\ Λ

llerivatariiiu η;ti ordiuis fiuietloiiuiii

/'(c"v('os mx) et /'(e"*Sin inxjEXPO SITIO

y

QUAM '

V EMIA AM PUSS. FACULT. JPHILOS. IJPSAL.

✓

P. P.

HJALMAR HOLMGRENPhil. Mag.

ET\

CAROLUS GERH. Μ ETZEN.ν Ostrogothi.

Ν!

IN AIJDIT. GUSTAV. D. XIX MAJI MDCCCXLVHL

Η. A. »1. S.

P. I.I

UP8ALIAE

EXCt'D. HEG. ACAD. TYI'OG Η Al'Hl.

O Q '

In diario, qiiod Archiv der Mathemali!.· und Physik inscrihitur, a Prof.Grunert edilo, Th. VII, fi. 2, Cel. Schlömilch geometris admonet dedefectu, quo ratione derivatarum habila superioris ordinis laboret calcu-lus differentialis, quippe qui careat thcorematibus generalibus, e quibusproficisci debeat liarum functionum disciplina. Cognila1 enim erant anteaderivatae cujuslibet ordinis functionum spcciali tanlum forma. Quam ino-piam ut sublevaret, sibi proposuit, ut derivatas inveniret n:ti ordinisfunctionum generalium v. c. f{x") , /{c*), & c. Atque viam rationemquepropriam et peculiarem sequenti contigit, ut derivatas functionum nomi-natarum in loco citato proponeret. Postea Cel. Prof. Malmsten idenah'actavit problema, et rem eo perduxit, ut non solum derivatas n:ti or¬dinis functionum generalium supra reliquas oinnes functiones, quas dicuntsimplices, sed eliam derivatas multarum aliarum functionum formaj etiamgeneralioris erueret^j. Jam in animum induximus, ut viam patefactamscquentes derivatas inveniremus n:ti ordinis functionum *Cös mx) et

y(ertTSin mx) , nondum, quoad cognitum habemus, propositas.

8. i.

Proposita igitur functione quadam tt=f(e Cosm.v) , derivatae hocmodo formari possunt: >

*) Opus, in quo hoc argumentum tractavit, ad Acari. Seient. Petropolit. transmissum, moxcdiltun cxspcctatnr.

I

— — e */·\α Cos mx -|- m Cos(m.v -[- ~J](l ~ιι

= c J.'[a Cosmx+L2amCos(mxJr *)+m?Cos (mx+ ™)~\+(?axJ\"\a2 (,^Cos 2mx

-4-JCos0.mx)+ßmCos^2mA<+|)+m2(iCos(2»iAr+ —^Cos ψ)] .

d:iH αχ

e j' La Coshia+3« 2mCos (mx+~) +5a»rCos(wKV+^r)+m3Cos (mx+ ™) J

+e2aA/.//[rt3(|Cos2niA*+|Cos0.mA|)+|a?»iCos(2wiA:+5)+«m?(?Cos(2nLr+2^)-3Cos:)+\m 3Cos(2mjc+

+e/,"'vy'.///[rt3(|Cos3mA'+|Cosma')+rtQHi(|Cos(3HJA'+^)+|Cos(inAf+i).)-H<»i "(|Cosf3mx+2^)-|Cos^mx+2^j)+m3(iCos(»iA:-h0v)-|Cos(mA+'0ir))].

fPu -

——- — cax/Y[a Cosmx+Aa 3mCos(«iA-+?)+6a°m 3Cos(»iJc+2^)+4am3Cos(ma'+JJ)+»l ,Cos^HJ.X*+'1ir)]

+e.inxf'"\_a i(^Cos2iiiA-+^Cos0.iiiA:)+14λ 3mCos(2/n.v+J)+tt?m 2(2 IOos^hia·*2-?)-5Cos2^)+14am 3Cos(2mA*+3^)+m4(^Cos(2mAr+4^)+^Cos^)]

+c""'xfV"[a\lC0s5mx+lCosnux)+a3m(ßCos{5mx+*)+6CiO$(mx+lj)-frt?m?(9Cos(5wiA+^)-5Cos(»iA+25))+ai/i^6Cos(3fMA*+;^r)-6Cos>hv+5r))+fti'f|Cos 3jnA+4ij+®Cos(mAM-4^))]z

/

+c '"'7'./rfrt ^'CosImA-^CösSmA+JCosO.maO+ii'mC^Cos^/imx+J)+Cos(2»kv+J )+a?m2(|Cos(4/WA:+^r)+0.Cos(2mA:+ ?)-|Cos+«m3(iCos(4wA+7r)-Cos'2mA+'^r))+mi|(gCos(4wA+7r)-5Cos(2»Kv+''5')

+3C0S4a. Ί

d;C. <&c.

ubi, brevitatis caussa, f , f" , f" , &c. pro derivatis ipsius/'(e"vCosmx) respectu e^Cos mx posuimus.

Si ad foruiam barum derivatarum attendimus, facile völligere licet,derivatam n:li ordinis hoc modo componi posse:

^=r|V"7^)T«n-VS^+'cos(^'.m.v-+7) . . . .(I),1

}>=i s=o k=o

denotante Ρ·>£—l numerum integrum, qui proximus ζ—1 superet,et HAk+l functionem numerorum n, k, s et p, ab χ prorsus indepen-

* γ

dentem.

§. 2.

Formula (1) ratione babita relationis

Cos(p-2k.mx-\-s~) — (—l/Cos(—p-2k.mx (2)facile in hanc formam redigi potest:

-—= ''s'"C's+'Cos(^/c.»,*+ 'ξ) .... (5);d.Xn γ—ι s—o k=o S Ρ

etenim nihil aliud fecimus, nisi quod pro ns^'rlCos{p-2Lmx-\-s"), quod-k—γ j ^

cunque est k, excepto k=L·? , duos summae S Cos(p-2k.mx-\^s~) ter-k=0

minos, a primo et ultimo aeque distantes, substituimusj exsistente veroV

k uz ? terminus medius idem manet, qui nsA* + * Cos5? .

Comparatis formulis (i) et (5), evadit, nisi sit kzzz?,

Ί^',Γ 'Cos (p-n

= ;Cj+'Co«^4M*+?)+^+•CK-^··»·**"))unde fit, respectu r,elationis (2),

"Akf '== "Ct+14-f— ι v<"C"','+1s ρ s V ^ J s ρ

Quoniam pro y nisi sit k —ζ , duos coéfficientcs indeterinina-Los posuimus, liorum relalionem inter se quandam staluerc possumus.Ponamus igitur

(-*Υ'Κ"+' (4);η j~,k + iί ρ

quapropter erit, nisi sit kzzz?,η h+>

_2 » l+ ,* ρ V

Exsistente k ~ζ , obtinebilur• /+1<a; Cos;"

n^+xcos

Reslat. ut determinetur n(JkJrls ρ

(5).

(G)..

s. 5.

Formula igitur (3), η -f- ' pro η substilulo, suppcdilatl/"+ ι,ι j»=i»+J x=n-j-i

S S «"-'+lm'S»+^'Cos(^.»l.v+-ρ — ι x = o k—o '

Deinde eandem formulain differentiando oblinebimus

(7)·

ι fl p—n s=n k—ρ '= S P«^yW « s »C^+1 Cos(p-s~k. tnx + «) ;

/> = ι χ—o k —O '

+ iS + '/j rtCosmvV+mCos(m.v+^)3 ^ rt" sms S"C!y (^0s(p-zk,mx+s£)k-<)

+ S evaxj ^ $ an *ms S "(Jk ' lm(p—%kjCos(p-2k—).ρ — ι ' s—o k—o" Ρ ' 2

Quod si in termino 2:do dextri membri ponilur ρ —· 1 pro ρ } et sta-tuuntur conditiones

:cp = o, :c=o,:<:\=o w»evadet

/7n+1»# »=»+i , .(s—n k-p^ e' J 1 J S«W"",'V Sp*(7'+1Cos;/)-i2/:.»»*+?)

^ —ι /s=zo k—o

(+ Sa"-* + ,m* S'*^'Cosrn*Cos(/)-2Äw,»+ £)i =0 /f — O ..

s—n k~ρ+ Sa"-'m'+[ S V.+1,Cos(.»a-+7)Cos(/)-2A-/.»ii+?)

ί·= 0 /c =0

t + S an~sms 1 S (p-%k) HfC ^ Cos(0-2/1. ϊϊιλ:+^ —■——") ί *ί=0 fc=o 7 2 yJ

Est vero

CosmxCos^-2Å*-/. mx+ s™)~ \COs(p-2k .mx +'?) -{- ^ Cos(^-2Ä"-2. mx+ *?)Cos (m X + l)Cos(p-2k'1 . 7ΏΧ+ Ψ)

z=: ~ Cos(/?-2Ä:. ^ Cos[/;-2A--2. mx+^'+^1*).

Ratione liaruia identitatum habita, et comparatione cum (7) inslitu-ia, ad hanc pervenimus aequationem:

'S \r"J« S+'m' S»+;Cl+,Cos(^I.inx+'·?) =

β—i s—o k—o S Ρ2

6

p=H 4-1 s—n Ii—ρS cpax/r) 1 Se"'*+,i»*Sp"C*+,Cos(p-2k.mx+%yp=i J s=o k=o s v

I + S an s+lm $ I "ett1 Cos(/j-2/t .»w + J)l s=o 1 k—o

* + S an S + 1ms S | Co$(p^2k-å.mx + ™)s=o k—o

I ' \

s—n k—ρO η-s s+i ^ jM fc+i r , (ί-Η')π\+ o a m o - Cos(p-sk.mx+ )

s—o k—o 2

s— η h—pr, ii-s S+l ς, , «W+i^ . T (·*+'Κ— o α m <3

»-χ Cos(yü-2/t-2. wix+ )S-0 & — o 2

•s—Η /~MC' »-i i-Hl -C , „,.»1 k+i~

, 7 (s +1 )tt+ »3 a m 3 (p— 2fy SC, Cos(o-2k .mx+

s — o h—O S V 2

In summis 5:a et 5:ta, quibus constituitur dextrum a3quationis mern-brum, si pro k ponitur k—I, et in summis 4:ta, 5:ta et 6:ta s- Ipro s, additis conditionibus

:<> = 0. 'cj+^o.^cj+^o, ..... (9),obtinebitur, faeili transformatione effecta ,

p=n+1 s=n+1 h-p ' ,

S r*/r) S «""+VS +;C*+lCos(/,-2S.m*+?) =Ρ = l S = 0 h=o

p=n +· ι « ί=»·'|-1 k—ρS e''7W S a""+'m SCos'p-tk.mx+1) X

ρ — ι s—o Λ =o

r»/iH ι ,η^-,/ί+ι in~k „ n^k+i , n^fc+i,P/> +ä«V, +ä+p.,+ L-J5-+,,-1+(p-2A)5.1C/ J.-.(40).Jam vero quum, quicunque ipsis ar, a et m tribuuntur valöres, huic

;eq ii a tioni satisSeri oporteat, sequitur, ut coeffieientes Cosinuum ejus-

dem are iis in utroque meinb ro äquales sint; quamobrem, respeetu re-iationis

Cos{-p-2k. mx+')f) zu (— 1 )s Cos (p-2k. mx+ s~) ,

bane, quam dicunt, scalam recursionis coefficientium obtinebimus:

"+,4+■+ -1 )'"+;qr'+,=p{χ;+-<')+ι (;<£>(- ·)+<)·ΙΟ ί ')-+Χ,+η)+:0

+ip-2k)(S_X+'+(-1 )'SX'k+'), .... .(II). ,

§. 4.

Ouoniam, ratione conditionis (4) habita, prodeunt relationes

XX'-i-lfX;', nisi Sil *= 5 ,

"cp';+1 = (— iy'X_,, nisi sit k—'χ , et

;c£ = (-')s:c^·, nisi Sit .k=tl (12),bis ipsius k valoribus exceptis, aequatio (11) in baue abit:n + ι Λ+ ι n^ + 1 ι n 1 , ι n fJ* , ι np!t+ 1 _ ι ngX 4-(n-^k) " + ' ( 8

S C^ — Ρ stp +2 s^p-l 2 S^p-l 2 s-l^p-l 2 M^|-| '/ - h-\^p ··(!«>)./ \ V

Comparatis iis utriusque membri aequationis (10) terminis, qui Cosconlinent, facile perspicitur, eandem aequationem (15) valere, etiamsi sitk — ζ , s quidem pari exsistente.

Praeterea, adbibiiis aequationibus (12), utruin sit k — ~,ank =

aequatio (11) in hanc eandem transformari potest:γ+ι JJ-}-1 /1+1 p-l > />+1 ρ-1 /1 + 1

"+,c »+p"c,r+Ki+(-,)sy'eT+!"er+ici+(-i)'"') "c~-i "cT+ "c^·s ρ s ρ 4 S p-l - s p-i 4V s-l p-l '■ S-l ρ s-i ρ

iEquatio haec ab aequatione (15), k — —· Tel k ~ posito, coM P+l

solummodo differt, quod coefficienlibus formaa C* , ubi * numerus im-. A ( s V '

par, earét. Quos coéfficientes, quum igitur nihil ad ceteros cxprimcn-dos valeant, et praeterea in ipsa derivatae formula ob faclorem Cos ™baud occurrant, nihilo aequales ponere possumus. itaque addita condi-tionc

nC+> — 0, ubi s impar , : . (14),Λ ρ

aequalio (15 generalein exbibet coefficientium ratiönem.

Cui' aequali-oni ut satisfiat, ponamus

Λ -2mp+i)£,.i<-'J<"2r',i Vr-tL·. («)-J5. 5.

Priusquam vero ultra progrediamur, ostendi oportet, haue formulaniconditionibus (4), (8),' (9) et (14) satisfacere.

Quod pri inum ut fiat, in formula (15) ρ - k pro k ponamus; ände Iii

Vf4'+' (- I ytlsPp-k ^ ,· . N ,λ ,( — ^ ^ (— 1). ί-2/Λ « (p-k)k- ,A r 2?Γ(ρ+ΐ) i=o r=o /v J · 'vel, si pro r ponamus i-r et attendamus ad aequationem cognitani

Pp-k ~ Ρk ι

η ,ψ-k^r ι (-1 nspii C? & i s ti-s

• =^γΓΓΓ> s. ,(-ι/[-(ί-2»·)]'/ ν.*ΥΛr 1' Γ[ρ-\-\ ) i—o r-i-p + k

Ί ■(-λ?η*ι>>> ''3"s, ,·.; , n-T1' 777777Vr2?4 '