A New Generalization of the Trapezoid Formula for n

(1)

A New Generalization of the Trapezoid Formula for n- Time Differentiable Mappings and Applications

This is the Published version of the following publication

Cerone, Pietro, Dragomir, Sever S, Roumeliotis, John and Sunde, Jadranka (1999) A New Generalization of the Trapezoid Formula for n-Time

Differentiable Mappings and Applications. RGMIA research report collection, 2 (5).

The publisher’s official version can be found at

Note that access to this version may require subscription.

Downloaded from VU Research Repository https://vuir.vu.edu.au/17242/

(2)

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(9)

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(11)

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4 q$

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+{d,^q.4. +4,^q+e{,^q.4

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5 4

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zkhuh

M⁵+{> q, @ 4 +5q. 4, +q. 4,⁵

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+5q. 4,

D^q.4. +4,^qE^q.45r zlwkD@{d> E@e{=

Qrz/ iurp +718,/

+5q. 4, +q. 4,⁵M⁵+{> q,

@ q⁵+D.E,

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D^5q.4.E^5q.4

D^q.4. +4,^qE^q.4₅l

@ q⁵+D.E,

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DE

D^5q.E^5q

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@ q⁵+D.E,

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. +5q. 4,DE^D^q+E,^q`⁵

Qrz/ vxevwlwxwlrq riD@{d> E@e{dqg wkh idfw wkdwD.E@ed jlyhv L+{> q, @ _+q.4,^M+{>q,^s_5q.4>dv suhvhqwhg lq +716,1 Vxevwlwxwlrq ri lghqwlw| +514, lqwr +717, jlyhv +715, dqg wkh uvw sduw ri wkh wkhruhp lv wkxv suryhg1

Wkh xsshu erxqg lv rewdlqhg e| wdnlqj hlwkhuL+d> q, ruL+e> q, vlqfhL+{> q, lv frqyh{1 Khqfh wkh wkhruhp lv frpsohwho| suryhg1

(13)

Fruroodu|71 Ohw wkh frqglwlrqv ri Wkhruhp 7 krog1 Wkhq wkh iroorzlqj uhvxow krogv

4 ed

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d i+w,gw^q4[

n@3

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i^+n,+d, . +4,ⁿi^+n,+e,l +719,

ed

5

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i^+q4,+e,i^+q4,+d, ed

5 4

q$

ed 5

_q.4 s 4

5q. 4

;?

=

q.45q > q hyhq 5> q rgg = Surri1 Wdnlqj{@ ^d.e₅ lq +715, jlyhv +715,/ zkhuh

L d.e

5 > q

@ 4

+q. 4,s 5q. 4

ed 5

_q.4q

7q⁵. +5q. 4, ^4 . +4,^q`⁵r⁴₅

= H{dplqlqj wkh deryh h{suhvvlrq iruqhyhq ruqrgg uhdglo| jlyhv wkh uhvxow +719,1 Uhpdun 71 Iruqhyhq/ wkhq wkh wklug whup lq wkh prgxoxv vljq ydqlvkhv dqg wkxv wkhuh lv qr shuwxuedwlrq wr wkh wudsh}rlgdo uxoh +719,1

Wkhruhp 81 Ohw wkh frqglwlrqv ri Wkhruhp 7 eh vdwlvhg1 Ixuwkhu/ vxssrvh wkdw i^+q, lv glhuhqwldeoh dqg eh vxfk wkdw

i^+q.4,

4=@ vxs

w5^d>e`

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Wkhq

mSW+{,m esd

45 i^+q.4,

4 4

q$L+{> q,>

+71:,

zkhuh SW+{, lv wkh shuwxuehg wudsh}rlgdo w|sh uxoh jlyhq e| wkh ohiw kdqg vlgh ri +715, dqg L+{> q,lv dv jlyhq e| +716,1

Surri1 Ohwi> j = ^d> e`$Ueh devroxwho| frqwlqxrxv dqgi³> j³ eh erxqghg1 Wkhq Fkhe|fkhy*v lqhtxdolw| krogv +vhh ^48/ s1 53:`,

mW+i> j,m +ed,⁵

45 vxs

w5^d>e`mi³+w,m vxs

w5^d>e`mj³+w,m=

Pdwlf/ Shfdulf dqg Xmhylf ^49` xvlqj dsuhpdwxuh Juxvv w|sh dujxphqw suryhg wkdw mW+i> j,m +esd,

45 vxs

w5^d>e`mj³+w,ms

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+71;,

Wkxv/ dvvrfldwlqj i^+q,+, z lwk j+, dqg +{w,^q zlwk i lq +71;, uhdglo| surgxfhv +71:, zkhuhL+{> q, lv dv jlyhq e| +716,1

Wkhruhp 91 Ohw wkh frqglwlrqv ri Wkhruhp 7 eh vdwlvhg1 Ixuwkhu/ vxssrvh wkdw i^+q, lv orfdoo| devroxwho| frqwlqxrxv rq+d> e,dqg ohwi^+q.4,5O5+d> e,= Wkhq

mSW+{,m ed

i^+q.4,

5 4

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(14)

46

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mW+i> j,m +ed,⁵

⁵ ni³n₅nj³n₅>

zkhuh

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nj³n₅s

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+7143,

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81 VV,WB|NA YA 6jiWB, YA|j}iB|NA

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Lq ^4`/ wkh dxwkruv frqvlghuhg wkh iroorzlqj jhqhudol}dwlrq ri wkh wudsh}rlg iru0 pxod

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p4[

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&

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frqwlqxrxv rq ^d> e`1 Wkhq zh kdyh ]e

d

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(15)

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= @

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= Surri1 Dsso| wkh lqhtxdolw| +614, rq wkh vxelqwhuydo ^{m> {m.4` wr jhw

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(16)

48

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z h kdyh

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= @

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? AA AA AA AA AA AA A=

+q.4,$4 p4S

m@3 vxs

w5^{m>{m.4`

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4 +qt. 4,^4@t

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(17)

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4

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4

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{m.{m.4

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d

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(18)

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