MATSUOKA  Kenichi

NOMACHI  Sumio

Muroran-IT Academic Resource Archive

Some Formulas Derived from Finite Integration著者 NOMACHI  Sumio, MATSUOKA  Kenichijournal orpublication titleMemoirs of the Muroran Institute ofTechnology. Science and engineeringvolume 7number 3page range 855-858year 1972-09-15URL http://hdl.handle.net/10258/3560Sorne Forrnulas Derived frorn Finite Integration Sumio G. Nomachi and Kenichi G. Matsuoka Abstract Thc inite sinc snics can somctimcs hc rcprcf¥entcd hy a function of clof¥C>c] from号 Anrlit may cOl1veniently work 011 the !inite differece equaiion Some五nitecosine series which can take functions of closed form， are formulated in this paper L お10therfunction Let take mother function F"x: 叶.x(α+170)=ιwhere 1Jl = 1， 2， 3，・‘，n1 = lmagmary umt， then we have 九一1=C叶in whic ??? ? ?? ??↑???????? ???????????????????????? ??↑??? ?? ??????????????????????? ????、?????So the五niteintegration of the above becomes ? ???? ?????? ? ?。rStαadz-?!?{:坐守三)上cos~{(プ71)L21)2cosh(α+子i)-2(225) 856 Sumio G. Nomachi ancl Kenichi G. Matsuoka Considering that X FL必r3FLdrラwe have at once 1 1 ( ) sinh(na+mrri)sinh(α+竿 i)I: /I:.x =ー引cosh(nα十1Jlri)十1(十 7 一つ一一一一一一一ι.-、 ノ 2~coshlα+一一二 i)-1? I ¥ n / I The imaginary part of the above yields 川、 (-1)'''sinh nαsinf宇I1 I: Ji'".xf =一一つ一一一一てど←，、ェ1 ノ 21cosh α-cos二二二!from which we have η1 … ( -1)吋 i山 αsinf子I: sinh xa' si十二~::_'- x=- 一一 一二云---1 山 21 cosh α一 cos 二三~)¥ n / The inversion theorem leads to the following formula; η(-lyvin?目的子.T sinhx.a 511L二誠一=而よ?????In like manner， the re<ll part can be written as follows; rr~l -r. i 1 L 1¥m. _ 1. ， ..) ， (-1)地 sinhnα.sinhαRe l I: Ji'".xf =ーす)(-1)川 coshna+ 1(十一一一一一一 ~-¥ ， 、思 1 ) AJ、 ノ 21 cosh α-cos二三二 lThe inversion formula: つ n φ判押f(.T) = -=I: Boる[f(x)]cos 一_:_'x，11- 'i!I， -~'O Iι where R払川o[山]口→」4封一1い竹(いh叫州[山]寸州+;ヤf仰Rκ仏川ηJιI[山]= 眠札川Jルべ，[山Iι山]= ~ {弘[f(x)]寸.-(-l)"f附leads to the following results; (226) Some Formulas Derived frum Finite lntegration 2u(-mOB4Fz C∞吋O悶d叫s1油hnβ1円n( 1 mπ¥ -sinh α.sinh nα iうIcosn r -cos←一一一|1 (_l)n-"， 2n (cosh α-1) 2n (cosh α+1) ， which is differentiated with respect toα， to be as fol1ows; and 2 '.=! (-1)'" cos字 x.sinhα xsinhxa一一一 一一一ム~， _ (πV -sinh α.sinh nα η市円1 21 coshα一cos-一一|¥ n / n cosh xα.coshηαcosh xa.cosh α 一一証己E瓦弓盃F云瓦 sinh2α，sln五五百sinhα 2n (cosh αー 1J2(_1)n-"， sinh α 2n (cosh α+W' 2. Formulas of Polynomial Function Lettingαbe zero in the Eqs. (2) and (3)， we have ハ 1(-1)'" cos竿 x ~ 五五，(. ぷγ=言瓦(2n2+2ー が)Ij I _L-CO日一一一一 l¥ n / 一τ中川2十町山)-8~ {1+ ( -1 )'( 2 ) ( 3 ) (4 ) 2 " .:} (-mosJ7x 1 f" n2 -1 1 + ( -1)勾汁721 / 附¥=三万iX2一一γ 一一ーす':L____( . ( 5 ) ， 2 [l-cos二二一| 、 -¥ n / Putting n-x for .1: into the Eq. (5)， we have 2umfT(71-z)1J2722-11+(-1)ηつ社)2 (亡…亙1=2ntl:~-3一一一言一_，l' (6) ¥刀/which becomes upon the action of L1;-1 リ n-) グ日押 I I I -751ml一(n-x)=~ P+(-l)" '"J (7 ) By the well known finite integration formulr， we obtain (227) 857 858 Sumio G. Nomachi and Kenichi G. Matsuoka 五COST→ (n十 tIt)}-l-iiH(-1))which is same with the Eq. (7). 3. Advanced Formulas I _ 1lI7r ¥ Denoting that D禍=2( l-eo白一云一)a吋 ('()shα=] トーミ+机 山 equation(2) can be written as follows ; 2 ~1 (-1)禍 (D叫十~+が) mπ _ "s'"__''' nβ1 (D明十~)2十年2 し山刀ルcosh xa. sinh長.sinh耐 1 (-1)n x (t;ーが+4)=瓦而五弓函E瓦.瓦函示-~slnh示一五京王新一一五夜干存干r/}--The real part of the above， yields 2 干~ (-1)叫 (D禍+ご) mπ 1 n';:;:l (D明+ご)2+η2 しり" n ゐー (cosh2s -eos 27) (cosh 2吋-cos2n7) x[耐 1s(n十x+1)問山十1-x)ー 切叫(n-x一恥osr(n-1日)+co叫 (n十1-x)cos r(n+ 1十x)-co叫 (n-1付 ).cosr(n-1-.r)] t; (-1)" "'(.ご+4)一五(t;2+示)-云{(t;+ 4)2十約， ( 8 ) in which α=戸+ri，and the imaginary part， 2 n~l (-1)叫 COSTx 1 一一一一一一一一一一一一一-nγレ1 (D叫 +~)2+ 可2 - 1J(cosh 2s-cos 27)・(cosh2ns-cos 2n7) ×ぺや[卜s凶戸仰(n十1十廿叫Z劫れ)同s釘ln仰 +1ト一づx刈)-s凶 s(n-1一♂刈れ)片sln巾 1十刈十s討II削 (n什+1一Z功れ)同sln仰 +1 十刊叫Z刈)-s凶 s(n 一1 十x刈れ刷E吟巾刊)同川s幻i川n(一1)ηト一m+認可1l)+可~~+4芹子7 ・ ( 9 ) (Reeeived May 20， 1972) References l) S. G. Nomaehi: On Finite Fourier Sine Series wilh RespeCl 10 Finite Differences， the Memoirs of the Muroran Institute of Technology， Vol. 5， No. 1， 1965. 2) S. G. Nomachi: A Note on Finite Fourier Transforme Concerning Finite lntegration、lheMemori of the Murora Institute of Technology， Vol. 5， No. 2， 1966目3) F. B. Hildebrand: Finite幽DifferenceEquations and Simulations， Prentice-Hall， 1968. (228) 

Some Formulas Derived from Finite Integration

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