%/%") .C!'4'*,:(/!)(," 0C9"% (")-8*!)(," #9* 9" 0,/!("% O"(< !"# .% $!# ,P .% 0,/!("%

0C(")-8*!)(," %#) "," 7,*"-+ .! #)*!)-8(% J !0,')%* %#) 0% #-'!*%* .% 0,/!("%

Q

!"#$%&' () *+%!,-'. -/$0%+1&"%$,0 02*+&$32'

!"#$%&'()%#*$ +$ ,-*#.#..)$% /) 0*($+ #$%+(1&!#)#(+ )2..# '()$!+ 32+ 4*..#0/+ !+

1)$#5(+ 6 4*27*#( 2%#/#.+( 2$+ +84(+..#*$ ).914%*%#32+ limx→∞ f(x) = g(x):;)( +8+14/+ <

I =∫ +∞

af(x)dx =

∫ b

af(x)dx +

∫ +∞b

f(x)dx

≈∫ b

af(x)dx +

∫ +∞b

g(x)dx.

=) 4(+1#5(+ #$%&'()/+ +.% ,)/,2/&+ $21&(#32+1+$% +% /) .+,*$!+ 4+2% >%(+ ,)/,2/&+

)$)/9%#32+1+$%: ?$ 4()%#32+@ *$ .")..2(+ 32+ /) 7)/+2( #$%+(1&!#)#(+ b ,-*#.#+

+.% .2A.)11+$% '()$!+ +$ 7&(#B)$% 32+ /+ (&.2/%)% !2 ,)/,2/ +.% .2A.)11+$%

4(*,-+ !+ ,+/2# *0%+$2 +$ ,-*#.#..)$% 2b ,*11+ 7)/+2( #$%+(1&!#)#(+:

!"! #$%&'()' *+,(' &-(.,/0$-1%

C$ )2%(+ 4(*0/51+ .+ 4*.+ /*(.32+ /+ !*1)#$+ !"#$%&'()%#*$ ,*$%#+$% 2$+

.#$'2/)(#%& !+ /) D*$,%#*$ 6 #$%&'(+(: ;/2.#+2(. .%()%&'#+. 4+27+$% >%(+ )!*4%&+. <

E: F&)/#.+( 2$+ %()$.D*(1)%#*$ )/'&0(#32+ 32# D)..+ !#.4)()G%(+ /) .#$'2/)H

(#%& !2 !*1)#$+ !"#$%&'()%#*$ I+:': #$%&'()%#*$ 4)( 4)(%#+@ ,-)$'+1+$% !+

7)(#)0/+@ +%,:J ;)( +8+14/+ <

I =∫ 1

0lnxdx

=∫ 0

−∞ ueudu (changement de variable)

= [ueu]0−∞ −

∫ 0

−∞ eudu (integration par parties)

= −∫ b

−∞ 0du +∫ 0

beudu (eu ∼ 0 et b < 0 choisi tel qu′en 2.1.1)

= −1

: K$7+(.+( /) D*$,%#*$ I.# +//+ +.% 2$#!#1+$.#*$$+//+J:

∫ b

a

f(x)dx = bf(b) − af(a) −∫ f(b)

f(a)

f−1(y)dy

;)( +8+14/+ <

∫ 3

0

x−1/2dx =√

3 − 0 −∫ 1/

√3

+∞y−2dy

*2 +$,*(+

∫ 1

0

lnxdx =

∫ −∞

0

eudu

L: M&4)(+( /"#$%+(7)//+ !"#$%&'()%#*$: ="#$%&(>% !+ ,+%%+ *4&()%#*$ +.% !"2%#/#.+(

2$ 4). !"#$%&'()%#*$ 4/2. 4+%#% I2$ &,-)$%#//*$$)'+ 4/2. .+((&J )2%*2( !+

/) .#$'2/)(#%& .)$. )/*2(!#( /) %*%)/#%& !2 ,)/,2/: ;)( +8+14/+ <

∫ 1

0

x−1/2dx =

∫ 1

ǫ

x−1/2dx +

∫ ǫ

0

x−1/2dx

! "#$%&'() *+,))-./() 012 34(*$,56+()7 $4,89:()7

2-;8)&5<

! !" 2%134*'& 567

N$ ,-+(,-+ 6 ,)/,2/+( /"#$%&'()/+ I =∫ b

af(x)dx: ?//+ 4+2% >%(+ )44(*,-&+

4)( /"2$+ !+. %(*#. 1)$#5(+. .2#7)$%+.:

! "#$%& ' ()"*+ ,-.&/012130 41 56789:;1 <= 56789:;1>"5? ' @A==>@A=@

!"#$%&' () *+%!,-'. -/$0%+1&"%$,0 02*+&$32'

!"#$%& '&( )&*+,-.%&(

r0 = (b − a)f(a) ou r0 = (b − a)f(b) ! " #

$%&'(%)* '+% ,-./'.*(0+ 1% /. 20+3*(0+ f "

!"#$%& '&( /",012&(

T0 =b − a

2[f(a) + f(b)] ! "4#

$%&'(%)* 1%'5 ,-./'.*(0+6 1% /. 20+3*(0+ f "

!"#$%& '& 34#0(!-

S1 =1

3

b − a

2

[

f(a) + f(b) + 4f

(

a + b

2

)]

! "7#

$%&'(%)* *)0(6 ,-./'.*(0+6 1% /. 20+3*(0+ f " "445647868 /. 9,*:01% 1% ;(9<60+

+% 6=.<</(&'% &'% <0') n <.()"

>/ %6* 60'-%+* (+*,)%66.+* 1% 6'?1(-(6%) /=(+*%)-.//% [a, b] %+ n 60'6 (+*%)-.//%6

&'% +0'6 30+6(1@)%)0+6 (3( 1% 9A9% /0+B'%') h = (b − a)/n" >/ %6* C +0*%) &'%

3%**% 0<,).*(0+ +,3%66(*% 1% <0'-0() ,-./'%) /. 20+3*(0+ f %+ 3:.3'+ 1%6 n + 1<0(+*6 xi &'( 1,/(9(*%+* 3%6 (+*%)-.//%6 D

xi = a + hi, h =b − a

n, fi ≡ f(xi), i = 0, 1 . . . n

E+ <.)*(3'/(%)8 0+ )%9.)&'%). &'% f0 = f(a) %* fn = f(b)"F.+6 *0'*% /. 6'(*%8 +0'6 .//0+6 <06%)

n = 2q, q ∈ N.

G% 3:0(5 +=%6* /% 2)'(* 1=.'3'+% +,3%66(*, 9.*:,9.*(&'%8 9.(6 30+6*(*'% '+ '6.B%

30').+* %+ ).(60+ +0*.99%+* 1%6 6(9</(H3.*(0+6 1.+6 /=,3)(*')% 1% 3%)*.(+%6

%5<)%66(0+6"

E+ .<</(&'.+* /%6 20)9'/%6 ! " #8 ! "4# %* ! "7# 6') /%6 n = 2q(+*%)-.//%68 0+

1,1'(* '+% +0'-%//% 20)9'/.*(0+ 1% 3%6 9,*:01%6 %+ 20+3*(0+ 1% n !0' q# D

rq =b − a

2q

2q−1∑

i=0

fi ou rq = h2q−1∑

i=0

fi+1, q > 0 ! "I#

Tq =1

2

b − a

2q

(

f0 + fn + 22q−1∑

i=1

fi

)

, q > 0 ! "J#

Sq =1

3

b − a

2q

f0 + fn + 2

2q−1−1∑

i=1

f2i + 4

2q−1

∑

i=1

f2i−1

, q > 1 ! "K#

⋆ 9 :;<864=5= >8?= @/5A5=B5=)

L+ +0*% &'% /%6 *)0(6 20)9'/%6 ! "I#8 ! "J# %* ! "K# 6% 3./3'/%+* *0'*%6 %+ n = 2q

,&'.*(0+6"

! "#$%& ' ()"*+ ,-.&/012130 41 56789:;1 <= 56789:;1>"5? ' @A==>@A=@

!"#$%&' () *+%!,-'. -/$0%+1&"%$,0 02*+&$32'

� � �

��

����

� � �

� �

����

� � �

��

����

! ! "#$%&'&() *+' ,$-.(*+'

!" #$% #&%#&'( )(# "*+,,-./01+%0.& Iq '*(&# 0&%23-+"# I #$% '*+(%+&% 1#0""#(-#

)(# "*0&%#-4+""# '*0&%23-+%0.& + 2%2 $(5'040$2 (& 3-+&' &.15-# '# 6.0$ 708#8 "+

,-290$0.& 9-.:% +4#9 q;8 <2+&1.0&$= > ?&#$$# 23+"#= "#$ %-.0$ 12%@.'#$ &*.A-#&%

,+$ "+ 1B1# ,-290$0.&8 C+ ,-290$0.& '# "+ 12%@.'# #$% #$%012# ,+- !"#$#% '#

$.& #--#(- '2?&0# > "*.-'-# q ,+-

εq = I − Iq

C# 9+"9(" '# εq ,#(% $# 9+"9("#- ,+- 1+D.-+%0.& #& (%0"0$+&% "# %@2.-E1# '#$

+99-.0$$#1#&%$ ?&0$ )(0 #$% -+,,#"2 90F'#$$.($8

!"#$%&' (' )#**'

.456 789 :48;6548 f : [a, b] → R ;4865879 <7= 78 5869=>?@@9 :9=AB [a, b] 96 CB=5>?D@9<7= ]a, b[ 69@@9 E79 f(a) = f(b)) "@4=< 5@ 9F5<69 78 =B9@ c ∈]a, b[ 69@ E79 f ′(c) = 0)

+"&#,-.$/.0#, 1 G(0$)(# f #$% 9.&%0&(#= 0" #/0$%# m,M ∈ R %#"$ )(# f([a, b]) =[m, M ]8 H0 m = M = f #$% 9.&$%+&%# #% f ′

#$% 0'#&%0)(#1#&% &(""# $(- ]a, b[8 H0&.&=m < M #% +".-$ m < f(a) 7f +'1#% (& 10&01(1 $(- [a, b]; #%I.( f(a) < M 7f+'1#% (& 1+/01(1;8 C# -+0$.&&#1#&% #$% "# 1B1# '+&$ %.($ "#$ 9+$8 H(,,.F

$.&$ '.&9 $+&$ ,#-%# '# 32&2-+"0%2 )(# m < f(a) J +".-$ 0" #/0$%# c ∈]a, b[ %#" )(#f(c) = m8

∀x ∈]a, c[, f(x) > f(c) =⇒ f(x)−f(c)x−c 6 0

∀x ∈]c, b[, f(x) > f(c) =⇒ f(x)−f(c)x−c > 0

K.11# f #$% '2-04+5"# #& c= "# ,+$$+3# > "+ "010%# '+&$ "#$ '#(/ 0&23+"0%2$

90F'#$$($ '.&&# -#$,#9%04#1#&%

f ′(c) = limx→c−f(x)−f(c)

x−c 6 0

f ′(c) = limx→c+f(x)−f(c)

x−c > 0.

G+- 9.&%0&(0%2 '# f #& c= 0" 40#&% f ′(c) = 08 ¤

! "#$%&'#$()(*+ ,-#+.#(/ 0 1234 5 67

!" #$%&' ( )*#+, -./'0123241 52 6789:;<2 = 6789:;<2>#6? ( @A >@A @

!"#$%&' () *+%!,-'. -/$0%+1&"%$,0 02*+&$32'

!"!##$%"& ' ()*!"+,& -&. $//"!%..&,&0(. 10%.

.456 789 :48;6548 f : [a, b] → R ;4865879 <7= 78 5869=>?@@9 :9=AB [a, b] 96 CB=5>?D@9<7= ]a, b[) $@ 9E5<69 ?@4=< 78 =B9@ c ∈]a, b[ 69@ F79

f ′(c) =f(b) − f(a)

b − a! "#$

2*,!0.("$(%!0 -3 /!"!##$%"& %& '()* +, -./'0+12,2 &3 4./-*0./ g(x) =

f(x)− f(a) + (x− a) f(b)−f(a)b−a " 5&&, 61207, &,' 89:.*8;',' +( *81.2;<, +, =.&&,

>(0 '?3::&0>(, +./- ,* +.//, 0<<1+03*,<,/* &?1/./-1 +( *81.2;<, +,' 3--2.0'@

',<,/*' 7/0'" ¤

A.(2 &3 '(0*,B ./ (*0&0',23 &, *81.2;<, +,' 3--2.0'',<,/*' 7/0' :.(2 C('*07,2

>(, '0 f ,'* -./*0/(, '(2 [a, b] ,* +12063D&, '(2 ]a, b[ E

f(b) − f(a)

b − a6 sup

]a,b[

f ′(x)

!!"#! $" %& '()*+$" $", !"-)&./%",

! "!#$%&'"# () *')!#"#+

ε(r)q (x) =

∫ x

a

f(u)du − x − a

2qf(a), ∀x ∈]a, b].

, (-%$&$. q = 0 /h = b − a01 () &+$"2+. &. 3.##. 4%!3#"%! 5.'# 6#$. 7)8%$+. .!'#"("9)!# (. #:+%$;7. &.9 )33$%"99.7.!#9 <!"9 =

|dε(r)0 (x)/dx| = |f(x) − f(a)| 6 (x − a)M1, M1 = sup[a,b] f

′(x)

⇒ |ε(r)0 (x)| 6 1

2M1(x − a)2 6 12M1(b − a)2.

>%'$ q > 11 '! 3)(3'( &. rq 3%!9"9#. .! 2q3)(3'(9 &. r0 9'$ &.9 "!#.$2)((.9 &.

(%!?'.'$ h = (b − a)/2q1 &. 9%$#. *'. (-.$$.'$ .9# 3%77"9. 2q

4%"9@

|ε(r)q (x)| 6 2q × 1

2M1

(

b − a

2q

)2

= M1(b − a)2

2q+1.

!!"#! $" %& '()*+$" $", )!&012",

A. () 767. 7)!";$.1 %! 3%!9"&;$. () 4%!3#"%!

ε(T )0 (x) =

∫ x

a

f(u)du − x − a

2(f(a) + f(x))

, (-%$&$. B1 () &+$"2+. 9.3%!&. &. 3.##. 4%!3#"%! 5.'# 6#$. 7)8%$+. =

|d2ε(T )0 /dx2| = | − x−a

2 f ′′(x)|6 M2

x−a2 , M2 = sup[a,b] f

′′(x).

C( 9-.! 9'"# 5)$ "!#+?$)#"%!9 9'33.99"2.9 *'.

|ε(T )0 | 6 M2

(x − a)3

126 M2

(b − a)3

12.

! "#$%& ' ()"*+ ,-.&/012130 41 56789:;1 <= 56789:;1>"5? ' @A==>@A=@

!"#$%&' () *+%!,-'. -/$0%+1&"%$,0 02*+&$32'

!"#$$#%$ &'%$ q > 1 (# )*+)%+# )',,# &$-)-.#,,#/0 1

|ε(T )q | 6 2q × 1

12M2

(

b − a

2q

)3

= M21

12

(b − a)3

22q.

!!"#! $" %& '()*+$" $" ,-'./+0

2'%$ %03+3(#$ +# ,4,# &$3/)3&# 5%# &$-)-.#,,#/06 '/ $#.-7/30 +* 8'/)03'/

#$$#%$ #/ 8*3(*/0 3/0#$9#/3$ +# &'3/0 ,3+3#% c = (a + b)/2 1

ε(S)1 (x) =

∫ c+x

c−x

f(x)dx − 2x

6[f(c − x) + f(c + x) + 4f(c)] ,

*9#) x ∈]c − a; b − c[: !#( .-$39-#( (%))#((39#( .# )#00# 8'/)03'/ ('/0 1

dε(S)1 (x)/dc = 2

3 [f(c + x) + f(c − x)] − 13 [f ′(c + x) + f ′(c − x)] − 4

3f(c)

d2ε(S)1 (x)/dx2 = 1

3 [f ′(c + x) + f ′(c − x)] − x3 [f ′′(c + x) + f ′′(c − x)]

d3ε(S)1 (x)/dx3 = −x

3

[

f (3)(c + x) − f (3)(c − x)]

;"*&$<( +# 0=-'$<,# .#( *))$'3((#,#/0( 7/3(6

|d3ε(S)1 (x)/dx3| 6

2x2

3sup

]c−x,c+x[

f (4)(x) 62x2

3M4, M4 ≡ sup

]a,b[

f (4)

#0 3+ ("#/ (%30 &*$ 3/0->$*03'/( (%))#((39#( 5%#

|ε(S)1 (x)| 6

x5

90M4,

&%3(6 #/ &'(*/0 x = (b − a)/2 1

|ε(S)1 | 6

(b − a)5

2880M4.

?/7/6 '/ >-/-$*+3(# @ +"'$.$# q .# +* ,4,# 8*A'/ 5%# &'%$ +#( ,-0='.#( &$-)-B

.#/0#( 1

|ε(S)1 | 6

(b − a)5

2880 × 24qM4.

!"#(03,*03'/ .# +"#$$#%$ )',,3(# &*$ )=*)%/# .# )#( ,-0='.#( &#$,#0 .#

)',&*$#$ +#%$ &$-)3(3'/: C+ #(0 ."%(*># ."#D&$3,#$ +* &$-)3(3'/ #/ 8'/)03'/ .%

&*( ."-)=*/03++'//*># h = (b − a)/2q: E/# )',&*$*3('/ .# )#( ,-0='.#( #(0

.'//-# .*/( +# 0*F+#*% .# (G/0=<(# :H:

!"#$%&! ' ( E/# ,-0='.# ."'$.$# k #(0 #D*)0# &'%$ %/ &'+G/I,# ."'$.$#

3/8-$3#%$ '% ->*+ @ k: ?/ &*$03)%+3#$6 +* ,-0='.# .# J3,&('/ #(0 #D*)0# &'%$ %/#

8'/)03'/ )%F35%#6 #0 )#6 4567456 89:; 76 <=8 >/?@A=B;:779BB=C6: 2*$ #D#,&+# 1

∫ 1

0

(x3 + x)dx =1

6[2 + 0 + 4(

1

8+

1

2)] ≡ S1 =

3

4

!"#$%&! ) ( !#( )*+)%+( .# +* &$-)3(3'/ ."%/# ,-0='.# /# .'//# 5%"%/#

3/.3)*03'/ (%$0'%0 %03+# &'%$ )',&*$#$ .#%D ,-0='.#(: !"#K)*)30- $-#++# ."%/#

,-0='.# &#%0 9*$3#$ (%39*/0 +* 8'/)03'/ @ 3/0->$#$: !"-9*+%*03'/ .# +* &$-)3(3'/

8*30 3/0#$9#/3$ +"=G&'0=<(# 5%# +#( 8'/)03'/( @ 3/0->$#$ ('/0 &+%(3#%$( 8'3( .-$3B

9*F+#(: L#00# -9*+%*03'/ #(0 .'/) ."*%0*/0 ,#3++#%$# 5%# +#( 8'/)03'/( @ 3/0->$#$

('/0 $->%+3<$#(:

! "#$%& ' ()"*+ ,-.&/012130 41 56789:;1 ! <= 56789:;1>"5? ' @A==>@A=@

!"#$%&' () *+%!,-'. -/$0%+1&"%$,0 02*+&$32'

εq !"#"$ #$ %&$""$'"

($)*+,-%$

(b−a)2

2q+1 M1 ∝ h

."+/01$2

112

(b−a)3

22q M2 ∝ h2

345/2!,

(b−a)5

2880×24q M4 ∝ h4

!"#$ 67 8 (9)+/4*'%+*4: #$2 $;/"$224!,2 $* !"#"$2 #$ %&$""$'" #$2 59*<!#$2

(.36 =, ,!*$ Mi ≡ sup]a,b[ f(i)(x) $* αi ∈ R

! !" #$%&'() () *'+,)-.

>% "$22!"* )%+4"$5$,* #$ )$ ?'4 /"9)0#$ ?'$ %+ 2'4*$ {Tq}q∈N )!,2*4*'9$ #$2

$2*45+*4!,2 @ %&!"#"$ q #&',$ 4,*9-"+%$ /+" %+ 59*<!#$ #$2 *"+/01$2A )!,B$"-$

B$"2 %&4,*9-"+%$ $;+)*$A %$ /+2 #&9)<+,*4%%!,,+-$ 9*+,* #4B429 /+" #$'; @ )<+?'$

4*9"+*4!,6 C,$ /"$540"$ 5+,40"$ #&+59%4!"$" %+ /"9)424!, )!,242*$ #!,) @ "9#'4"$

h6 D+ )!,*"$/+"*4$ #$ )$**$ !/9"+*4!, $2* %&+'-5$,*+*4!, #' ,!5E"$ #&9B+%'+F

*4!,2 #$ :!,)*4!, $* #!,) #&',$ +'-5$,*+*4!, #$2 $""$'"2 #&+""!,#42 ,'59"4?'$2

$* #' *$5/2 #$ )+%)'%6

=, "$5+"?'$ ?'$ {Tq}q∈N $2* ',$ 2'4*$ $, h2$* )!,B$"-$ #!,) /%'2 B4*$

?'$ %+ 2'4*$ {rq}q∈N ?'4 B+"4$ $, h 5+42 5!4,2 B4*$ ?'$ {Sq}q∈N ?'4 $2* $,

h46 =, )<$")<$"+ #!,) #$2 59*<!#$2 4,*"4,20?'$5$,* /%'2 /"9)42$26 >% $;42*$

#$'; /!224E4%4*92 /!'" )$%+6 D+ /"$540"$ )!,242*$ @ /!'"2'4B"$ %+ /"!-"$224!,

4,4*49$ /+" %$2 59*<!#$2 (.3 G %+ :!"5'%$ #$2 "$)*+,-%$2 +//"!)<$ %+ :!,)*4!,

f /+" ',$ )!,2*+,*$ H!"#"$ IJA )$%%$ #$2 *"+/01$2 /+" ',$ :!,)*4!, +K,$ H!"#"$

7JA )$%%$ #$ 345/2!, /+" ', +") #$ /+"+E!%$ H!"#"$ J6 6 6 L//"!)<$" f /+" ',

/!%M,N5$ #&!"#"$ k /$"5$* #&!E*$,4" ',$ :!"5'%$ #$ ?'+#"+*'"$ #&!"#"$ 2k6O&$2* %$ /"4,)4/$ #$2 :!"5'%$2 #4*$2 #$ !"#$%&'$#!(6 D&4,)!,B9,4$,* #$ )$2

59*<!#$2 2!,* %+ %!'"#$'" #$2 $;/"$224!,2 ?'&$%%$2 :!,* 4,*$"B$,4" $* %$2 -"+,#$2

B+%$'"2 ?'$ /"$,,$,* 2$2 )!$K)4$,*2A "9#'42+,* %$'" 2*+E4%4*9 ,'59"4?'$6 O$2

)!,*"+4,*$2 5+"-4,+%42$,* ?'$%?'$ /$' %&'2+-$ #$ )$2 :!"5'%$2 ?'4 ,&$;)0#$ ?'$

"+"$5$,* %&!"#"$ P6

D+ 2$)!,#$ /!224E4%4*9 $2* )*+,,-.-/!/ %+ )!,B$"-$,)$ #&',$ 59*<!#$ (.3 /+"

%$ 2)<95+ #&$;*"+/!%+*4!, #$ (4)<+"#2!, G )&$2* %+ 59*<!#$ #$ (!5E$"-6

!"#$%&'$"(&) *+ ,(-.$#*/&)

D$ 2)<95+ #$ (4)<+"#2!,

2&+//%4?'$ @ *!'*$ +//"!;45+*4!, A(h) @ %&!"#"$

k #&',$ ?'+,*4*9 A Q I ,&$2* /+2 ,9)$22+4"$5$,* ',$ 4,*9-"+%$A $%%$ /$'* R*"$ %+

2!%'*4!, #&',$ 9?'+*4!, #4S9"$,*4$%%$ /+" $;$5/%$6 D&$;*"+/!%+*4!, 2&+//'4$ 2'"

%$ :+4* ?'$ %&$""$'" +22!)49$ @ %&+//"!;45+*4!, A(h) /$'* R*"$ $;/"459$ /+" ',

#9B$%!//$5$,* #$ .+M%!" )$%# .!( ,$!0,1!%#( %! )-2!%)!%# 2+( )! h G

A = Ak(h) +∑

i>k αihi . . .

= Ak(h) + O(hk).

=, ,!*$ O(hk) ',$ 2!55$ #$ *$"5$2 #&!"#"$2 2'/9"4$'"2 @ k H!, ,$ )<$")<$"+

/+2 @ #!,,$" %&$;/"$224!, #$2 αiJ D&!"#"$ #$ %&$""$'" $2* #!,,9 /+" %$ *$"5$

#!54,+,*A )$%'4 #$ #$-"9 %$ /%'2 :+4E%$ H/'42?'$ h < 1J6

! "#$%& '() *%+,-(.&/0 12034#5#((#6 788797:;<=

!" #$%&' ( )*#+, -./'0123241 52 6789:;<2 => 6789:;<2?#6@ ( A>>? A>

!"#$%&' () *+%!,-'. -/$0%+1&"%$,0 02*+&$32'

!" #$%&'( )" *+$%(,)#-. )&/.+0 1." #1+0" Aq )&/.+" 2(, "#0 )-..& 2(,

34"52,"##+-. 6

Aj+1 =1

2j − 1

(

2jAj(h/2) − Aj(h))

, j > k. 7 89:

!"#"$%&%"'

456 758956 {Ak+n} :;<=>6 ?@8 A@ 85A@7>B= :5 8;CD885=C5 E()FG 6B=7 :56 @??8BH>I9@7>B=6 :5 A J A/B8:85 k + n K

A = Ak+n + O(hk+n), n > 0

()*"'$&!+&%"' ;122-#-.# <1" 3( 2,-2-#+0+-. #-+0 =,(+" (1 ,(.> j > k8 ?3-,#@

A = Aj(h) + αjhj + O(hj+1). 7 8AB:

C(, (+33"1,#@ 3"# αi &0(.0 +.)&2".)(.0# )" h -. 2"10 0-1D-1,# &$,+,"

A = Aj(h/2) + αj(h/2)j + O(

(h/2)j+1)

= Aj(h/2) + 2−jαj(h)j + O(hj+1). 7 8AA:

E. )+#2-#" )-.$ (="$ 3"# &<1(0+-.# 7 8AB: "0 7 8AA: )" )"15 &<1(0+-.# F )"15

+.$-..1"# 7A "0 αj #-.0 +.$-..1"# 0(.)+# <1" Aj(h) "0 Aj(h/2) #-.0 $(3$13&#@ "0h "#0 1. 2(,('G0,":8 H. #-1#0,(I(.0 '"'J," F '"'J," 7 8AB: F 7 8AA: '130+23+&"

2(, 2j@ -. #" ,('G." F 1. #I#0G'" )41." #"13" &<1(0+-. F 1." +.$-..1" 6

A = 12j−1

(

2jA(h/2) − A(h))

+ O(hj+1)

= Aj+1 + O(hj+1). ¤

,"!*-./ 0/ 1"*2/!3

K(.# 3" $(# )41. $(3$13 )4+.0&>,(3"@ -. (223+<1" 34"50,(2-3(0+-. )" *+$%(,)#-.

F 1." '&0%-)" *L;@ >&.&,(3"'".0 3( '&0%-)" )"# 0,(2GM"#8 E. )&/.+0 3" #$%&'(

)" *-'J",> ". 2-#(.0 R1,q = Tq, q > 0 "0 2(, 3( ,"3(0+-. )" *+$%(,)#-.

Rp+1,q =1

4p − 1(4pRp,q+1 − Rp,q) , ∀p > 1 7 8A :

!"#!%)&)$

R2,q = Sq+1 7 8AN:

()*"'$&!+&%"' ;-+0 q > 0@

R2,q =1

3(4Tq+1 − Tq)

=1

3

{

2b − a

2q+1

[

f(a) + f(b) + 22q+1−1

∑

i=1

f(a + i b−a2q+1 )

]

−1

2

b − a

2q

[

f(a) + f(b) + 22q−1∑

i=1

f(a + i b−a2q )

]

}

=1

3

b − a

2q+1

[

f(a) + f(b) + 2∑

fpairs + 4∑

fimpairs

]

= Sq+1. ¤

! "#$%& ' ()"*+ ,-.&/012130 41 56789:;1 ! <= 56789:;1>"5? ' @A==>@A=@

!"#$%&' () *+%!,-'. -/$0%+1&"%$,0 02*+&$32'

h = b − a h/2 h/4 h/2q

R1,0 → R1,1 → R1,2 · · · → R1,q

ւ ւ ւR2,0 → R2,1 R2,p−1

ւ ւR3,0 R3,p−2

··· ւ

Rp,0

!"#$ ! " #$%&'( )* +,'-*./

0*1 Rp,q 2,.'*34 )*1 1564*1 1567(34 p ,5 1567(34 q! 83 ( 75 95* $*::*1 1567(34

p $,37*./*34 -*(5$,5; ;:51 764* 95* $*::*1 1567(34 q! <&(3',631= 6: 3>*?614* ;(1

)* .*:(46,3 16';:* *34.* Rp,q *4 Rp,q+1 @1(52 ;,5. p = 0 956 .*3) 2,.'5:* )*1

4.(;AB*1C! D3 ;.(4695*= ,3 $%,6164 ),3$ :( 1564* )*1 {Rj,0}j>0 ),34 ,3 ,-46*34

:*1 4*.'*1 )* :( '(36A.* 1567(34* E ,3 $(:$5:* )>(-,.) R1,0, R1,1, R1,2, . . . R1,p

;561 :*1 {R1,j}j=0,...p−1= ;561 :*1 {R2,j}j=0,...p−2= *4$! F5195>G Rp−1,0 *4 Rp−1,1

956 ),33*34 Rp,0! H*44* ;.,$&)5.* *14 6::514.& ;(. :* 4(-:*(5 ! !

I: 3>*14 ;(1 .*$,''(3)& )>546:61*. p + 1 2,61 :( 2,.'5:* )*1 4.(;AB*1 ;,5.

$(:$5:*. :( ;.*'6A.* :6/3* @:*1 R1,q = TqC! D3 *J*4= $%(95* Tq 2(64 (;;*: G 2q

())646,31 *4 2q + 1 &7(:5(46,31 )* :( 2,3$46,3 f ;(.'6 :*195*::*1 2q−1 + 1 ,34

)&FG &4& $(:$5:&*1 G :>,.).* q − 1 @:*1 &7(:5(46,31 )*1 ;,6341 )>63)6$*1 ;(6.1C! 83

',34.* 95* :*1 Tq ;*57*34 K4.* $(:$5:&1 )* ;.,$%* *3 ;.,$%* E

Tq+1 =1

2Tq +

b − a

2q+1

2q

∑

j=1

f

(

a + (2i − 1)b − a

2q+1

)

. @ !LMC

!"#$%&'(&)#$

R1,q+1 = Tq+1

= 12

b−a2q+1

[

f(a) + f(b) + 22q+1−1

∑

i=1

f(a + i b−a2q )

]

= 12

b−a2q+1

[

f(a) + f(b) + 22q−1∑

i=1

f(a + 2i b−a2q+1 ) + 2

2q∑

i=1

f(

a + (2i − 1) b−a2q+1

)

]

= 12

b−a2q+1

[

f(a) + f(b) + 22q−1∑

i=1

f(a + i b−a2q )

]

+ 12

b−a2q+1

[

22q∑

i=1

f(

a + (2i − 1) b−a2q+1

)

]

. ¤

! "#$%& ' ()"*+ ,-.&/012130 41 56789:;1 ! <= 56789:;1>"5? ' @A==>@A=@