JACOBUS mCOL. GMNL1IND

DE

DERIVATA n:li ORDINIS

FDNCTIONIS CUJUSLIBET ^m

QUANTITATUM.

COMMENTATIO

QUAM

EX SPECIALI REGIS GRATIA

ET CONSENSU AMPLISS. FACULT. PHILOS. ÜPSAL

p. P.

JACOBUS mCOL. GMNL1IND

PHIL. CAISD. STIP. KORBERG.

ET

PETRÜS ACiiSTSTÜS GRENHOLM

SIORRLANDI

IN AUDITORIO GüSTAYIANO DIE III MAJI MDCCCXLVIII.

Η. Α. M. 8.

P. I.

ÜPSALIAE

WAHLSTRÖM ET C.

HANDLANDEN

HÖGÄDLE HEHR

PEHR

GUSTAF DAHLSTRÖM

med tacksamhet tillegnadt

af

PEHR AUGUST GRENHOLM.

llikade Föräldrar

\

tillegiiadt

af

sonlig kärlek ock tacksamhet.

Greometrse

functionum quarumlibet quantitatum⁹ quae essent functiones speciales guantitatis cujusdam variabilis independentis, quac- rerej et id magno quidem ^cum successu. Sed est problema generalius, quo derivatae n:ti ordinis funetionum quarumlibet unius, duarum, trium, . . , m quantitatum., quse sint functiones quselibet quantitatis cujusdam variabilis independentis, quse- rantur. Hoc problema solvendum suscepimus, contigitque, ut expressiones generales barum derivatarum sub forma comhi-

natoria inveniremug.

§. 1 Theorema I.

Si u=f (if9 z, , . v) est functio

quselibet

variabilium i/? z,... v, et y= φ(x)9 ^{,ζ =}ψ

(χ)

functiones quselibet quantitatis

variabilis independentis

autem non in _numero m quantitatum 2/, z, . . . v:

erit deri-

vata w.ti ordinis:

d"n /-» Γ(η+1)

j dxn

*J

1

"....[Γ(5)]*

ι"

1

'

Γ(α+1)Γ(α+1)...Γ(μ+i)T(b +1)Γ(β+i)....T(b+i)....

1

*

....Γ(Α+ΐ)Γ(*+ΐ)7^τ(^+ί)Γ(Ζ+ΐ)Γ(Λ+ΐ)...Γ(ΐ+ΐ)

/dny\a/daz V /dnv

γ fd^yS* /άη~χζ\ß /dn"V\h

\dxnJ \dxnJ \dxnJ \dxn~lJ \dxil'lJ \dxn~1)'

ffd*

äγ /d2v\^ /dy\} /dz γ /du V

'""'Wv

\dij \faj''\dx)'

fit + Τ+^...+ tjj

dytdzT...dvt ?

indicante signo γ integrale combinatorium, i. e. summam omnium terminörum7 qui ea sunt forma7 quam exbibet expres- sio subsigno / contenta, ubi α, a,...a, /9,..

ib?....

.. * bj Z? λ7 ^... lj f, Ty ... t numerl integri positivi yel ^=o

esse pofcerunt, et conditiouibus:

(2)

^Ä+a+",+a^+(n'4X6+/s+-«+b)+»"+2(A+x4-...+b)+Z+^...+l==n,

j^t=.a+b+..,+k+l7 τζ=α+β+..,+χ+λ,^.... t=a-f-b+...-fk+l

satisfaeere debebunt.

Demonstratio. Ratione inita inveniemus primae derivatas:

du du dy du dz du du

5

d2u d2u /dy\* diu

/ilzV2^ d2u /JuV

d2u dy dz

dx2 dy2

\dxj

d2u dy dv d2u dz dv

...+ 2 ζ. μ... +2 1-...

dydv dx dx dzdv dx dx

dud2y du d2z du d2v

+

dy dx2 dz dx2

+

^dx2'

etc. etc.

Quibus quum conveniant formulae (1) et (2), restat, ut demonstremus, si ^verae sint hae formulae, ^veras etiam fore for-

mulas:

'dn+iu _ρ Γ(η⁺2)

dx*+t

~~^f[r(n+2)].«

1

•••'ir(5)],fc+■*+'"++,λ+···+,i

1

r(1a+l)r(ltt+l)...r(1a+l)ir(1i>+l)r(1/?+l)...r^b+1) ^....

1 '

(5)^ * ....Γ(1/£+ΐ)Γ(1«+1)...Γ(11ί+ί)Γ(1/+1)Γ(1λ+Ι)...Γ(11+1)'

sdaJriy^ia/

daJrtz^i«

da+iV\insdny\ibsdnzyß ,dav\ih

^dxn+i' ^

^dxa+l' ^dx ^dxn' ^-dxnyl

fd2y\tksd2Z\tx

sd'v^/dyybdzy* ^sdvy*·

'^-dx2

^dx2 ^dx2' ^dx^

^dx^

d,t+iT+"'

+

dyltdzlT...dvlt

4

et:

(w-fl) (iÄ+f&f· ·

·+ι&)+ίί(ι&4"ΐ^+·

·+ι1)^4*

^(1Ä4'i5i+

+ib)

+ +«· ·+il^=Μ+4,

\t^{= -j-}ih-\-^{· · ·4" Ι" srr} +j/S+^····+jii + ^j ···.

it ⁼ift+jb+n.+fli+jlj

quse e formulis

(1)

(2), Substitute

i. e. ut demonstremus, si derivata n:ti ordinis terminum quem-

libet, qui ea sit forma, quam in formula (1) exbibet expres- sio sub signo J contenta, ubi conditionibus (2) est satisfacien- dum, nee ullum alium terminum babeat, derivatam (w+ l):i

ordinis terminum quemlibet, qui ea erit forma, quam in for¬

mula (5) exbibet expressio sub signo J" contenta, ubi condi¬

tionibus (4) erit satisfaciendura, nec ullum alium terminum

habituram.

Facile igitur patet, differentiato termino quolibet, qui ea sit forma, quam in dextro membro aequationis (1) exbibet ex¬

pressio sub signo

J

ubi

(2)

«'+!*+··· +|t

d u

dyltdzlT,.. dvlt

§

ubi conditionibus (4) erit satisfaciendum^, nee

uilum aliinn

minum. Tum contendimus, si verae sint formulae (1) et (2),

derivatam (n⁺l):i ordinis termiuum

quemlibet habituram,

non tantum, praeter factorem numericum, ea erit forma, quam

exbibet expressio (5), ubi conditionibus (4) erit

satisfaciendum,

sed etiam factorem numericum, quem in dextro

membro

mulae (5) huic expressioni adjunximus, continebit. Nam si

verae sunt formulae (1), et ta>o, derivata n;ti ordinis

ha-

bebit terminum:

Γ(η⁺1)

+ + I« Η +Ia[r(u+l)]i& +ijS+....+ tb ....

*

1 /dn+iy\ä-1/day\ib +tl

'

7^«)...

' \äxr+i)"\dx*J

ubi brevitatis caussa praetermisimus factores, quos, eadem

forma manente, habet dextrum membrum formulae (5) sub signo J eontentos. Quo termino differentiato,

neglectisque

tenninis, qui, praeter factorem numericum, non erunt

aequales

(n+l)1ur(w+l)

Eodem modo, si 1«>o, e termino:

Γ(η+1)

pT(»+'2)]ie+ ia~l +···+ ιβ[Γ(»+

1)]«6+'ι^'+ 1 + ···+ ib

1

fda + lz\a'lfdnz + 1

.,.r(fCt)..,r(tß⁺2}

" \d^~0'' \dxn) ^''

colligeunis terminum:

(n+1) tctr(n-\-l)

• ?

si _,a>o, e termino:

Γ(η+1)

[F(n+2)]iw+ i«+^··· +,a-l JJt6+ ,β +... + tb+ 1 ^#

1

fdn +

'

...r(,a) ^...r(,b+2)

~'\dx"

···'

terminum:

(w+1) tar(w+l)

.... ?

si _jC in expressione (i>) est exponens derivalae ^{-—- ? ty} expo- d:cn_1

, ,

, d11"1ζ ^. dnIi;

nens denvatae ——,...,c exponens denvatae ^, et ,6>o.

d*0"1 rfo"1 '

e termino:

F(n+1)

...[Γ^+Ι)!6'1 + .!* +···+ |b [r(w)]rc+ * +,y+...+ ,c....

1 /dtty\ih-4 /dn-1^\ic+ l

'

...

r(,&)...r(lC+2).T7*·*J '''ydx^J ""

terminum:

ηtbr(n+l)

....,

si _,ß>o, iisdem manentibus ,c, ,y,...,c, e termino:

r(n+i)

.. . [F(n+l)i&+ ti3-1 + ···+ ib r(n)\c +»y+ i +...+

t /dnz\ti9"1 /da-iz\>y+i

7

terminum:

ηχβΓ(η+1)_'

·■·.··?

si ib>o, iisdem manentibus l^j.·· »c? e termino:

Γ(η+1)

...[r(w+l)]»6+iß+· ··+ib-4 [r(»i)]ic+(y+·..+ »«+

4

1

(ίΙαν\ι^~1

dn"xv\vc

4

'

...r(1b)...r(1c+2)...."\d^/ "'\άχη~ν

terminum:

ηjb^it+i)

....j

si xk>0) e termino:

Γ(η+1)

....[Γ(3)]ί*-4 +ι*+···+ib[r(2)Jiz"4+ ιλ +···+ι1

*

ι

f49y\k~l

dy\i^

*

'

...

Γ(1Α)...Γ(1/+2)..."'\^"ν \dxj

terminum:

2 tkr(n+i)

.... ·

si tK>o9 e termino:

r(n+i)

.,..[Γ(3)]^+ ^4 +... +«b[r(2)]^+ ,λ+ i+...+ ,1

·

1 /ίΡζ\ιχ~1 /ί/ζ\ιλ+4

....r(tk) ^,.. (t^+2)^...

*" \dx*J

8

terminum:

2t*r(n+l)

....,

si jli> ö, é terminö:

....[Γ(5)]Λ +

> +

+ + ti +

+11 +1

1 /(Vv\t^~l (diΛι1+1

*

....r(tk)... r(t

1+2)..." * \dx*l ''' \dxj

terminum:

2tltr(n+l)

'L— ■-- · '■1 · · · ·

y

···

si ii>o, e termiuo:

r(«+i) i

(

ιd^J

d.'-«+ .'+··

+

terminum: 'ty' dz'...dv'

in- y ' '

7

si ιλ>ο, e termino:

Γ(η+1)

*

....[Γ(2)]ιζ+ιλ-1

+ '1 * ""

■+■,*·»t

u

. it- iT-l -tlt

terminum: ay az .,*at

χλr(n+i)

•— ... ·;