Disquisitiones generales circa seriem infinitam ...

by Carolus Fridericus Gauss

Carolus Fridericus Gauss

Disquisitiones generales circa seriem infinitam $1+{\tfrac {\alpha \beta }{1.\gamma }}x+{\tfrac {\alpha (\alpha +1)\beta (\beta +1)}{1.2.\gamma (\gamma +1)}}xx+{\tfrac {\alpha (\alpha +1)(\alpha +2)\beta (\beta +1)(\beta +2)}{1.2.3.\gamma (\gamma +1)(\gamma +2)}}x^{3}+{\text{ etc.}}$

1814

editio: ex Carl Friedrich Gauss Werke, Band III, K. Gesellschaft der Wissenschaften zu Göttingen, 1876
fons: librum vide

INTRODUCTIO.

1.

Series, quam in hac commentatione perscrutari suscipimus, tamquam functio quatuor quantitatum $\alpha ,$ $\beta ,$ $\gamma ,$ $x$ spectari potest, quas ipsius elementa vocabimus, ordine suo elementum primum $\alpha ,$ secundum $\beta ,$ tertium $\gamma ,$ quartum $x$ distinguentes. Manifesto elementum primum cum secundo permutare licet: quodsi itaque brevitatis caussa seriem nostram hoc signo $F(\alpha ,\beta ,\gamma ,x)$ denotamus, habebimus $F(\alpha ,\beta ,\gamma ,x)=F(\beta ,\alpha ,\gamma ,x).$

2.

Tribuendo elementis $\alpha ,$ $\beta ,$ $\gamma$ valores determinatus, series nostra in functionem unicae variabilis $x$ transit, quae manifesto post terminum $1-\alpha ^{tum}$ vel $1-\beta ^{tum}$ abrumpitur, si $\alpha -1$ vel $\beta -1$ est numerus integer negativus, in casibus reliquis vero in infinitum excurrit. In casu priori series exhibet functionem algebraicam rationalem, in posteriori autem plerumque functionem transscendentem. Elementum tertium $\gamma$ debet esse neque numerus negativus integer neque $=0,$ ne ad terminos infinite magnos delabamur.

3.

Coëfficientes potestatum $x^{m},$ $x^{m+1}$ in serie nostra sunt ut

$1+{\tfrac {\gamma +1}{m}}+{\tfrac {\gamma }{mm}}:1+{\tfrac {\alpha +\beta }{m}}+{\tfrac {\alpha \beta }{mm}}$

adeoque ad rationem aequalitatis eo magis accedunt, quo maior assumitur $m$. Si itaque etiam elemento quarto $x$ valor determinatus tribuitur, ab huius indole convergentia seu divergentia pendebit. Quoties scilicet ipsi $x$ tribuitur valor realis, positivus seu negativus, unitate minor, series certo, si non statim ab initio, tamen post certum intervallum, convergens erit, atque ad summam finitam ex asse determinatam perducet. Idem eveniet per valorem imaginarium ipsius $x$ formae $a+b{\sqrt {-1}}$, quoties $aa+bb<1$. Contra pro valore ipsius $x$ reali unitateque maiori, vel pro imaginario formae $a+b{\sqrt {-1}}$, quoties $aa+bb>1$, series si non statim tamen post certum intervallum necessario divergens erit, ita ut de ipsius summa sermo esse nequeat. Denique pro valore $x=1$ (seu generalius pro valore formae $a+b{\sqrt {-1}}$, quoties $aa+bb=1$) seriei convergentia seu divergentia ab ipsarum $\alpha ,$ $\beta ,$ $\gamma$ indole pendebit, de qua, atque in specie de summa seriei pro $x=1$, in Sect. tertia loquemur.

Patet itaque, quatenus functio nostra tamquam summa seriei definita sit, disquisitionem natura sua restrictam esse ad casus eos, ubi series revera convergat, adeoque quaestionem ineptam esse, quinam sit valor seriei pro valore ipsius $x$ unitate maiori. Infra autem, inde a Sectione quarta, functionem nostram altiori principio superstruemus, quod applicationem generalissimam patiatur.

4.

Differentiatio seriei nostrae, considerando solum elementum quartum $x$ tamquam variabile, ad functionem similem perducit, quum manifesto habeatur

${\tfrac {\operatorname {d} F(\alpha ,\beta ,\gamma ,x)}{\operatorname {d} x}}={\tfrac {\alpha \beta }{\gamma }}F(\alpha +1,\beta +1,\gamma +1,x)$

Idem valet de differentiationibus repetitis.

5.

Operae pretium erit, quasdam functiones, quas ad seriem nostram reducere licet, quarumque usus in tota analysi est frequentissimus, hic apponere. I. $(t+u)^{n}=t^{u}F(-n,\beta ,\beta ,-{\tfrac {u}{t}})$
ubi elementum $\beta$ est arbitrarium.

II. $(t+u)^{n}+(t-u)^{n}=2t^{n}F(-{\tfrac {1}{2}}n,-{\tfrac {1}{2}}n+{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {uu}{tt}})$

III. $(t+u)^{n}+t^{n}=2t^{n}F(-n,\omega ,2\omega ,-{\tfrac {u}{t}})$
denotante $\omega$ quantitatem infinite parvam.

IV. $(t+u)^{n}-(t-u)^{n}=2nt^{n-1}uF(-{\tfrac {1}{2}}n+{\tfrac {1}{2}},-{\tfrac {1}{2}}n+1,{\tfrac {3}{2}},{\tfrac {uu}{tt}})$

V. $(t+u)^{n}-t^{n}=nt^{n-1}uF(1-n,1,2,-{\tfrac {u}{t}})$

VI. $\log(1+t)=tF(1,1,2,-t)$

VII. $\log {\tfrac {1+t}{1-t}}=2tF({\tfrac {1}{2}},1,{\tfrac {3}{2}},tt)$

VIII. $e^{t}=F(1,k,1,{\tfrac {t}{k}})=1+tF(1,k,2,{\tfrac {t}{k}})=1+t+{\tfrac {1}{2}}ttF(1,k,3,{\tfrac {t}{k}})$ etc.
denotante $e$ basin logarithmorum hyperbolicorum, $k$ numerum infinite magnum.

IX. $e^{t}+e^{-t}=2{F}(k,k^{\prime },{\tfrac {1}{2}},{\tfrac {tt}{4kk^{\prime }}})$
denotantibus $k,k^{\prime }$ numeros infinite magnos.

X. $e^{t}-e^{-t}=2tF(k,k^{\prime },{\tfrac {3}{2}},{\tfrac {tt}{4kk^{\prime }}})$

XI. $\sin t=tF(k,k^{\prime },{\tfrac {3}{2}},-{\tfrac {tt}{4kk^{\prime }}})$

XII. $\cos t=F(k,k^{\prime },{\tfrac {1}{2}},-{\tfrac {tt}{4kk^{\prime }}})$

XIII. $t=\sin t.F({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {3}{2}},{\sin t}^{2})$

XIV. $t=\sin t.\cos t.F(1,1,{\tfrac {3}{2}},{\sin t}^{2})$

XV. $t=\operatorname {tang} t.F({\tfrac {1}{2}},1,{\tfrac {3}{2}},-{\operatorname {tang} t}^{2})$

XVI. $\sin nt=n\sin t.F({\tfrac {1}{2}}n+{\tfrac {1}{2}},-{\tfrac {1}{2}}n+{\tfrac {1}{2}},{\tfrac {3}{2}},{\sin t}^{2})$

XVII. $\sin nt=n\sin t.\cos t.F({\tfrac {1}{2}}n+1,-{\tfrac {1}{2}}n+1,{\tfrac {3}{2}},{\sin t}^{2})$

XVIII. $\sin nt=n\sin t.{\cos t}^{n-1}F(-{\tfrac {1}{2}}n+1,-{\tfrac {1}{2}}n+{\tfrac {1}{2}},{\tfrac {3}{2}},-{\operatorname {tang} t}^{2})$

XIX. $\sin nt=n\sin t.{\cos t}^{-n-1}F({\tfrac {1}{2}}n+1,{\tfrac {1}{2}}n+{\tfrac {1}{2}},{\tfrac {3}{2}},-{\operatorname {tang} t}^{2})$

XX. $\cos nt=F({\tfrac {1}{2}}n,-{\tfrac {1}{2}}n,{\tfrac {1}{2}},{\sin t}^{2})$

XXI. $\cos nt=\cos t.F({\tfrac {1}{2}}n+{\tfrac {1}{2}},-{\tfrac {1}{2}}n+{\tfrac {1}{2}},{\tfrac {1}{2}},{\sin t}^{2})$

XXII. $\cos nt={\cos t}^{n}F(-{\tfrac {1}{2}}n,-{\tfrac {1}{2}}n+{\tfrac {1}{2}},{\tfrac {1}{2}},-{\operatorname {tang} t}^{2})$

XXIII. $\cos nt={\cos t}^{-n}F({\tfrac {1}{2}}n+{\tfrac {1}{2}},{\tfrac {1}{2}}n,{\tfrac {1}{2}},-{\operatorname {tang} t}^{2})$

6.

Functiones praecedentes sunt algebraicae atque transscendentes a logarithmis circuloque pendentes. Neutiquam vero harum caussa disquisitionem nostram generalem suscipimus, sed potius in gratiam theoriae functionum transscendentium altiorum promovendae, quarum genus amplissimum series nostra complectitur. Huc, inter infinita alia, pertinent coëfficientes ex evolutione functionis $(aa+bb-2ab\cos \varphi )^{-n}$ in seriem secundum cosinus angulorum $\varphi ,2\varphi ,3\varphi$ etc. progredientem orti, de quibus in specie alia occasione fusius agemus. Ad formam seriei nostrae autem illi coëfficientes pluribus modis reduci possunt. Scilicet statuendo

$(aa+bb-2ab\cos \varphi )^{-n}=\Omega =A+2A^{\prime }\cos \varphi +2A^{\prime \prime }\cos 2\varphi +2A^{\prime \prime \prime }\cos 3\varphi +$ etc.

habemus primo

${\begin{alignedat}{2}&A&&=a^{-2n}F(n,n,1,{\tfrac {bb}{aa}})\\&A^{\prime }&&=na^{-2n-1}bF(n,n+1,2,{\tfrac {bb}{aa}})\\&A^{\prime \prime }&&={\tfrac {n(n+1)}{1.2}}a^{-2n-2}bbF(n,n+2,3,{\tfrac {bb}{aa}})\\&A^{\prime \prime \prime }&&={\tfrac {n(n+1)(n+2)}{1.2.3}}a^{-2n-3}b^{3}F(n,n+3,4,{\tfrac {bb}{aa}})\\\end{alignedat}}$

etc.

Si enim $aa+bb-2ab\cos \varphi$ consideratur tamquam productum ex $a-br$ in $a-br^{-1}$ (designante $r$ quantitatem $\cos \varphi +\sin \varphi .{\sqrt {-1}})$, fit $\Omega$ aequalis producto

ex $a^{-2n}$

in $1+n{\tfrac {br}{a}}+{\tfrac {n(n+1)}{1.2}}\cdot {\tfrac {bbrr}{aa}}+{\tfrac {n(n+1)(n+2)}{1.2.3}}\cdot {\tfrac {b^{3}r^{3}}{a^{3}}}+$ etc.

in $1+n{\tfrac {br^{-1}}{a}}+{\tfrac {n(n+1)}{1.2}}\cdot {\tfrac {bbr^{-2}}{aa}}+{\tfrac {n(n+1)(n+2)}{1.2.3}}\cdot {\tfrac {b^{3}r^{-3}}{a^{3}}}+$ etc.

Quod productum quum identicum esse debeat cum

$A+A^{\prime }(r+r^{-1})+A^{\prime \prime }(rr+r^{-2})+A^{\prime \prime \prime }(r^{3}+r^{-3})$

valores supra dati sponte prodeunt.

Porro habemus secundo

${\begin{aligned}&A=(aa+bb)^{-n}F({\tfrac {1}{2}}n,{\tfrac {1}{2}}n+{\tfrac {1}{2}},1,{\tfrac {4aabb}{(aa+bb)^{2}}})\\&A^{\prime }=n(aa+bb)^{-n-1}abF({\tfrac {1}{2}}n+{\tfrac {1}{2}},{\tfrac {1}{2}}n+1,2,{\tfrac {4aabb}{(aa+bb)^{2}}})\\&A^{\prime \prime }={\tfrac {n(n+1)}{1.2}}(aa+bb)^{-n-2}aabbF({\tfrac {1}{2}}n+1,{\tfrac {1}{2}}n+{\tfrac {3}{2}},3,{\tfrac {4aabb}{(aa+bb)^{2}}})\\&A^{\prime \prime \prime }={\tfrac {n(n+1)(n+2)}{1.2.3}}(aa+bb)^{-n-3}a^{3}b^{3}F({\tfrac {1}{2}}n+{\tfrac {3}{2}},{\tfrac {1}{2}}n+2,4,{\tfrac {4aabb}{(aa+bb)^{2}}})\\\end{aligned}}$

etc.

qui valores facile deducuntur ex

$\Omega (aa+bb)^{n}=1+n(r+r^{-1}){\tfrac {ab}{aa+bb}}+{\tfrac {n(n+1)}{1.2}}(r+r^{-1})^{2}{\tfrac {aabb}{(aa+bb)^{2}}}+$ etc.

Tertio fit

${\begin{alignedat}{2}&A&&=(a+b)^{-2n}F(n,{\tfrac {1}{2}},1,{\tfrac {4ab}{(a+b)^{2}}})\\&A^{\prime }&&=n(a+b)^{-2n-2}abF(n+1,{\tfrac {3}{2}},3,{\tfrac {4ab}{(a+b)^{2}}})\\&A^{\prime \prime }&&={\tfrac {n(n+1)}{1.2}}(a+b)^{-2n-4}aabbF(n+2,{\tfrac {5}{2}},5,{\tfrac {4ab}{(a+b)^{2}}})\\&A^{\prime \prime \prime }&&={\tfrac {n(n+1)(n+2)}{1.2.3}}(a+b)^{-2n-6}a^{3}b^{3}F(n+3,{\tfrac {7}{2}},7,{\tfrac {4ab}{(a+b)^{2}}})\\\end{alignedat}}$

etc.

Denique fit quarto

${\begin{alignedat}{2}&A&&=(a-b)^{-2n}F(n,{\tfrac {1}{2}},1,-{\tfrac {4ab}{(a-b)^{2}}})\\&A^{\prime }&&=n(a-b)^{-2n-2}abF(n+1,{\tfrac {3}{2}},3,-{\tfrac {4ab}{(a-b)^{2}}})\\&A^{\prime \prime }&&={\tfrac {n(n+1)}{1.2}}(a-b)^{-2n-4}aabbF(n+2,{\tfrac {5}{2}},5,-{\tfrac {4ab}{(a-b)^{2}}})\\&A^{\prime \prime \prime }&&={\tfrac {n(n+1)(n+2)}{1.2.3}}(a-b)^{2n-6}a^{3}b^{3}F(n+3,{\tfrac {7}{2}},7,-{\tfrac {4ab}{(a-b)^{2}}})\end{alignedat}}$

etc.

Valores illi atque hi facile eruuntur ex

${\begin{aligned}\Omega (a+b)^{2n}&=(1-{\tfrac {4ab\cos {\frac {1}{2}}\varphi ^{2}}{(a+b)^{2}}})^{-n}\\&=1+n{\tfrac {ab}{(a+b)^{2}}}(r^{\frac {1}{2}}+r^{-{\frac {1}{2}}})^{2}+{\tfrac {n(n+1)}{1.2}}\cdot {\tfrac {aabb}{(a+b)^{2}}}(r^{\frac {1}{2}}+r^{-{\frac {1}{2}}})^{2}+{\text{etc. }}\\\Omega (a-b)^{2n}&=(1+{\tfrac {4ab\sin {\frac {1}{2}}\varphi ^{2}}{(a-b)^{2}}})^{-n}\\&=1+n{\tfrac {ab}{(a-b)^{2}}}(r^{\frac {1}{2}}-r^{-{\frac {1}{2}}})^{2}+{\tfrac {n(n+1)}{1.2}}\cdot {\tfrac {aabb}{(a-b)^{2}}}(r^{\frac {1}{2}}-r^{-{\frac {1}{2}}})^{4}+{\text{etc. }}\end{aligned}}$

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SECTIO PRIMA.
Relationes inter functiones contiguas.

7.

Functionem ipsi $F^{\prime }(\alpha ,\beta ,\gamma ,x)$ contiguam vocamus, quae ex illa oritur, dum elementum primum, secundum, vel tertium unitate vel augetur vel diminuitur, manentibus tribus reliquis elementis. Functio itaque primaria $F(\alpha .\beta ,\gamma ,x)$ sex contiguas suppeditat, inter quarum binas ipsamque primariam aequatio persimplex linearis datur. Has aequationes, numero quindecim, hic in conspectum producimus, brevitatis gratia elementum quartum quod semper subintelligitur $=x$ omittentes, functionemque primariam simpliciter per $F$ denotantes.

[1]	${\textstyle 0=(\gamma -2\alpha -(\beta -\alpha )x)F+\alpha (1-x)F(\alpha +1,\beta ,\gamma )-(\gamma -\alpha )F(\alpha -1,\beta ,\gamma )}$
[2]	${\textstyle 0=(\beta -\alpha )F+\alpha F(\alpha +1,\beta ,\gamma )-\beta F(\alpha ,\beta +1,\gamma )}$
[3]	${\textstyle 0=(\gamma -\alpha -\beta )F+\alpha (1-x)F(\alpha +1,\beta ,\gamma )-(\gamma -\beta )F(\alpha ,\beta -1,\gamma )}$
[4]	${\textstyle 0=\gamma (\alpha -(\gamma -\beta )x)F-\alpha \gamma (1-x)F(\alpha +1,\beta ,\gamma )+(\gamma -\alpha )(\gamma -\beta )xF(\alpha ,\beta ,\gamma +1)}$
[5]	${\textstyle 0=(\gamma -\alpha -1)F+\alpha F(\alpha +1,\beta ,\gamma )-(\gamma -1)F(\alpha ,\beta ,\gamma -1)}$
[6]	${\textstyle 0=(\gamma -\alpha -\beta )F-(\gamma -\alpha )F(\alpha -1,\beta ,\gamma )+\beta (1-x)F(\alpha ,\beta +1,\gamma )}$
[7]	${\textstyle 0=(\beta -\alpha )(1-x)F-(\gamma -\alpha )F(\alpha -1,\beta ,\gamma )+(\gamma -\beta )F(\alpha ,\beta -1,\gamma )}$
[8]	${\textstyle 0=\gamma (1-x)F-\gamma F(\alpha -1,\beta ,\gamma )+(\gamma -\beta )xF(\alpha ,\beta ,\gamma +1)}$
[9]	${\textstyle 0=(\alpha -1-(\gamma -\beta -1)x)F+(\gamma -\alpha )F(\alpha -1,\beta ,\gamma )-(\gamma -1)(1-x)F(\alpha ,\beta ,\gamma -1)}$
[10]	${\textstyle 0=(\gamma -2\beta +(\beta -\alpha )x)F+\beta (1-x)F(\alpha ,\beta +1,\gamma )-(\gamma -\beta )F(\alpha ,\beta -1,\gamma )}$
[11]	${\textstyle 0=\gamma (\beta -(\gamma -\alpha )x)F-\beta \gamma (1-x)F(\alpha ,\beta +1,\gamma )-(\gamma -\alpha )(\gamma -\beta )F(\alpha ,\beta ,\gamma +1)}$
[12]	${\textstyle 0=(\gamma -\beta -1)F+\beta F(\alpha ,\beta +1,\gamma )-(\gamma -1)F(\alpha ,\beta ,\gamma -1)}$
[13]	${\textstyle 0=\gamma (1-x)F-\gamma F(\alpha ,\beta -1,\gamma )+(\gamma -\alpha )xF(\alpha ,\beta ,\gamma +1)}$
[14]	${\textstyle 0=(\beta -1-(\gamma -\alpha -1)x)F+(\gamma -\beta )F(\alpha ,\beta -1,\gamma )-(\gamma -1)(1-x)F(\alpha ,\beta ,\gamma -1)}$
[15]	${\textstyle {\begin{aligned}0=\gamma (\gamma -1-(2\gamma -\alpha -\beta -1)x)F+(\gamma -\alpha )(\gamma -\beta )xF(\alpha ,\beta ,\gamma +1)&\\-\gamma (\gamma -1)(1-x){F}(\alpha ,\beta ,\gamma -1)&\end{aligned}}}$

8.

Ecce iam demonstrationem harum formularum. Statuendo

${\frac {(\alpha +1)(\alpha +2)\ldots (\alpha +m-1)\beta (\beta +1)\ldots (\beta +m-2)}{1.2.3.\ldots .m.\gamma (\gamma +1).\ldots .(\gamma +m-1)}}=M$

erit coëfficiens potestatis $x^{m}$

in $F\;\;.\quad .\quad .\quad .\quad$	$.\quad .\quad \alpha (\beta +m-1)M$
in $F(\alpha ,\beta -1,\gamma )\quad$	$.\quad .\quad \alpha (\beta -1)M$
in $F(\alpha +1,\beta ,\gamma )\quad$	$.\quad .\quad (\alpha +m)(\beta +m-1)M$
in $F(\alpha ,\beta ,\gamma -1)\quad$	$.\quad .\quad {\frac {\alpha (\beta +m-1)(\gamma +m-1)M}{\gamma -1}}$

coëfficiens autem potestatis $x^{m-1}$ in $F(\alpha +1,\beta ,\gamma ),$ seu coëfficiens potestatis $x^{m}$ in $xF(\alpha +1,\beta ,\gamma )$

$=m(\gamma +m-1)M$

Hinc statim demanat veritas formularum 5 et 3; permutando $\alpha$ cum $\beta ,$ oritur ex 5 formula 12, atque ex his duabus per eliminationem 2. Perinde per eandem permutationem ex 3 oritur 6; ex 6 et 12 combinatis oritur 9, hinc per permutationem 14, quibus combinatis habetur 7; denique ex 2 et 6 eruitur 1, atque hinc permutando 10. Formula 8 simili modo ut supra formulae 5 et 3, e consideratione coëfficientium derivari potest (eodemque modo, si placeret, omnes 15 formulae erui possent), vel elegantius ex iam notis sequenti modo. Mutando in formula 5 elementum $\alpha$ in $\alpha -1,$ atque $\gamma$ in $\gamma +1,$ prodit

$0=(\gamma -\alpha +1)F(\alpha -1,\beta ,\gamma +1)+(\alpha -1)F(\alpha ,\beta ,\gamma +1)-\gamma F(\alpha -1,\beta ,\gamma )$

Mutando vero in formula 9 tantummodo $\gamma$ in $\gamma +1,$ fit

$0=(\alpha -1-(\gamma -\beta )x)F(\alpha ,\beta ,\gamma +1)+(\gamma -\alpha +1)F(\alpha -1,\beta ,\gamma +1)-\gamma (1-x)F(\alpha ,\beta ,\gamma )$

E subtractione harum formularum statim oritur 8, atque hinc per permutationem 13. Ex 1 et 8 prodit 4, hincque permutando 11. Denique ex 8 et 9 deducitur 15.

9.

Si $\alpha ^{\prime }-\alpha ,$ $\beta ^{\prime }-\beta ,$ $\gamma ^{\prime }-\gamma ,$ nec non $\alpha ^{\prime \prime }-\alpha ,$ $\beta ^{\prime \prime }-\beta ,$ $\gamma ^{\prime \prime }-\gamma$ sunt numeri integri (positivi seu negativi), a functione $F(\alpha ,\beta ,\gamma )$ ad functionem $F(\alpha ^{\prime },\beta ^{\prime },\gamma ^{\prime }),$ et perinde ab hac usque ad functionem $F(\alpha ^{\prime \prime },\beta ^{\prime \prime },\gamma ^{\prime \prime })$ transire licet per seriem similium functionum, ita ut quaelibet contigua sit antecedenti et consequenti, mutando scilicet primo elementum unum e.g. $\alpha$ continuo unitate, donec a $F(\alpha ,\beta ,\gamma )$ perventum sit ad $F(\alpha ^{\prime },\beta ,\gamma ),$ dein mutando elementum secundum, donec perventum sit ad $F(\alpha ^{\prime },\beta ^{\prime },\gamma ),$ denique mutando elementum tertium, donec perventum sit ad $F(\alpha ^{\prime },\beta ^{\prime },\gamma ^{\prime }),$ et perinde ab hac usque ad $F(\alpha ^{\prime \prime },\beta ^{\prime \prime },\gamma ^{\prime \prime }).$ Quum itaque per art. 7 habeantur aequationes lineares inter functionem primam, secundam atque tertiam, et generaliter inter ternas quascunque consequentes huius seriei, facile perspicitur, hinc per eliminationem deduci posse aequationem linearem inter functiones $F(\alpha ,\beta ,\gamma ),$ $F(\alpha ^{\prime },\beta ^{\prime },\gamma ^{\prime }),$ $F(\alpha ^{\prime \prime },\beta ^{\prime \prime },\gamma ^{\prime \prime }),$ ita ut generaliter loquendo e duabus functionibus, quarum tria elementa prima numeris integris differunt, quamlibet aliam functionem eadem proprietate gaudentem derivare liceat, siquidem elementum quartum idem maneat. Ceterum hic nobis sufficit, hanc veritatem insignem generaliter stabilivisse, neque hic compendiis immoramur, per quae operationes ad hunc finem necessariae quam brevissimae reddantur.

10.

Propositae sint e.g. functiones

$F(\alpha ,\beta ,\gamma ),F(\alpha +1,\beta +1,\gamma +1),F(\alpha +2,\beta +2,\gamma +2)$

inter quas aequationem linearem invenire oporteat. Iungamus ipsas per functiones contiguas sequenti modo:

${\begin{aligned}&F(\alpha ,\beta ,\gamma )=F\\&F(\alpha +1,\beta ,\gamma )=F^{\prime }\\&F(\alpha +1,\beta +1,\gamma )=F^{\prime \prime }\\&F(\alpha +1,\beta +1,\gamma +1)=F^{\prime \prime \prime }\\&F(\alpha +2,\beta +1,\gamma +1)=F^{\prime \prime \prime \prime }\\&F(\alpha +2,\beta +2,\gamma +1)=F^{\prime \prime \prime \prime \prime }\\&F(\alpha +2,\beta +2,\gamma +2)=F^{\prime \prime \prime \prime \prime \prime }\end{aligned}}$

Habemus itaque quinque aequationes lineares (e formulis 6, 13, 5 art. 7):

I.	$0=(\gamma -\alpha -1)F-(\gamma -\alpha -1-\beta )F^{\prime }-\beta (1-x)F^{\prime \prime }$
II.	$0=\gamma F^{\prime }-\gamma (1-x)F^{\prime \prime }-(\gamma -\alpha -1)xF^{\prime \prime \prime }$
III.	$0=\gamma {F}^{\prime \prime }-(\gamma -\alpha -1){F}^{\prime \prime \prime }-(\alpha +1){F}^{\prime \prime \prime \prime }$
IV.	$0=(\gamma -\alpha -1){F}^{\prime \prime \prime }-(\gamma -\alpha -2-\beta ){F}^{\prime \prime \prime \prime }-(\beta +1)(1-x){F}^{\prime \prime \prime \prime \prime }$
V.	$0=(\gamma +1){F}^{\prime \prime \prime \prime }-(\gamma +1)(1-x){F}^{\prime \prime \prime \prime \prime }-(\gamma -\alpha -1)x{F}^{\prime \prime \prime \prime \prime }$

Ex I et II prodit, eliminando $F^{\prime }$

VI. ${\phantom {\text{II}}}0=\gamma {F}-\gamma (1-x){F}^{\prime \prime }-(\gamma -\alpha -\beta -1)x{F}^{\prime \prime \prime }$

Hinc atque ex III, eliminando ${F}^{\prime \prime }$

VII. ${\phantom {\text{I}}}0=\gamma {F}-(\gamma -\alpha -1-\beta x){F}^{\prime \prime \prime }-(\alpha +1)(1-x){F}^{\prime \prime \prime \prime }$

Porro ex IV atque V, eliminando ${F}^{\prime \prime \prime \prime \prime }$

VIII. $0=(\gamma +1){F}^{\prime \prime \prime }-(\gamma +1){F}^{\prime \prime \prime \prime }+(\beta +1)x{F}^{\prime \prime \prime \prime \prime \prime }$

Hinc atque ex VII, eliminando ${F}^{\prime \prime \prime \prime },$

IX. ${\phantom {\text{II}}}0=\gamma (\gamma +1)F-(\gamma +1)(\gamma -(\alpha +\beta +1)x){F}^{\prime \prime \prime }-(\alpha +1)(\beta +1)x(1-x){F}^{\prime \prime \prime \prime \prime \prime }$

11.

Si omnes relationes inter ternas functiones ${F}(\alpha ,\beta ,\gamma ),$ ${F}(\alpha +\lambda ,\beta +\mu ,\gamma +\nu ),$ $F(\alpha +\lambda ^{\prime },\beta +\mu ^{\prime },\gamma +\nu ^{\prime }),$ in quibus $\lambda ,$ $\mu ,$ $\nu ,$ $\lambda ^{\prime },$ $\mu ^{\prime },$ $\nu ^{\prime }$ vel $=0$ vel $=+1$ vel $=-1,$ exhaurire vellemus, formularum multitudo usque ad 325 ascenderet. Haud inutilis foret talis collectio, saltem simpliciorum ex his formulis: hoc vero loco sufficiat, paucas tantummodo apposuisse, quas vel ex formulis art. 7, vel si magis placet, simili modo ut duae priores ex illis in art. 8 erutae sunt, quivis nullo negotio sibi demonstrare poterit.

[16]	$F(\alpha ,\beta ,\gamma )-F(\alpha ,\beta ,\gamma -1)$	$=-{\frac {\alpha \beta x}{\gamma (\gamma -1)}}F(\alpha +1,\beta +1,\gamma +1)$
[17]	$F(\alpha ,\beta +1,\gamma )-F(\alpha ,\beta ,\gamma )$	$={\frac {\alpha x}{\gamma }}F(\alpha +1,\beta +1,\gamma +1)$
[18]	$F(\alpha +1,\beta ,\gamma )-F(\alpha ,\beta ,\gamma )$	$={\frac {\beta x}{\gamma }}F(\alpha +1,\beta +1,\gamma +1)$
[19]	$F(\alpha ,\beta +1,\gamma +1)-F(\alpha ,\beta ,\gamma )$	$={\frac {\alpha (\gamma -\beta )x}{\gamma (\gamma +1)}}F(\alpha +1,\beta +1,\gamma +2)$
[20]	$F(\alpha +1,\beta ,\gamma +1)-F(\alpha ,\beta ,\gamma )$	$={\frac {\beta (\gamma -\alpha )x}{\gamma (\gamma +1)}}F(\alpha +1,\beta +1,\gamma +2)$
[21]	$F(\alpha -1,\beta +1,\gamma )-F(\alpha ,\beta ,\gamma )$	$={\frac {(\alpha -\beta -1)x}{\gamma }}F(\alpha ,\beta +1,\gamma +1)$
[22]	$F(\alpha +1,\beta -1,\gamma )-F(\alpha ,\beta ,\gamma )$	$={\frac {(\beta -\alpha -1)x}{\gamma }}F(\alpha +1,\beta ,\gamma +1)$
[23]	$F(\alpha -1,\beta +1,\gamma )-F(\alpha +1,\beta -1,\gamma )$	$={\frac {(\alpha -\beta )x}{\gamma }}F(\alpha +1,\beta +1,\gamma +1)$

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SECTIO SECUNDA.
Fractiones continuae.

12.

Designando

${\frac {F(\alpha ,\beta +1,\gamma +1,x)}{F(\alpha ,\beta ,\gamma ,x)}}\;$ per $\;G(\alpha ,\beta ,\gamma ,x)$

fit

${\frac {F(\alpha +1,\beta ,\gamma +1,x)}{F(\alpha ,\beta ,\gamma ,x)}}={\frac {F(\beta ,\alpha +1,\gamma +1,x)}{F(\beta ,\alpha ,\gamma ,x)}}=G(\beta ,\alpha ,\gamma ,x)$

et proin, dividendo aequationem 19 per ${F}(\alpha ,\beta +1,\gamma +1,x),$

$1-{\frac {1}{G(\alpha ,\beta ,\gamma ,x)}}={\frac {\alpha (\gamma -\beta )}{\gamma (\gamma +1)}}xG(\beta +1,\alpha ,\gamma +1,x)$

sive

[24]	$G(\alpha ,\beta ,\gamma ,x)={\frac {1}{1-{\frac {\alpha (\gamma -\beta )}{\gamma (\gamma +1)}}xG(\beta +1,\alpha ,\gamma +1,x)}}$

et quum perinde fiat

$G(\beta +1,\alpha ,\gamma +1,x)={\frac {1}{1-{\frac {(\beta +1)(\gamma +1-\alpha )}{(\gamma +1)(\gamma +2)}}xG(\alpha +1,\beta +1,\gamma +2,x)}}$

etc., resultabit pro $G(\alpha ,\beta ,\gamma ,x)$ fractio continua

[25]	$\displaystyle {\frac {F(\alpha ,\beta +1,\gamma +1,x)}{F(\alpha ,\beta ,\gamma ,x)}}={\cfrac {1}{1-{\cfrac {ax}{1-{\cfrac {bx}{1-{\cfrac {cx}{1-{\cfrac {dx}{1-{\text{etc.}}}}}}}}}}}}$

ubi

${\begin{alignedat}{2}&a={\frac {\alpha (\gamma -\beta )}{\gamma (\gamma +1)}}&\qquad &b={\frac {(\beta +1)(\gamma +1-\alpha )}{(\gamma +1)(\gamma +2)}}\\&c={\frac {(\alpha +1)(\gamma +1-\beta )}{(\gamma +2)(\gamma +3)}}&\qquad &d={\frac {(\beta +2)(\gamma +2-\alpha )}{(\gamma +3)(\gamma +4)}}\\&e={\frac {(\alpha +2)(\gamma +2-\beta )}{(\gamma +4)(\gamma +5)}}&\qquad &f={\frac {(\beta +3)(\gamma +3-\alpha )}{(\gamma +5)(\gamma +6)}}\end{alignedat}}$

etc., cuius lex progressionis obvia est.

Porro ex aequationibus 17, 18, 21, 22 sequitur

[26]	${\frac {F(\alpha ,\beta +1,\gamma ,x)}{F(\alpha ,\beta ,\gamma ,x)}}={\frac {1}{1-{\frac {\alpha x}{\gamma }}G(\beta +1,\alpha ,\gamma ,x)}}$
[27]	${\frac {F(\alpha +1,\beta ,\gamma ,x)}{F(\alpha ,\beta ,\gamma ,x)}}={\frac {1}{1-{\frac {\beta x}{\gamma }}G(\alpha +1,\beta ,\gamma ,x)}}$
[28]	${\frac {F(\alpha -1,\beta +1,\gamma ,x)}{F(\alpha ,\beta ,\gamma ,x)}}={\frac {1}{1-{\frac {(\alpha -\beta -1)x}{\gamma }}G(\beta +1,\alpha -1,\gamma ,x)}}$
[29]	${\frac {F(\alpha +1,\beta -1,\gamma ,x)}{F(\alpha ,\beta ,\gamma ,x)}}={\frac {1}{1-{\frac {(\beta -\alpha -1)x}{\gamma }}G(\alpha +1,\beta -1,\gamma ,x)}}$

unde, substitutis pro functione $G$ eius valoribus in fractionibus continuis, totidem fractiones continuae novae prodeunt.

Ceterum sponte patet, fractionem continuam in formula 25 abrumpi, si e numeris $\alpha ,$ $\beta ,$ $\gamma -\alpha ,$ $\gamma -\beta$ aliquis fuerit integer negativus, alioquin vero in infinitum excurrere.

13.

Fractiones continuae in art. praec. erutae maximi sunt momenti, asserique potest, vix ullas fractiones continuas secundum legem obviam progredientes ab analystis hactenus erutas esse, quae sub nostris tamquam casus speciales non sint contentae. Imprimis memorabilis est casus is, ubi in formula 25 statuitur $\beta =0,$ unde $F(\alpha ,\beta ,\gamma ,x)=1,$ adeoque, scribendo $\gamma -1$ pro $\gamma$

[30]	$F(\alpha ,1,\gamma )=1+{\frac {\alpha }{\gamma }}x+{\frac {\alpha (\alpha +1)}{\gamma (\gamma +1)}}xx+{\frac {\alpha (\alpha +1)(\alpha +2)}{\gamma (\gamma +1)(\gamma +2)}}x^{3}+{\text{etc.}}$
	$={\cfrac {1}{1-{\cfrac {ax}{1-{\cfrac {bx}{1-{\cfrac {cx}{1-{\cfrac {dx}{1-{\text{etc.}}}}}}}}}}}}$

ubi

${\begin{alignedat}{2}&a={\frac {\alpha }{\gamma }}&&b={\frac {\gamma -\alpha }{\gamma (\gamma +1)}}\\&c={\frac {(\alpha +1)\gamma }{(\gamma +1)(\gamma +2)}}\qquad &&d={\frac {2(\gamma +1-a)}{(\gamma +2)(\gamma +3)}}\\&e={\frac {(\alpha +2)(\gamma +1)}{(\gamma +3)(\gamma +4)}}&&f={\frac {3(\gamma +2-\alpha )}{(\gamma +4)(\gamma +5)}}\\\end{alignedat}}$

etc.

14.

Operae pretium erit, quosdam casus speciales huc adscripsisse. Ex formula I art. 5 sequitur, statuendo $t=1,$ $\beta =1$

[31]	$(1+u)^{n}={\cfrac {1}{1-{\cfrac {nu}{1+{\cfrac {{\frac {n+1}{2}}u}{1-{\cfrac {{\frac {n-1}{2.3}}u}{1+{\cfrac {{\frac {2(n+2)}{3.4}}u}{1-{\cfrac {{\frac {2(n-2)}{4.5}}u}{1+{\text{etc.}}}}}}}}}}}}}}$

E formulis VI, VII art. 5 sequitur

[32]	$\log(1+t)={\cfrac {t}{1+{\cfrac {{\frac {1}{2}}_{}t}{1+{\cfrac {{\frac {1}{6}}_{}t}{1+{\cfrac {{\frac {2}{6}}_{}t}{1+{\cfrac {{\frac {2}{10}}_{}t}{1+{\cfrac {{\frac {3}{10}}_{}t}{1+{\cfrac {{\frac {3}{14}}_{}t}{1^{}+{\text{etc.}}}}}}}}}}}}}}}}$
[32]	$\log {\frac {1+t}{1-t}}={\cfrac {2t}{1-{\cfrac {{\frac {1}{3}}tt}{1-{\cfrac {{\frac {2.2}{3.5}}_{}tt}{1-{\cfrac {{\frac {3.3}{5.7}}_{}tt}{1-{\cfrac {{\frac {4.4}{7.9}}_{}tt}{1^{}-{\text{etc.}}}}}}}}}}}}$

Mutando hic signa $-$ in $+$ prodit fractio continua pro $\operatorname {arc.tang} t.$

Porro habemus

[34]	$e^{t}={\cfrac {1}{1-{\cfrac {t}{1+{\cfrac {{\frac {1}{3}}t}{1-{\cfrac {{\frac {1}{6}}t}{1+{\cfrac {{\frac {1}{6}}t}{1-{\cfrac {{\frac {1}{10}}t}{1+{\cfrac {{\frac {1}{10}}t}{1-{\text{etc.}}}}}}}}}}}}}}}}$

[35]	$t={\cfrac {\sin t\cos t}{1-{\cfrac {{\frac {1.2}{1.3}}{\sin t}^{2}}{1-{\cfrac {{\frac {1.2}{3.5}}{\sin t}^{2}}{1-{\cfrac {{\frac {3.4}{5.7}}{\sin t}^{2}}{1-{\cfrac {{\frac {3.4}{7.9}}{\sin t}^{2}}{1-{\cfrac {{\frac {5.6}{9.11}}{\sin t}^{2}}{1-{\text{etc.}}}}}}}}}}}}}}$

Statuendo $\alpha =3,$ $\gamma ={\tfrac {5}{2}},$ e formula 30 sponte sequitur fractio continua in Theoria motus corporum coelestium art. 90 proposita. Ibidem duae aliae fractiones continuae prolatae sunt, quarum evolutionem hacce occasione supplere visum est. Statuendo

$Q=1-{\cfrac {{\frac {5.8}{7.9}}x}{1-{\cfrac {{\frac {1.4}{9.11}}x}{1-{\frac {7.10}{11.13}}x{\text{ etc.}}}}}}$

fit l.c. $x-\xi ={\frac {x}{1+{\frac {2x}{35Q}}}}={\frac {xQ}{Q+{\frac {2}{35}}x}},$ adeoque

$\xi ={\frac {{\frac {2}{35}}xx}{Q+{\frac {2}{35}}x}}$

quae est formula prior: posterior sequenti modo eruitur. Statuendo

$R=1-{\cfrac {{\frac {1.4}{7.9}}x}{1-{\cfrac {{\frac {5.8}{9.11}}x}{1-{\cfrac {{\frac {3.6}{11.13}}x}{1-{\frac {7.10}{13.15}}x{\text{ etc.}}}}}}}}$

erit per formulam 25

${\frac {1}{R}}=G({\tfrac {1}{2}},{\tfrac {3}{2}},{\tfrac {7}{2}},x),\quad$ atque $\quad {\frac {1}{Q}}=G({\tfrac {5}{2}},-{\tfrac {1}{2}},{\tfrac {7}{2}},x)$

Hinc

${\begin{aligned}&RF({\tfrac {1}{2}},{\tfrac {5}{2}},{\tfrac {9}{2}},x)=F({\tfrac {1}{2}},{\tfrac {3}{2}},{\tfrac {7}{2}},x)\\&QF({\tfrac {5}{2}},{\tfrac {1}{2}},{\tfrac {9}{2}},x)=F({\tfrac {5}{2}},-{\tfrac {1}{2}},{\tfrac {7}{2}},x)\end{aligned}}$

sive permutando elementum primum cum secundo

$QF({\tfrac {1}{2}},{\tfrac {5}{2}},{\tfrac {9}{2}},x)=F(-{\tfrac {1}{2}},{\tfrac {5}{2}},{\tfrac {7}{2}},x)$

Sed per aequationem 21 habemus

$F(-{\tfrac {1}{2}},{\tfrac {5}{2}},{\tfrac {7}{2}},x)-F({\tfrac {1}{2}},{\tfrac {3}{2}},{\tfrac {7}{2}},x)=-{\tfrac {4}{7}}xF({\tfrac {1}{2}},{\tfrac {5}{2}},{\tfrac {9}{2}},x)$

unde fit $Q={R}-{\tfrac {4}{7}}x,$ quo valore in formula supra data substituto prodit

$\xi ={\frac {{\frac {2}{35}}xx}{R-{\frac {18}{35}}x}}$

quae est formula posterior.

Statuendo in formula 30, $\alpha ={\tfrac {m}{n}},$ $x=-\gamma nt,$ fit pro valore infinite magno ipsius $\gamma$

[36]	$F({\tfrac {m}{n}},1,\gamma ,-\gamma nt)=1-mt+m(m+n)tt-m(m+n)(m+2n)t^{3}+{\text{etc. }}$
	$\quad ={\cfrac {1}{1+{\cfrac {mt}{1+{\cfrac {nt}{1+{\cfrac {(m+n)t}{1+{\cfrac {2nt}{1+{\cfrac {(m+2n)t}{1+3nt{\text{ etc.}}}}}}}}}}}}}}$

---

SECTIO TERTIA.
De summa seriei nostrae statuendo elementum quartum $=1,$ ubi simul quaedam aliae functiones transscendentes discutiuntur.

15.

Quoties elementa $\alpha ,$ $\beta ,$ $\gamma$ omnia sunt quantitates positivae, omnes coëfficientes potestatum elementi quarti $x$ positivi evadunt: quoties vero ex illis elementis unum alterumve negativum est, saltem inde ab aliqua potestate $x^{m}$ omnes coëfficientes eodem signo affecti erunt, si modo $m$ accipitur maior quam valor absolutus elementi negativi maximi. Porro hinc sponte patet, seriei șummam pro $x=1$ finitam esse non posse, nisi coëfficientes saltem post certum terminum in infinitum decrescant, vel, ut secundum morem analystarum loquamur, nisi coëfficiens termini $x^{\infty }$ sit $=0.$ Ostendemus autem, et quidem, in gratiam eorum, qui methodis rigorosis antiquorum geometrarum favent, omni rigore,

primo, coëfficientes (siquidem series non abrumpatur), in infinitum crescere, quoties fuerit $\alpha +\beta -\gamma -1$ quantitas positiva.

secundo, coëfficientes versus limitem finitum continuo convergere, quoties fuerit $\alpha +\beta -\gamma -1=0.$

tertio, coëfficientes in infinitum decrescere, quoties fuerit $\alpha +\beta -\gamma -1$ quantitas negativa.

quarto, summam seriei nostrae pro $x=1,$ non obstante convergentia in casu tertio, infinitam esse, quoties fuerit $\alpha +\beta -\gamma$ quantitas positiva vel $=0.$

quinto, summam vero finitam esse, quoties $\alpha +\beta -\gamma$ fuerit quantitas negativa.

16.

Hanc disquisitionem generalius adaptabimus seriei infinitae $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc. ita formatae, ut quotientes ${\tfrac {M^{\prime }}{M}},$ ${\tfrac {M^{\prime \prime }}{{M}^{\prime }}},$ ${\tfrac {M^{\prime \prime \prime }}{M^{\prime \prime }}}$ etc. resp. sint valores fractionis

${\frac {t^{\lambda }+At^{\lambda -1}+Bt^{\lambda -2}+Ct^{\lambda -3}+\,{\text{etc.}}}{t^{\lambda }+at^{\lambda -1}+bt^{\lambda -2}+ct^{\lambda -3}+\,{\text{etc.}}}}$

pro $t=m,$ $t=m+1,$ $t=m+2$ etc. Brevitatis caussa huius fractionis numeratorem per $P,$ denominatorem per $p$ denotabimus: praeterea supponemus, $P,$ $p$ non esse identicas, sive differentias $A-a,$ $B-b,$ $C-c$ etc. non omnes simul evanescere.

I. Quoties e differentiis $A-a,$ $B-b,$ $C-c$ etc. prima quae non evanescit est positiva, assignari poterit limes aliquis $l,$ quem simulac egressus est valor ipsius $t,$ valores functionum $P$ et $p$ certo semper evadent positivi, atque $P>p.$ Manifestum est, hoc evenire, si pro $l$ accipiatur radix maxima realis aequationis $p(P-p)=0;$ si vero haec aequatio nullas omnino radices reales habeat, proprietatem illam pro omnibus valoribus ipsius $t$ locum habere. Quapropter in serie ${\tfrac {M^{\prime }}{M}},$ ${\tfrac {M^{\prime \prime }}{M^{\prime }}},$ ${\tfrac {M^{\prime \prime \prime }}{M^{\prime \prime }}}$ etc. saltem post certum intervallum (si non ab initio) omnes termini erunt positivi atque maiores unitate; quodsi itaque nullus neque $=0$ neque infinite magnus evadit, perspicuum est, seriem $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc. si non ab initio tamen post certum intervallum omnes suos terminos eodem signo affectos continuoque crescentes habituram esse.

Eadem ratione, si e differentiis $A-a,$ $B-b,$ $C-c$ etc. prima quae non evanescit est negativa, series $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc. si non ab initio tamen post certum intervallum omnes suos terminos eodem signo affectos continuoque decrescentes habebit.

II. Si iam coëfficientes $A,$ $a$ sunt inaequales, termini seriei $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc. ultra omnes limites sive in infinitum vel crescent vel decrescent, prout differentia $A-a$ est positiva vel negativa: hoc ita demonstramus. Si $A-a$ est quantitas positiva, accipiatur numerus integer $h$ ita, ut fiat $h(A-a)>1,$ statuaturque ${\tfrac {M^{h}}{m}}=N,$ ${\tfrac {M^{\prime h}}{m+1}}=N^{\prime },$ ${\tfrac {M^{\prime \prime h}}{m+2}}=N^{\prime \prime },$ ${\tfrac {M^{\prime \prime \prime h}}{m+3}}=N^{\prime \prime \prime }$ etc., nec non $tP^{h}=Q,$ $(t+1)p^{h}=q.$ Tunc patet, ${\tfrac {N^{\prime }}{N}},$ ${\tfrac {N^{\prime \prime }}{N^{\prime }}},$ ${\tfrac {N^{\prime \prime \prime }}{N^{\prime \prime }}}$ etc. esse valores fractionis ${\tfrac {Q}{q}}$ ponendo $t=m,$ $t=m+1,$ $t=m+2$ etc., ipsas $Q,$ $q$ vero esse functiones algebraicas formae huius

${\begin{aligned}&Q=t^{\lambda h+1}+hAt^{\lambda h}+{\text{etc. }}\\&q=t^{\lambda h+1}+(ha+1)t^{\lambda h}+{\text{etc. }}\end{aligned}}$

Quare quum per hyp. differentia $hA-(ha+1)$ sit quantitas positiva, termini seriei $N,$ $N^{\prime },$ $N^{\prime \prime },$ $N^{\prime \prime \prime }$ etc. si non ab initio tamen post certum intervallum continuo crescent (per I); hinc termini seriei $m{N},(m+1){N}^{\prime },(m+2){N}^{\prime \prime },$ $(m+3)N^{\prime \prime \prime }$ etc. necessario ultra omnes limites crescent, et proin etiam termini seriei $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime }$ etc., quippe quorum potestates exponente $h$ illis sunt aequales. Q.E.P.

Si $A-a$ est quantitas negativa, accipere oportet integrum $h$ ita, ut $h(a-A)$ fiat maior quam 1, unde per ratiocinia similia termini seriei

$mM^{h},\quad (m+1)M^{\prime h},\quad (m+2)M^{\prime \prime h},\quad (m+3)M^{\prime \prime \prime h}\quad {\text{etc.}}$

post certum intervallum continuo decrescent. Quamobrem termini seriei $M^{h},$ $M^{\prime h},$ $M^{\prime \prime h}$ etc. adeoque etiam termini huius $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc. necessario in infinitum decrescent. Q.E.S.

III. Si vero coëfficientes primi $A,$ $a$ sunt aequales, termini seriei $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc. versus limitem finitum continuo convergent, quod ita demonstramus. Supponamus primo, terminos seriei post certum intervallum continuo crescere, sive e differentiis $B-b,$ $C-c$ etc. primam quae non evanescat esse positivam. Sit $h$ integer talis, ut $h+b-B$ fiat quantitas positiva, statuamusque

$M\left({\frac {m}{m-1}}\right)^{h}=N,\quad M^{\prime }\left({\frac {m+1}{m}}\right)^{h}=N^{\prime },\quad M^{\prime \prime }\left({\frac {m+2}{m+1}}\right)^{h}=N^{\prime \prime }$ etc.

atque $(tt-1)^{h}P=Q,$ $t^{2h}p=q,$ ita ut ${\tfrac {N^{\prime }}{N}},$ ${\tfrac {N^{\prime \prime }}{N^{\prime }}}$ etc. sint valores fractionis ${\tfrac {Q}{q}}$ ponendo $t=m,$ $t=m+1$ etc. Quum itaque habeatur

$Q=t^{\lambda +2h}+At^{\lambda +2h-1}+(B-h)t^{\lambda +2h-2}$ etc.

$q=t^{\lambda +2h}+At^{\lambda +2h-1}+bt^{\lambda +2h-2}$ etc.

atque per hyp. $B-h-b$ sit quantitas negativa, termini seriei $N,$ $N^{\prime },$ $N^{\prime \prime },$ $N^{\prime \prime \prime }$ etc. post certum saltem intervallum continuo decrescent, adeoque termini seriei $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc., qui illis resp. semper sunt minores, dum continuo crescunt, tantummodo versus limitem finitum convergere possunt. Q.E.P.

Si termini seriei $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc. post certum intervallum continuo decrescunt, accipere oportet pro $h$ integrum talem, ut $h+B-b$ sit quantitas positiva, evinceturque per ratiocinia prorsus similia, terminos seriei

$M\left({\frac {m-1}{m}}\right)^{h},\quad M^{\prime }\left({\frac {m}{m+1}}\right)^{h},\quad M^{\prime \prime }\left({\frac {m+1}{m+2}}\right)^{h}$ etc.

post certum intervallum continuo crescere, unde termini seriei $M,$ $M^{\prime },$ $M^{\prime \prime }$ etc., qui illis resp. semper sunt maiores, dum continuo decrescunt, necessario tantummodo versus limitem finitum decrescere possunt. Q.E.S.

IV. Denique quod attinet ad summam seriei, cuius termini sunt $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc., in casu eo, ubi hi in infinitum decrescunt, supponamus primo, $A-a$ cadere inter $0$ et $-1$, sive $A+1-a$ esse vel quantitatem positivam vel $=0.$ Sit $h$ integer positivus, arbitrarius in casu eo, ubi $A+1-a$ est quantitas positiva, vel talis qui reddat quantitatem $h+m+A+B-b$ positivam in casu eo ubi $A+1-a=0.$ Tunc erit

$P(t-(m+h-1))$	$=t^{\lambda +1}+(A+1-m-h)t^{\lambda }+(B-A(m+h-1))t^{\lambda -1}$ etc.
$p(t-(m+h))$	$=t^{\lambda +1}+(a-m-h)t^{\lambda }+(b-a(m+h))t^{\lambda -1}$ etc.

ubi vel $A+1-m-h-(a-m-h)$ erit quantitas positiva, vel, si haec fit $=0,$ saltem $B-A(m+h-1)-(b-a(m+h))$ positiva erit. Hinc (per I) pro quantitate $t$ assignari poterit valor aliquis $l,$ quem simulac transgressa est, valores fractionis ${\tfrac {P(t-(m+h-1))}{p(t-(m+h))}}$ semper fient positivi atque unitate maiores. Sit $n$ integer maior quam $l$ simulque maior quam $h,$ sintque termini seriei $M,$ $M^{\prime },$ $M^{\prime \prime },M^{\prime \prime \prime }$ etc. qui respondent valoribus $t=m+n,$ $t=m+n+1,$ $t=m+n+2$ etc., hi $N,$ $N^{\prime },$ $N^{\prime \prime },$ $N^{\prime \prime \prime }$ etc. Erunt itaque

${\frac {(n+1-h)N^{\prime }}{(n-h){N}}},\quad {\frac {(n+2-h)N^{\prime \prime }}{(n+1-h)N^{\prime }}},\quad {\frac {(n+3-h)N^{\prime \prime \prime }}{(n+2-h)N^{\prime \prime }}}\quad$ etc.

quantitates positivae unitate maiores, unde

$N^{\prime }>{\frac {(n-h)N}{n+1-h}},\quad N^{\prime \prime }>{\frac {(n-h)N}{n+2-h}},\quad N^{\prime \prime \prime }>{\frac {(n-h)N}{n+3-h}}\quad$ etc.

adeoque summa seriei $N+N^{\prime }+N^{\prime \prime }+N^{\prime \prime \prime }+$ etc. maior summa seriei

$(n-h)N\left({\frac {1}{n-h}}+{\frac {1}{n+1-h}}+{\frac {1}{n+2-h}}+{\frac {1}{n+3-h}}+{\text{etc.}}\right)$

quotcunque termini colligantur. Sed posterior series, crescente terminorum numero in infinitum, omnes limites egreditur, quum summa seriei $1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{4}}+{\text{ etc.}}$ quam infinitam esse constat etiam infinita maneat, si ab initio termini $1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+{\text{etc.}}+{\tfrac {1}{n-1-h}}$ rescindantur. Quare summa seriei $N+N^{\prime }+N^{\prime \prime }+N^{\prime \prime \prime }+{\text{etc.}},$ adeoque etiam summa huius $M+M^{\prime }+M^{\prime \prime }+M^{\prime \prime \prime }+$ etc., cuius pars est illa, ultra omnes limites crescit.

V. Quoties autem $A-a$ est quantitas negativa absolute maior quam unitas, summa seriei $M+M^{\prime }+M^{\prime \prime }+M^{\prime \prime \prime }+$ etc. in infinitum continuatae certo erit finita. Sit enim $h$ quantitas positiva minor quam $a-A-1,$ demonstrabiturque per ratiocinia similia, assignari posse valorem aliquem $l$ quantitatis $t,$ ultra quem fractio ${\tfrac {Pt}{p(t-h-1)}}$ semper adipiscatur valores positivos unitate minores. Quodsi iam pro $n$ accipitur numerus integer ipsis $l,$ $m,$ $h+1$ maior, terminique seriei $M,$ $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime }$ etc., valoribus $t=n,$ $t=n+1,$ $t=n+2$ etc. respondentes, designantur per $N,$ $N^{\prime },$ $N^{\prime \prime }$ etc., erit

$N^{\prime }<{\frac {n-h-1}{n}}\cdot N,\quad N^{\prime \prime }<{\frac {(n-h-1)(n-h)}{n(n+1)}}\cdot N^{\prime }\quad$ etc.

adeoque summa seriei $N+N^{\prime }+N^{\prime \prime }+$ etc., quotcunque termini colligantur, minor producto ex $N$ in summam totidem terminorum seriei

$1+{\frac {n-h-1}{n}}+{\frac {(n-h-1)(n-h)}{n(n+1)}}+{\frac {(n-h-1)(n-h)(n-h+1)}{n(n+1)(n+2)}}$ etc.

Huius vero summa pro quolibet terminorum numero facile assignari potest; est scilicet

terminus primus	$={\tfrac {n-1}{h}}-{\tfrac {n-h-1}{h}}$ |-	summa duorum terminorum	$={\tfrac {n-1}{h}}-{\tfrac {(n-h-1)(n-h)}{hn}}$
summa trium terminorum	$={\tfrac {n-1}{h}}-{\tfrac {(n-h-1)(n-h)(n-h+1)}{hn(n+1)}}$ etc.

et quum pars altera (per II) formet seriem ultra omnes limites decrescentem, summa illa in infinitum continuata statui debet $={\tfrac {n-1}{h}}.$ Hinc $N+N^{\prime }+N^{\prime \prime }$ etc. in infinitum continuata semper manebit minor quam ${\tfrac {N(n-1)}{h}},$ et proin $M+M^{\prime }+M^{\prime \prime }$ etc. certo ad summam finitam converget. Q. E. D.

VI. Ut ea, quae generaliter de serie $M,$ $M^{\prime },$ $M^{\prime \prime }$ etc. demonstravimus, ad coëfficientes potestatum $x^{m},$ $x^{m+1},$ $x^{m+2}$ etc. in serie ${F}(\alpha ,\beta ,\gamma ,x),$ applicentur, statuere oportebit $\lambda =2,$ $A=\alpha +\beta ,$ $B=\alpha \beta ,$ $a=\gamma +1,$ $b=\gamma ,$ unde quinque assertiones in art. praec. propositae sponte demanant.

17.

Disquisitio itaque de indole summae seriei $F(\alpha ,\beta ,\gamma ,1)$ natura sua restringitur ad casum, quo $\gamma -\alpha -\beta$ est quantitas positiva, ubi illa summa semper exhibet quantitatem finitam. Praemittimus autem observationem sequentem. Si coëfficientes seriei $1+ax+bxx+cx^{3}+$ etc. $=S$ inde a certo termino ultra omnes limites decrescunt, productum

$(1-x)S=1+(a-1)x+(b-a)xx+(c-b)x^{3}+$ etc.

pro $x=1$ statuere oportet $=0,$ etiamsi summa ipsius seriei $S$ infinite magna evadat. Quoniam enim collectis duobus terminis summa fit $=a,$ collectis tribus $=b,$ collectis quatuor $=c$ etc., limes summae in infinitum continuatae est $=0.$ Quoties itaque $\gamma -\alpha -\beta$ est quantitas positiva, statuere oportet $(1-x)F(\alpha ,\beta ,\gamma -1,x)=0$ pro $x=1,$ unde per aequationem 15 art. 7 habebimus

	$0=\gamma (\alpha +\beta -\gamma )F(\alpha ,\beta ,\gamma ,1)+(\gamma -\alpha )(\gamma -\beta )F(\alpha ,\beta ,\gamma +1,1),$ sive
[37]	$F(\alpha ,\beta ,\gamma ,1)={\frac {(\gamma -\alpha )(\gamma -\beta )}{\gamma (\gamma -\alpha -\beta )}}F(\alpha ,\beta ,\gamma +1,1)$

Quare quum perinde fiat

${\begin{aligned}&F(\alpha ,\beta ,\gamma +1,1)={\frac {(\gamma +1-\alpha )(\gamma +1-\beta )}{(\gamma +1)(\gamma +1-\alpha -\beta )}}F(\alpha ,\beta ,\gamma +2,1)\\&F(\alpha ,\beta ,\gamma +2,1)={\frac {(\gamma +2-\alpha )(\gamma +2-\beta )}{(\gamma +2)(\gamma +2-\alpha -\beta )}}F(\alpha ,\beta ,\gamma +3,1)\end{aligned}}$

et sic porro, erit generaliter, $k$ denotante integrum positivum quemcunque

$F(\alpha ,\beta ,\gamma ,1)$ aequalis producto ex ${F}(\alpha ,\beta ,\gamma +k,1)$

in $\quad (\gamma -\alpha )(\gamma +1-\alpha )(\gamma +2-\alpha )\ldots (\gamma +k-1-\alpha )$
in $\quad (\gamma -\beta )(\gamma +1-\beta )(\gamma +2-\beta )\ldots (\gamma +k-1-\beta )$
diviso per productum
ex $\quad \gamma (\gamma +1)(\gamma +2)\ldots (\gamma +k-1)$
in $\quad (\gamma -\alpha -\beta )(\gamma +1-\alpha -\beta )(\gamma +2-\alpha -\beta )\ldots (\gamma +k-1-\alpha -\beta )$

18.

Introducamus abhinc sequentem notationem:

[38]	$\Pi (k,z)={\frac {1\;.\;2\;.\;3\ldots \ldots k}{(z+1)(z+2)(z+3)\ldots (z+k)}}k^{z}$

ubi $k$ natura sua subintelligitur designare integrum positivum, qua restrictione $\Pi (k,z)$ exhibet functionem duarum quantitatum $k,$ $z$ prorsus determinatam. Hoc modo facile intelligetur, theorema in fine art. praec. propositum ita exhiberi posse

[39]	$F(\alpha ,\beta ,\gamma ,1)={\frac {\Pi (k,\gamma -1).\Pi (k,\gamma -\alpha -\beta -1)}{\Pi (k,\gamma -\alpha -1).\Pi (k,\gamma -\beta -1)}}.F(\alpha ,\beta ,\gamma +k,1)$

19.

Operae pretium erit, indolem functionis $\Pi (k,z)$ accuratius perpendere. Quoties $z$ est integer negativus, functio manifesto valorem infinite magnum obtinet, simulac ipsi $k$ tribuitur valor satis magnus. Pro valoribus integris ipsius ${z}$ non negativis autem habemus

${\begin{aligned}&\Pi (k,0)=1\\&\Pi (k,1)={\frac {1}{1+{\frac {1}{k}}}}\\&\Pi (k,2)={\frac {1\;.\;2}{(1+{\frac {1}{k}})(1+{\frac {2}{k}})}}\\&\Pi (k,3)={\frac {1\;.\;2\;.\;3}{(1+{\frac {1}{k}})(1+{\frac {2}{k}})(1+{\frac {3}{k}})}}\end{aligned}}$

etc. sive generaliter

[40]	$\Pi (k,z)={\frac {1\;.\;2\;.\;3\;\ldots \ldots \;z}{(1+{\frac {1}{k}})(1+{\frac {2}{k}})(1+{\frac {3}{k}})\ldots (1+{\frac {z}{k}})}}$

Generaliter autem pro quovis valore ipsius ${z}$ habemus

[41]	$\Pi (k,z+1)=\Pi (k,z)\cdot {\frac {1+z}{1+{\frac {1+z}{k}}}}$
[42]	$\Pi (k+1,z)=\Pi (k,z)\cdot \left\{{\frac {\left(1+{\frac {1}{k}}\right)^{z+1}}{1+{\frac {1+z}{k}}}}\right\}$

adeoque, quum $\Pi (1,z)={\frac {1}{z+1}},$

[43]	$\Pi (k,z)={\frac {1}{z+1}}\cdot {\frac {2^{z+1}}{1^{z}\cdot (2+z)}}\cdot {\frac {3^{z+1}}{2^{z}(3+z)}}\cdot {\frac {4^{z+1}}{3^{z}(4+z)}}\cdots {\frac {k^{z+1}}{(k-1)^{z}(k+z)}}$

20.

Imprimis vero attentione dignus est limes, ad quem pro valore dato ipsius $z$ functio $\Pi (k,z)$ continuo converget, dum $k$ in infinitum crescit. Sit primo $h$ valor finitus ipsius $k$ maior quam $z,$ patetque, si $k$ transire supponatur ex $h$ in $h+1,$ logarithmum ipsius $\Pi (k,z)$ accipere incrementum, quod per seriem convergentem sequentem exprimatur

${\frac {z(1+z)}{2(h+1)^{2}}}+{\frac {z(1-zz)}{3(h+1)^{3}}}+{\frac {z(1+z^{3})}{4(h+1)^{4}}}+{\frac {z(1-z^{4})}{5(h+1)^{5}}}+{\text{etc.}}$

Si itaque $k$ e valore $h$ transit in $h+n,$ logarithmus ipsius $\Pi (k,z)$ accipiet incrementum

${\begin{alignedat}{3}&&&{\tfrac {1}{2}}z(1+z)&\left({\frac {1}{(h+1)^{2}}}+{\frac {1}{(h+2)^{2}}}+{\frac {1}{(h+3)^{2}}}+{\text{etc.}}+{\frac {1}{(h+n)^{2}}}\right)\\&+&{\tfrac {1}{3}}&z(1-zz)&\left({\frac {1}{(h+1)^{3}}}+{\frac {1}{(h+2)^{3}}}+{\frac {1}{(h+3)^{3}}}+{\text{etc.}}+{\frac {1}{(h+n)^{3}}}\right)\\&+&{\tfrac {1}{4}}&z(1+z^{3})&\left({\frac {1}{(h+1)^{4}}}+{\frac {1}{(h+2)^{4}}}+{\frac {1}{(h+3)^{4}}}+{\text{etc.}}+{\frac {1}{(h+n)^{4}}}\right)\\&+&&{\text{etc.}}&\end{alignedat}}$

quod semper finitum manere, etiamsi $n$ in infinitum crescat, facile demonstrari potest. Quare nisi iam in $\Pi (h,z)$ factor infinitus affuerit, i. e nisi $z$ sit numerus integer negativus, limes ipsius $\Pi (k,z)$ pro $k=\infty$ certo erit quantitas finita. Manifesto itaque $\Pi (\infty ,z)$ tantummodo a $z$ pendet, sive functionem ipsius $z$ ex asse determinatam exhibet, quae abhinc simpliciter per $\Pi z$ denotabitur. Definimus itaque functionem $\Pi z$ per valorem producti

${\frac {1\;.\;2\;.\;3\;.\ldots .\;k\;.\;k^{z}}{(z+1)(z+2)(z+3)\ldots (z+k)}}$

pro $k=\infty ,$ aut si mavis per limitem producti infiniti

${\frac {1}{z+1}}\cdot {\frac {2^{z+1}}{1^{z}(2+z)}}\cdot {\frac {3^{z+1}}{2^{z}(3+z)}}\cdot {\frac {4^{z+1}}{3^{z}(4+z)}}{\text{ etc. }}$

21.

Ex aequatione 41 statim sequitur aequatio fundamentalis

[44]	$\Pi (z+1)=(z+1)\Pi z$

unde generaliter, designante $n$ integrum positivum quemcunque

[45]	$\Pi (z+n)=(z+1)(z+2)(z+3)\ldots \ldots (z+n)\Pi z$

Pro valore integro negativo ipsius $z$ erit valor functionis $\Pi z$ infinite magnus; pro valoribus integris non negativis habemus

${\begin{aligned}&\Pi \,0=1\\&\Pi \,1=1\\&\Pi \,2=2\\&\Pi \,3=6\\&\Pi \,4=24{\text{ etc.}}\end{aligned}}$

atque generaliter

[46]

$\Pi z=1.2.3\ldots z$

Sed male haec proprietas functionis nostrae tamquam ipsius definitio venditaretur, quippe quae natura sua ad valores integros restringitur, et praeter functionem nostram infinitis aliis (e.g. $\cos 2\pi z.\Pi z,$ $\cos \pi z^{2n}\Pi z$ etc., denotante $\pi$ semiperipheriam circuli, cuius radius $=1$) communis est.

22.

Functio $\Pi (k,z),$ etiamsi generalior videatur quam $\Pi z,$ tamen abhinc nobis superflua erit, quum facile ad posteriorem reducatur. Colligitur enim e combinatione aequationum 38, 45, 46.

[47]	$\Pi (k,z)={\frac {k^{z}\Pi k.\Pi z}{\Pi (k+z)}}$

Ceterum nexus harum functionum cum is, quas clar. Kramp facultates numericas nominavit, per se obvius est. Scilicet facultas numerica, quam hic auctor per $a^{bIc}$ designat, in signis nostris est

$={\frac {c^{b}b^{{\frac {a}{c}}-1}\Pi b}{\Pi (b,{\frac {a}{c}}-1)}}={\frac {c^{b}\Pi ({\frac {a}{c}}+b-1)}{\Pi ({\frac {a}{c}}-1)}}$

Sed consultius videtur, functionem unius variabilis in analysin introducere, quam functionem trium variabilium, praesertim quum hanc ad illam reducere liceat.

23.

Continuitas functionis $\Pi z$ interrumpitur, quoties ipsius valor fit infinite magnus, i.e. pro valoribus integris negativis ipsius $z.$ Erit itaque illa positiva a $z=-1$ usque ad $z=\infty ,$ et quum pro utroque limite $\Pi z$ obtineat valorem infinite magnum, inter ipsos dabitur valor minimus, quem esse $=0{,}8856024$ atque respondere valori $z=0{,}4616321$ invenimus. Inter limites $z=-1$ et $z=-2,$ valor functionis $\Pi z$ fit negativus, inter $z=-2$ atque $z=-3$ iterum positivus et sic porro, uti ex aequ. 44 sponte sequitur. Porro patet, si omnes valores functionis $\Pi z$ inter limites arbitrarios unitate differentes e.g. a $z=0$ usque ad $z=1$ pro notis habere liceat, valorem functionis pro quovis alio valore reali ipsius $z$ adiumento aequationis 45 facile inde deduci posse. Ad hunc finem construximus tabulam, ad calcem huius sectionis annexam, quae ad figuras viginti exhibet logarithmos briggicos functionis $\Pi z,$ pro $z=0$ usque ad $z=1$ per singulas partes centesimas summa cura computatos, ubi tamen monendum, figuram ultimam vigesimam interdum una duabusve unitatibus erroneam esse posse.

24.

Quum limes functionis $F(\alpha ,\beta ,\gamma +k,1),$ crescente $k$ in infinitum, manifesto sit unitas, aequatio 39 transit in hanc

[48]	$F(\alpha ,\beta ,\gamma ,1)={\frac {\Pi (\gamma -1).\Pi (\gamma -\alpha -\beta -1)}{\Pi (\gamma -\alpha -1).\Pi (\gamma -\beta -1)}}$

quae formula exhibet solutionem completam quaestionis, quae obiectum huius sectionis constituit. Sponte hinc sequuntur aequationes elegantes:

[49]	$F(\alpha ,\beta ,\gamma ,1)=F(-\alpha ,-\beta ,\gamma -\alpha -\beta ,1)$
[50]	$F(\alpha ,\beta ,\gamma ,1).F(-\alpha ,\beta ,\gamma -\alpha ,1)=1$
[51]	$F(\alpha ,\beta ,\gamma ,1).F(\alpha ,-\beta ,\gamma -\beta ,1)=1$

in quarum prima $\gamma ,$ in secunda $\gamma -\beta ,$ in tertia $\gamma -\alpha$ debet esse quantitas positiva.

25.

Applicemus formulam 48 ad quasdam ex aequationibus art. 5. Formula XIII, statuendo $t=90^{0}={\tfrac {1}{2}}\pi ,$ fit ${\tfrac {1}{2}}\pi =F({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {3}{2}},1),$ sive aequivalet aequationi notae

${\tfrac {1}{2}}\pi =1+{\tfrac {1.1}{2.3}}+{\tfrac {1.1.3}{2.4.5}}+{\tfrac {1.1.3.5}{2.4.6.7}}+$ etc.

Quare quum per formulam 48 habeatur $F({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {3}{2}},1)={\frac {\Pi {\tfrac {1}{2}}.\Pi (-{\tfrac {1}{2}})}{\Pi {0}.\Pi {0}}},$ atque sit $\Pi {0}=1,$ $\Pi {\tfrac {1}{2}}={\tfrac {1}{2}}\Pi (-{\tfrac {1}{2}}),$ fit $\pi =(\Pi (-{\tfrac {1}{2}}))^{2}$ sive

[52]	$\Pi (-{\tfrac {1}{2}})={\sqrt {\pi }}$
[53]	$\Pi {\tfrac {1}{2}}={\tfrac {1}{2}}{\sqrt {\pi }}$

Formula XVI art. 5, quae aequivalet aequationi notae

$\sin nt=n\sin t-{\tfrac {n(nn-1)}{2\;.\;3}}{\sin t}^{3}+{\tfrac {n(nn-1)(nn-9)}{2\;.\;3\;.\;4\;.\;5}}{\sin t}^{5}-$ etc.

atque generaliter pro quovis valore ipsius $n$ locum habet, si modo $t$ limites $-90^{\circ }$ et $+90^{\circ }$ non transgrediatur, dat pro $t={\tfrac {1}{2}}\pi$

$\sin {\tfrac {n\pi }{2}}={\frac {n\Pi {\tfrac {1}{2}}.\Pi (-{\tfrac {1}{2}})}{\Pi (-{\tfrac {1}{2}}n).\Pi {\tfrac {1}{2}}n}}$

unde deducitur formula elegans

$\Pi {\tfrac {1}{2}}n.\Pi (-{\tfrac {1}{2}}n)={\frac {{\tfrac {1}{2}}n\pi }{\sin {\tfrac {1}{2}}n\pi }},$ sive statuendo $n=2z$

[54]	$\Pi (-z).\Pi (+z)={\frac {z\pi }{\sin z\pi }}$
[55]	$\Pi (-z).\Pi (z-1)={\frac {\pi }{\sin z\pi }}$

nec non scribendo $z+{\tfrac {1}{2}}$ pro $z$

[56]	$\Pi (-{\tfrac {1}{2}}+z).\Pi (-{\tfrac {1}{2}}-z)={\frac {\pi }{\cos z\pi }}$

E combinatione formulae 54 cum definitione functionis $\Pi$ sequitur, ${\tfrac {z\pi }{\sin z\pi }}$ esse limitem producti infiniti

${\frac {(1\;.\;2\;.\;3\;.\;4\;\ldots \;k)^{2}}{(1-zz)(4-zz)(9-zz)\ldots (kk-zz)}}$

crescente $k$ in infinitum, adeoque

$\sin z\pi =z\pi \left(1-zz\right)\left(1-{\tfrac {zz}{4}}\right)\left(1-{\tfrac {zz}{9}}\right)$ etc. in inf.

similique modo ex 56 deducitur

$\cos z\pi =\left(1-4zz\right)\left(1-{\tfrac {4zz}{9}}\right)\left(1-{\tfrac {4zz}{25}}\right)$ etc. in inf.

formulae notissimae, quae ab analystis per methodos prorsus diversas erui solent.

26.

Designante $n$ numerum integrum, valor expressionis

${\frac {n^{nz}\Pi (k,z).\Pi (k,z-{\frac {1}{n}}).\Pi (k,z-{\frac {2}{n}})\ldots \ldots \Pi (k,z-{\frac {n-1}{n}})}{\Pi (nk,nz)}}$

rite collectus invenitur

$={\frac {(1\;.\;2\;.\;3\;\ldots \;k)^{n}n^{nk}}{1\;.\;2\;.\;3\;\ldots \;nk.k^{{\frac {1}{2}}(n-1)}}}$

adeoque a $z$ est independens, sive idem manebit, quicunque valor ipsi $z$ tribuatur. Exhiberi poterit itaque, quoniam $\Pi (k,0)=\Pi (nk,0)=1,$ per productum

$\Pi (k,-{\tfrac {1}{n}}).\Pi (k,-{\tfrac {2}{n}}).\Pi (k,-{\tfrac {3}{n}})\ldots \Pi (k,-{\tfrac {n-1}{n}})$

Crescente igitur $k$ in infinitum, nanciscimur

${\frac {n^{nz}\Pi z.\Pi (z-{\tfrac {1}{n}}).\Pi (z-{\tfrac {2}{n}})\ldots \Pi (z-{\tfrac {n-1}{n}})}{\Pi nz}}=\Pi (-{\tfrac {1}{n}}).\Pi (-{\tfrac {2}{n}}).\Pi (-{\tfrac {3}{n}})\ldots \Pi (-{\tfrac {n-1}{n}})$

Productum ad dextram, in se ipsum ordine factorum inverso multiplicatum, producit, per form. 55,

${\frac {\pi }{\sin {\frac {1}{n}}\pi }}.{\frac {\pi }{\sin {\frac {2}{n}}\pi }}.{\frac {\pi }{\sin {\frac {3}{n}}\pi }}\cdots {\frac {\pi }{\sin {\frac {n-1}{n}}\pi }}={\frac {(2\pi )^{n-1}}{n}}$

Unde habemus theorema elegans

[57]	${\frac {n^{nz}\Pi z.\Pi (z-{\frac {1}{n}}).\Pi (z-{\frac {2}{n}})\ldots \ldots \Pi (z-{\frac {n-1}{n}})}{\Pi nz}}={\frac {(2\pi )^{{\frac {1}{2}}(n-1)}}{\sqrt {n}}}$

27.

Integrale $\int x^{\lambda -1}(1-x^{\mu })^{\nu }\operatorname {d} x,$ ita acceptum, ut evanescat pro $x=0,$ exprimitur per seriem sequentem, siquidem $\lambda ,$ $\mu$ sunt quantitates positivae:

${\frac {x^{\lambda }}{\lambda }}-{\frac {\nu x^{\mu +\lambda }}{\mu +\lambda }}+{\frac {\nu (\nu -1)x^{2\mu +\lambda }}{1.2.(2\mu +\lambda )}}-{\text{etc.}}={\frac {x^{\lambda }}{\lambda }}F\left(-\nu ,{\frac {\lambda }{\mu }},{\frac {\lambda }{\mu }}+1,x^{\mu }\right)$

Hinc ipsius valor pro $x=1$ erit

$={\frac {\Pi {\frac {\lambda }{\mu }}.\Pi \nu }{\lambda \Pi ({\frac {\lambda }{\mu }}+v)}}$

Ex hoc theoremate omnes relationes, quas ill. Euler olim multo labore evolvit, sponte demanant. Ita e.g. statuendo

$\int {\frac {\operatorname {d} x}{\sqrt {(1-x^{4})}}}=A,\quad \int {\frac {xx\operatorname {d} x}{\sqrt {(1-x^{4})}}}=B$

erit $A={\tfrac {\Pi {\frac {1}{4}}.\Pi (-{\frac {1}{2}})}{\Pi (-{\frac {1}{4}})}},$ $B={\tfrac {\Pi {\frac {3}{4}}.\Pi (-{\frac {1}{2}})}{3\Pi {\frac {1}{4}}}}={\tfrac {\Pi (-{\frac {1}{4}}).\Pi (-{\frac {1}{2}})}{4\Pi {\frac {1}{4}}}},$ adeoque $AB={\tfrac {1}{4}}\pi .$ Simul hinc sequitur, quoniam $\Pi {\tfrac {1}{4}}.\Pi (-{\tfrac {1}{4}})={\tfrac {{\frac {1}{4}}\pi }{\sin {\frac {1}{4}}\pi }}={\frac {\pi }{\sqrt {8}}},$

$\Pi {\tfrac {1}{4}}={\sqrt[{4}]{{\tfrac {1}{8}}\pi AA}}={\sqrt[{4}]{\frac {\pi ^{3}}{128BB}}},\quad \Pi (-{\tfrac {1}{4}})={\sqrt[{4}]{\frac {\pi ^{3}}{8AA}}}={\sqrt[{4}]{2\pi BB}}$

Valor numericus ipsius $A,$ computante Stirling, habetur $=1{,}3110287771\,4605987,$ valor ipsius $B,$ secundum eundem auctorem, $=0{,}5990701173\,6779611,$ ex nostro calculo, artificio peculiari innixo, $=0{,}5990701173\,6779610372.$

Generaliter facile ostendi potest, valorem functionis $\Pi {z},$ si $z$ sit quantitas rationalis $={\tfrac {m}{\mu }},$ denotantibus $m,$ $\mu$ integros, ex $\mu -1$ valoribus determinatis talium integralium pro $x=1$ deduci posse, et quidem permultis modis diversis. Accipiendo enim pro $\lambda$ numerum integrum atque pro $\nu$ fractionem, cuius denominator $=\mu ,$ valor illius integralis semper reducitur ad tres $\Pi z,$ ubi $z$ est fractio cum denominatore $=\mu ;$ quodvis vero huiusmodi $\Pi z$ vel ad $\Pi (-{\tfrac {1}{\mu }}),$ vel ad $\Pi (-{\tfrac {2}{\mu }}),$ vel ad $\Pi (-{\tfrac {3}{\mu }})$ etc. vel ad $\Pi (-{\tfrac {\mu -1}{\mu }})$ reduci potest per formulam 45, siquidem $z$ revera est fractio; si enim $z$ est integer, $\Pi {z}$ per se constat. Ex illis vero integralium valoribus, generaliter loquendo, quodvis $\Pi (-{\tfrac {m}{\mu }}),$ si $m<\mu ,$ per eliminationem erui potest.^[1] Quin adeo semissis talium integralium sufficiet, si formulam 54 simul in auxilium vocamus. Ita e.g. statuendo

$\int {\frac {\operatorname {d} x}{\sqrt[{5}]{(1-x^{5})}}}=C,\quad$$\int {\frac {\operatorname {d} x}{\sqrt[{5}]{(1-x^{5})^{2}}}}=D,\quad$$\int {\frac {\operatorname {d} x}{\sqrt[{5}]{(1-x^{5})^{3}}}}=E,\quad$$\int {\frac {\operatorname {d} x}{\sqrt[{5}]{(1-x^{5})^{4}}}}=F,$

erit

$C=\Pi {\tfrac {1}{5}}.\Pi (-{\tfrac {1}{5}}),\quad$$D={\frac {\Pi {\tfrac {1}{5}}.\Pi (-{\tfrac {2}{5}})}{\Pi (-{\frac {1}{5}})}},\quad$$E={\frac {\Pi {\tfrac {1}{5}}.\Pi (-{\tfrac {3}{5}})}{\Pi (-{\tfrac {2}{5}})}},\quad$$F={\frac {\Pi {\tfrac {1}{5}}.\Pi {(-{\tfrac {4}{5}})}}{\Pi (-{\frac {3}{5}})}}$

Hinc propter $\Pi {\tfrac {1}{5}}={\tfrac {1}{5}}\Pi (-{\tfrac {4}{5}}),$ habemus

$\Pi (-{\tfrac {1}{5}})={\sqrt[{5}]{\frac {5C^{4}}{DEF}}},\quad$$\Pi (-{\tfrac {2}{5}})={\sqrt[{5}]{\frac {25C^{3}D^{3}}{EEFF}}},\quad$$\Pi (-{\tfrac {3}{5}})={\sqrt[{5}]{\frac {125CCDDEE}{F^{3}}}},\quad$$\Pi (-{\tfrac {4}{5}})={\sqrt[{5}]{625CDEF}}$

Formulae 54, 55 adhuc suppeditant

$C={\frac {\pi }{\sin {\tfrac {1}{5}}\pi }},\quad$${\frac {D}{F}}={\frac {\sin {\tfrac {1}{5}}\pi }{\sin {\tfrac {2}{5}}\pi }}$

ita ut duo integralia $D,E,$ vel $E$ et ${F}$ sufficiant, ad omnes valores $\Pi (-{\tfrac {1}{5}}),$ $\Pi (-{\tfrac {2}{5}})$ etc. computandos.

28.

Statuendo $y=\nu x,$ atque $\mu =1,$ ${\tfrac {\Pi \lambda .\Pi \nu }{\lambda \Pi (\lambda +\nu )}}$ erit valor integralis $\int {\frac {y^{2-1}(1-{\frac {y}{\nu }})^{\nu }\operatorname {d} y}{\nu ^{2}}}$ ab $y=0$ usque ad $y=\nu ,$ sive valor integralis $\int y^{\lambda -1}(1-{\tfrac {y}{\nu }})^{\nu }\operatorname {d} y$ inter eosdem limites $={\frac {\nu ^{\lambda }\Pi \lambda .\Pi \nu }{\lambda \Pi (\lambda +\nu )}}={\frac {\Pi (\nu ,\lambda )}{\lambda }}$ (form. 47), siquidem $\nu$ denotet integrum. Iam crescente $\nu$ in infinitum, limes ipsius $\Pi (\nu ,\lambda )$ erit $=\Pi \lambda ,$ limes ipsius $(1-{\tfrac {y}{\nu }})^{\nu }$ autem $e^{-y},$ denotante $e$ basin logarithmorum hyperbolicorum. Quamobrem si $\lambda$ est positiva, ${\tfrac {\Pi \lambda }{\lambda }}$ sive $\Pi (\lambda -1)$ exprimet integrale $\int y^{\lambda -1}e^{-y}\operatorname {d} y$ ab $y=0$ usque ad $y=\infty ,$ sive scribendo $\lambda$ pro $\lambda -1,$ $\Pi \lambda$ est valor integralis $\int y^{\lambda }e^{-y}\operatorname {d} y$ ab $y=0$ usque ad $y=\infty ,$ si $\lambda +1$ est quantitas positiva.

Generalius statuendo $y=z^{\alpha },$ $\alpha \lambda +\alpha -1=\beta ,$ transit $\int y^{\lambda }e^{-y}\operatorname {d} y$ in $\int \alpha z^{\beta }e^{-z^{\alpha }}\operatorname {d} z,$ quod itaque inter limites $z=0$ atque $z=\infty$ sumtum exprimetur per $\Pi ({\tfrac {\beta +1}{\alpha }}-1)$ sive

Valor integralis $\int z^{\beta }e^{-z^{\alpha }}\operatorname {d} z,$ a $z=0$ usque ad $z=\infty$ fit $={\frac {\Pi ({\frac {\beta +1}{\alpha }}-1)}{\alpha }}={\frac {\Pi {\frac {\beta +1}{\alpha }}}{\beta +1}}$ si modo $\alpha$ atque $\beta +1$ sunt quantitates positivae (si utraque est negativa, integrale per $-{\tfrac {\Pi {\frac {\beta +1}{\alpha }}}{\beta +1}}$ exprimetur). Ita e.g. pro $\beta =0,$ $\alpha =2,$ valor integralis $\int e^{-zz}\operatorname {d} z$ invenitur $=\Pi {\tfrac {1}{2}}={\tfrac {1}{2}}{\sqrt {\pi }}.$

29.

Ill. Euler pro summa logarithmorum

$\log 1+\log 2+\log 3+$ etc. $+\log z$

eruit seriem

$(z+{\tfrac {1}{2}})\log z-z+{\tfrac {1}{2}}\log 2\pi +{\frac {\mathfrak {A}}{1.2z}}-{\frac {\mathfrak {B}}{3.4z^{3}}}+{\frac {\mathfrak {C}}{5.6z^{3}}}-$ etc.

ubi ${\mathfrak {A}}={\tfrac {1}{6}},$ ${\mathfrak {B}}={\tfrac {1}{30}},$ ${\mathfrak {C}}={\tfrac {1}{42}}$ etc. sunt numeri Bernoulliani. Per hanc itaque seriem exprimitur $\log \Pi z;$ etiamsi enim primo aspectu haec conclusio ad valores integros restricta videatur, tamen rem propius contemplando invenietur, evolutionem ab Eulero adhibitam (Instit. Calc. Diff. Cap. vi. 159) saltem ad valores positivos fractos eodem iure applicari posse, quo ad integros: supponit enim tantummodo, functionem ipsius ${z},$ in seriem evolvendam, esse talem, ut ipsius diminutio, si ${z}$ transeat in ${z}-1,$ exhiberi possit per theorema Taylori, simulque ut eadem diminutio sit $=\log z.$ Conditio prior innititur continuitati functionis, adeoque locum non habet pro valoribus negativis ipsius ${z},$ ad quos proin seriem illam extendere non licet: conditio posterior autem functioni $\log \Pi z$ generaliter competit sine restrictione ad valores integros ipsius ${z}.$ Statuemus itaque

[58]	$\log \Pi z=(z+{\tfrac {1}{2}})\log z-z+{\tfrac {1}{2}}\log 2\pi +{\frac {\mathfrak {A}}{1.2z}}-{\frac {\mathfrak {B}}{3.4z^{3}}}+{\frac {\mathfrak {C}}{5.6z^{5}}}-{\frac {\mathfrak {D}}{7.8z^{7}}}+$etc.

Quum hinc quoque habeatur

$\log \Pi 2z=(2z+{\tfrac {1}{2}})\log 2z-2z+{\tfrac {1}{2}}\log 2\pi +{\frac {\mathfrak {A}}{1.2.2z}}-{\frac {\mathfrak {B}}{3.4.8z^{3}}}+{\frac {\mathfrak {C}}{5.6.32z^{5}}}-{\frac {\mathfrak {D}}{7.8.128z^{7}}}+$ etc.

atque per formulam 57, statuendo $n=2,$

$\log \Pi (z-{\tfrac {1}{2}})=\log \Pi 2z-\log \Pi z-(2z+{\tfrac {1}{2}})\log 2+{\tfrac {1}{2}}\log 2\pi ,$

fit

[59]	$\log \Pi (z-{\tfrac {1}{2}})=z\log z-z+{\tfrac {1}{2}}\log 2\pi -{\frac {\mathfrak {A}}{1.2.2z}}+{\frac {7{\mathfrak {B}}}{3.4.8z^{3}}}-{\frac {31{\mathfrak {C}}}{5.6.32z^{5}}}+{\frac {127{\mathfrak {D}}}{7.8.128z^{7}}}-$ etc.

Hae duae series pro valoribus magnis ipsius ${z}$ ab initio satis promte convergunt, ita ut summam approximatam commode satisque exacte colligere liceat: attamen probe notandum est, pro quovis valore dato ipsius $z,$ quantumvis magno, praecisionem limitatam tantummodo obtineri posse, quum numeri Bernoulliani seriem hypergeometricam constituant, adeoque series illae, si modo satis longe extendantur, certo e convergentibus divergentes evadant. Ceterum negari nequit, theoriam talium serierum divergentium adhuc quibusdam difficultatibus premi, de quibus forsan alia occasione pluribus commentabimur.

30.

E formula 38 sequitur

${\frac {\Pi (k,z+\omega )}{\Pi (k,z)}}={\frac {z+1}{z+1+\omega }}.{\frac {z+2}{z+2+\omega }}.{\frac {z+3}{z+3+\omega }}\ldots .{\frac {z+k}{z+k+\omega }}.k^{\omega }$

unde sumtis logarithmis, in series infinitas evolutis, prodit

[60]

${\begin{aligned}\log \Pi (k,z+\omega )=&\log \Pi (k,z)\\&+\omega \left(\log k-{\tfrac {1}{z+1}}-{\tfrac {1}{z+2}}-{\tfrac {1}{z+3}}-{\text{etc. }}-{\tfrac {1}{z+k}}\right)\\&+{\tfrac {1}{2}}\omega \omega \left({\tfrac {1}{(z+1)^{2}}}+{\tfrac {1}{(z+2)^{2}}}+{\tfrac {1}{(z+3)^{3}}}+{\text{etc.}}+{\tfrac {1}{(z+k)^{2}}}\right)\\&-{\tfrac {1}{3}}\omega ^{3}\left({\tfrac {1}{(z+1)^{3}}}+{\tfrac {1}{(z+2)^{3}}}+{\tfrac {1}{(z+3)^{3}}}+{\text{etc.}}+{\tfrac {1}{(z+k)^{3}}}\right)\\&+{\text{ etc. in inf.}}\end{aligned}}$

Series, hic in $\omega$ multiplicata, quae, si magis placet, ita etiam exhiberi potest,

$-{\tfrac {1}{z+1}}+\log 2-{\tfrac {1}{z+2}}+\log {\tfrac {3}{2}}-{\tfrac {1}{z+3}}+\log {\tfrac {4}{3}}-{\tfrac {1}{z+4}}+\log {\tfrac {5}{4}}-{\text{etc.}}+\log {\tfrac {k}{k-1}}-{\tfrac {1}{z+k}}$

e terminorum multitudine finita constat, crescente autem $k$ in infinitum, ad limitem certum converget, qui novam functionum transscendentium speciem nobis sistit, in posterum per $\Psi z$ denotandam.

Designando porro summas serierum sequentium, in infinitum extensarum,

${\begin{aligned}&{\tfrac {1}{(z+1)^{2}}}+{\tfrac {1}{(z+2)^{2}}}+{\tfrac {1}{(z+3)^{2}}}+{\text{etc. }}\\&{\tfrac {1}{(z+1)^{3}}}+{\tfrac {1}{(z+2)^{3}}}+{\tfrac {1}{(z+3)^{3}}}+{\text{etc. }}\\&{\tfrac {1}{(z+1)^{4}}}+{\tfrac {1}{(z+2)^{4}}}+{\tfrac {1}{(z+3)^{4}}}+{\text{etc. }}\\&{\text{ etc. }}\end{aligned}}$

resp. per ${P},$ ${Q},$ ${R}$ etc. (pro quibus signa functionalia introducere minus necessarium videtur), habebimus

[61]	$\log \Pi (z+\omega )=\log \Pi z+\omega \Psi z+{\tfrac {1}{2}}\omega \omega P-{\tfrac {1}{3}}\omega ^{3}Q+{\tfrac {1}{4}}\omega ^{4}R-{\text{etc.}}$

Manifesto functio $\Psi z$ erit functio derivata prima functionis $\log \Pi z,$ adeoque

[62]	${\frac {\operatorname {d} \Pi z}{\operatorname {d} z}}=\Pi z.\Psi z$

Perinde erit $P={\frac {\operatorname {d} \Psi z}{\operatorname {d} z}},Q=-{\frac {\mathrm {dd} \Psi _{z}}{2\operatorname {d} z^{2}}},R=+{\frac {\operatorname {d} ^{3}\Psi z}{2.3\operatorname {d} z^{3}}}$ etc.

31.

Functio $\Psi z$ aeque fere memorabilis est atque functio $\Pi z,$ quapropter insigniores relationes ad illam spectantes hic colligemus. E differentiatione aequationis 44 fit

[63]	$\Psi (z+1)=\Psi z+{\frac {1}{z+1}}$

unde

[64]	$\Psi (z+n)=\Psi z+{\frac {1}{z+1}}+{\frac {1}{z+2}}+{\frac {1}{z+3}}+$etc.$+{\frac {1}{z+n}}$

Huius adiumento a valoribus minoribus ipsius $z$ ad maiores progredi, vel a maioribus ad minores regredi licet: pro valoribus maioribus positivis ipsius $z$ functionis valores numerici satis commode per formulas sequentes e differentiatione aequationum 58, 59 oriundas computantur, de quibus tamen eadem sunt tenenda, quae in art. 29 circa formulas 58 et 59 monuimus:

[65]	$\Psi z=\log z+{\frac {1}{2z}}-{\frac {\mathfrak {A}}{2zz}}+{\frac {\mathfrak {B}}{4z^{4}}}-{\frac {\mathfrak {E}}{6z^{\mathrm {6} }}}+$etc.
[66]	$\Psi (z-{\tfrac {1}{2}})=\log z+{\frac {\mathfrak {A}}{2.2zz}}-{\frac {7{\mathfrak {B}}}{4.8z^{4}}}+{\frac {31{\mathcal {E}}}{6.32z^{6}}}-$ etc.

Ita pro $z=10$ computavimus

$\Psi z=2{,}3517525890\;\;6672110764\;\;743$

unde regredimur ad

$\Psi 0=-0{,}5772156649\;\;0153286060\;\;653$^[2]

Pro valore integro positivo ipsius ${z}$ fit generaliter

[67]	$\Psi z=\Psi 0+1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+$ etc. $+{\tfrac {1}{z}}$

Pro valore integro negativo autem manifesto $\Psi z$ fit quantitas infinite magna.

32.

Formula 55 nobis suppeditat $\log \Pi (-z)+\log \Pi (z-1)=\log \pi -\log \sin z\pi ,$ unde fit per differentiationem

[68]	$\Psi (-z)-\Psi (z-1)=\pi \operatorname {cotang} z\pi$

Et quum e definitione functionis $\Psi$ generaliter habeatur

[69]	$\Psi x-\Psi y=-{\frac {1}{x+1}}+{\frac {1}{y+1}}-{\frac {1}{x+2}}+{\frac {1}{y+2}}-{\frac {1}{x+3}}+$ etc.

oritur series nota

$\pi {\text{ cotang }}z\pi ={\frac {1}{z}}-{\frac {1}{1-z}}+{\frac {1}{1+z}}-{\frac {1}{2-z}}+{\frac {1}{2+z}}-{\frac {1}{3-z}}+$ etc.

Simili modo e differentiatione formulae 57 prodit

[70]	$\Psi z+\Psi (z-{\tfrac {1}{n}})+\Psi (z-{\tfrac {2}{n}})+$ etc. $+\Psi (z-{\tfrac {n-1}{n}})=n\Psi nz-n\log n$

adeoque statuendo $z=0$

[71]	$\Psi (-{\tfrac {1}{n}})+\Psi (-{\tfrac {2}{n}})+\Psi (-{\tfrac {3}{n}})+$ etc. $+\Psi (-{\tfrac {n-1}{n}})=(n-1)\Psi 0-n\log n$

Ita e.g. habetur

$\Psi (-{\tfrac {1}{2}})=\Psi 0-2\log 2=-1,9635100260\;\;2142347944\;\;099,$ unde porro

$\qquad \qquad \Psi {\tfrac {1}{2}}=+0,0364899739\;\;7857652055\;\;901.$

33.

Sicuti in art. praec. $\Psi (-{\tfrac {1}{2}})$ ad $\Psi 0$ et logarithmum reduximus, ita generaliter $\Psi (-{\tfrac {m}{n}}),$ designantibus $m,n$ integros, quorum minor $m,$ ad $\Psi 0$ et logarithmos reducemus. Statuamus ${\tfrac {2\pi }{n}}=\omega ,$ sitque $\varphi$ alicui angulorum $\omega ,$ $2\omega ,$ $3\omega \ldots (n-1)\omega$ aequalis; unde $1=\cos n\varphi =\cos 2n\varphi =\cos 3n\varphi$ etc., $\cos \varphi =\cos(n+1)\varphi =\cos(n+2)\varphi$ etc., $\cos 2\varphi =\cos(n+2)\varphi$ etc., nec non $\cos \varphi +\cos 2\varphi +\cos 3\varphi +$ etc. $+\cos(n-1)\varphi +1=0.$ Habemus itaque

$\cos \varphi .\Psi {\tfrac {1-n}{n}}=-n\cos \varphi +\cos \varphi .\log 2-{\tfrac {n}{n+1}}\cos(n+1)\varphi +\cos \varphi .\log {\tfrac {3}{2}}-$ etc.
$\cos 2\varphi .\Psi {\tfrac {2-n}{n}}=-{\tfrac {n}{2}}\cos 2\varphi +\cos 2\varphi .\log 2-{\tfrac {n}{n+2}}\cos(n+2)\varphi +\cos 2\varphi .\log {\tfrac {3}{2}}-$etc.

$\cos 3\varphi .\Psi {\tfrac {3-n}{n}}=-{\tfrac {n}{3}}\cos 3\varphi +\cos 3\varphi .\log 2-{\tfrac {n}{n+3}}\cos(n+3)\varphi +\cos 3\varphi .\log {\tfrac {3}{2}}-$etc.

etc. usque ad

$\cos(n-1)\varphi .\Psi (-{\tfrac {1}{n}})=-{\tfrac {n}{n-1}}\cos(n-1)\varphi +\cos(n-1)\varphi .\log 2-{\tfrac {n}{2n-1}}\cos(2n-1)\varphi$
$\qquad \qquad +\cos(n-1)\varphi .\log {\tfrac {3}{2}}-$ etc.

${\phantom {\cos(n-1)\varphi -}}\Psi 0=-{\tfrac {n}{n}}\cos n\varphi +\log 2-{\tfrac {n}{2n}}\cos 2n\varphi +\log {\tfrac {3}{2}}-$ etc.

atque per summationem

$\cos \varphi .\Psi {\tfrac {1-n}{n}}+\cos 2\varphi .\Psi {\tfrac {2-n}{n}}+\cos 3\varphi .\Psi {\tfrac {3-n}{n}}+$ etc. $+\cos(n-1)\varphi .\Psi (-{\tfrac {1}{n}})+\Psi 0$

$=-n(\cos \varphi +{\tfrac {1}{2}}\cos 2\varphi +{\tfrac {1}{3}}\cos 3\varphi +{\tfrac {1}{4}}\cos 4\varphi +$ etc. in infin.$)$

Sed habetur generaliter, pro valore ipsius $x$ unitate non maiori,

$\log(1-2x\cos \varphi +xx)=-2(x\cos \varphi +{\tfrac {1}{2}}xx\cos 2\varphi +{\tfrac {1}{3}}x^{3}\cos 3\varphi +{\text{etc.}})$

quae quidem series facile sequitur ex evolutione $\log(1-rx)+\log(1-{\tfrac {x}{r}}),$ denotante $r$ quantitatem $\cos \varphi +{\sqrt {-1}}.\sin \varphi .$ Hinc fit aequatio praecedens

[72]	$\cos \varphi .\Psi {\tfrac {1-n}{n}}+\cos 2\varphi .\Psi {\tfrac {2-n}{n}}+\cos 3\varphi .\Psi {\tfrac {3-n}{n}}+{\text{etc.}}+\cos(n-1)\varphi .\Psi (-{\tfrac {1}{n}})$
	$=-\Psi 0+{\tfrac {1}{2}}n\log(2-2\cos \varphi )$

Statuatur in hac aequatione deinceps $\varphi =\omega ,$ $\varphi =2\omega ,$ $\varphi =3\omega$ etc. usque ad $\varphi =(n-1)\omega ,$ multiplicentur singulae hae aequationes ordine suo per $\cos m\omega ,$ $\cos 2m\omega ,$ $\cos 3m\omega$ etc. usque ad $\cos(n-1)m\omega ,$ productorumque aggregato adiiciatur aequatio 71

$\Psi {\tfrac {1-n}{n}}+\Psi {\tfrac {2-n}{n}}+\Psi {\tfrac {3-n}{n}}+{\text{etc.}}+\Psi (-{\tfrac {1}{n}})=(n-1)\Psi 0-n\log n$

Quodsi iam perpenditur, esse

${\begin{aligned}1+\cos m\omega .\cos k\omega +\cos 2m\omega .\cos 2k\omega +\cos 3m\omega .\cos 3k\omega &\\+{\text{ etc.}}+\cos(n-1)m\omega .\cos(n-1)k\omega &=0\end{aligned}}$

denotante $k$ aliquem numerorum $1,$ $2,$ $3\ldots (n-1)$ exceptis his duobus $m$ atque $n-m,$ pro quibus summa illa fit $={\tfrac {1}{2}}n,$ patebit, ex summatione illarum aequationum prodire, post divisionem per ${\tfrac {n}{2}},$

[73]	$\Psi (-{\tfrac {m}{n}})+\Psi (-{\tfrac {n-m}{m}})=$ $2\Psi 0-2\log n+\cos m\omega .\log(2-2\cos \omega )+\cos 2m\omega .\log(2-2\cos 2\omega )$ $+\cos 3m\omega .\log(2-2\cos 3\omega )+{\text{etc.}}+\cos(n-1)m\omega .\log(2-2\cos(n-1)\omega )$

Manifesto terminus ultimus huius aequationis fit $=\cos m\omega .\log(2-2\cos \omega ),$ penultimus $=\cos 2m\omega .\log(2-2\cos 2\omega )$ etc., ita ut bini termini semper sint aequales, excepto, si $n$ est par, termino singulari $\cos {\tfrac {n}{2}}.m\omega \log(2-2\cos {\tfrac {n}{2}}\omega ),$ qui fit $=+2\log 2$ pro $m$ pari, vel $=-2\log 2$ pro $m$ impari. Combinando iam cum aequatione 73 hanc

$\Psi (-{\tfrac {m}{n}})-\Psi (-{\tfrac {n-m}{n}})=\pi \operatorname {cotang} {\tfrac {m}{n}}\pi$

habemus, pro valore impari ipsius $n,$ siquidem $m$ est integer positivus minor quam $n$

[74]

${\begin{aligned}\Psi (-{\tfrac {m}{n}})=&\Psi 0+{\tfrac {1}{2}}\pi \operatorname {cotang} {\tfrac {m\pi }{n}}-\log n+\cos {\tfrac {2m\pi }{n}}.\log(2-2\cos {\tfrac {2\pi }{n}})\\&+\cos {\tfrac {4m\pi }{n}}.\log(2-2\cos {\tfrac {4\pi }{n}})+\cos {\tfrac {6m\pi }{n}}.\log(2-2\cos {\tfrac {6\pi }{n}})+{\text{etc. }}\\&+\cos {\tfrac {(n-1)m\pi }{n}}.\log(2-2\cos {\tfrac {(n-1)\pi }{n}})\end{aligned}}$

Pro valore pari ipsius $n$ autem

[75]

${\begin{aligned}\Psi (-{\tfrac {m}{n}})=&\Psi 0+{\tfrac {1}{2}}\pi \operatorname {cotang} {\tfrac {m\pi }{n}}-\log n+\cos {\tfrac {2m\pi }{n}}\log(2-2\cos {\tfrac {2\pi }{n}})\\&+\cos {\tfrac {4m\pi }{n}}\log(2-2\cos {\tfrac {4\pi }{n}})+{\text{etc.}}+\cos {\tfrac {(n-2)m\pi }{n}}\log(2-2\cos {\tfrac {(n-2)\pi }{n}})\\&\pm \log 2\end{aligned}}$

ubi signum superius valet pro $m$ pari, inferius pro impari. Ita e.g. invenitur

${\begin{array}{ll}\Psi (-{\tfrac {1}{4}})=\Psi 0+{\tfrac {1}{2}}\pi -3\log 2,&\Psi (-{\tfrac {3}{4}})=\Psi 0-{\tfrac {1}{2}}\pi -3\log 2\\\Psi (-{\tfrac {1}{3}})=\Psi 0+{\tfrac {1}{2}}\pi {\sqrt {\tfrac {1}{3}}}-{\tfrac {3}{2}}\log 3,&\Psi (-{\tfrac {2}{3}})=\Psi 0-{\tfrac {1}{2}}\pi {\sqrt {\tfrac {1}{3}}}-{\tfrac {3}{2}}\log 3\end{array}}$

Ceterum combinatis his aequationibus cum aequatione 64 sponte patet, $\Psi z$ generaliter pro quovis valore rationali ipsius $z,$ positivo seu negativo per $\Psi {0}$ atque logarithmos determinari posse, quod theorema sane maxime est memorabile.

34.

Quum, per art. 28, $\Pi \lambda$ sit valor integralis $\int y^{\lambda }e^{-y}\operatorname {d} y,$ ab $y=0$ usque ad $y=\infty ,$ siquidem $\lambda +1$ est quantitas positiva, fit differentiando secundum $\lambda$

${\frac {\operatorname {d} \Pi \lambda }{\operatorname {d} \lambda }}={\frac {\operatorname {d} \int y^{\lambda }e^{-y}\operatorname {d} y}{\operatorname {d} \lambda }}=\int y^{\lambda }e^{-y}\log y\operatorname {d} y$

sive

[76]	$\Pi \lambda .\Psi \lambda =\int y^{\lambda }e^{-y}\log y.\operatorname {d} y,$ ab $y=0$ usque ad $y=\infty$

Generalius statuendo $y=z^{\alpha },$ $\alpha \lambda +\alpha -1=\beta ,$ valor integralis $\int z^{\varepsilon }e^{-z^{a}}\log z\operatorname {d} z,$ a $z=0$ usque ad $z=\infty ,$ fit

$={\tfrac {1}{\alpha \alpha }}\Pi ({\tfrac {\beta +1}{\alpha }}-1).\Psi ({\tfrac {\beta +1}{\alpha }}-1)={\tfrac {1}{\alpha (\beta +1)}}\Pi {\tfrac {\beta +1}{\alpha }}.\Psi {\tfrac {\beta +1}{\alpha }}-{\tfrac {1}{(\beta +1)^{2}}}\Pi {\tfrac {\beta +1}{\alpha }}$

siquidem simul $\beta +1$ atque $\alpha$ sunt quantitates positivae, vel aequalis eidem quantitati cum signo opposito, si utraque $\beta +1,$ $\alpha$ est negativa.

35.

At non solum productum $\Pi \lambda .\Psi \lambda ,$ verum etiam ipsa functio $\Psi \lambda$ per integrale determinatum exhiberi potest. Designante $k$ integrum positivum, patet valorem integralis $\int {\frac {x^{\lambda }-x^{\lambda +k}}{1-x}}.\operatorname {d} x,$ ab $x=0$ usque ad $x=1$ esse

$={\frac {1}{\lambda +1}}+{\frac {1}{\lambda +2}}+{\frac {1}{\lambda +3}}+{\text{etc.}}+{\frac {1}{\lambda +k}}$

Porro quum valor integralis $\int \left({\frac {1}{1-x}}-{\frac {kx^{k-1}}{1-x^{k}}}\right)\operatorname {d} x$ generaliter sit $=$ Const. $+\log {\frac {1-x^{k}}{1-x}},$ idem inter limites $x=0$ atque $x=1$ erit $=\log k,$ unde patet, valorem integralis $S=\int \left({\frac {1-x^{2}+x^{2+k}}{1-x}}-{\frac {kx^{k-1}}{1-x^{k}}}\right)\operatorname {d} x$ inter eosdem limites esse

$=\log k-{\frac {1}{\lambda +1}}-{\frac {1}{\lambda +2}}-{\frac {1}{\lambda +3}}-{\text{etc.}}-{\frac {1}{\lambda +k}}$

quam expressionem denotabimus per $\Omega .$ Discerpamus integrale $S$ in duas partes

$\int \left({\frac {1-x^{\lambda }}{1-x}}\right)\operatorname {d} x+\int \left({\frac {x^{\lambda +k}}{1-x}}-{\frac {kx^{k-1}}{1-x^{k}}}\right)\operatorname {d} x$

Pars prima $\int {\frac {1-x^{\lambda }}{1-x}}.\operatorname {d} x,$ statuendo $x=y^{k}$ mutatur in

$\int {\frac {ky^{k-1}-ky^{2k+k-1}}{1-y^{k}}}\operatorname {d} y$

unde sponte patet, illius valorem ab $x=0$ usque ad $x=1,$ aequalem esse valori integralis

$\int {\frac {kx^{k-1}-kx^{\lambda k+k-1}}{1-x^{k}}}\operatorname {d} x$

inter eosdem limites, quum manifesto literam $y$ sub hac restrictione in $x$ mutare liceat. Hinc fit integrale ${S},$ inter eosdem limites

$=\int \left({\frac {x^{\lambda +k}}{1-x}}-{\frac {kx^{\lambda k+k-1}}{1-x^{k}}}\right)\operatorname {d} x$

Hoc vero integrale, statuendo $x^{k}=z,$ transit in

$\int \left({\frac {z^{\frac {\lambda +1}{k}}}{k(1-z)^{\frac {1}{k}}}}-{\frac {z^{\lambda }}{1-z}}\right)\operatorname {d} z$

quod itaque inter limites $z=0$ atque $z=1$ sumtum aequale est ipsi $\Omega .$ Sed crescente $k$ in infinitum, limes ipsius $\Omega$ est $\Psi \lambda ,$ limes ipsius ${\tfrac {\lambda +1}{k}}$ est $0,$ limes ipsius $k(1-z)^{\frac {1}{k}}$ vero est $\log {\tfrac {1}{z}}$ sive $-\log z.$ Quare habemus

[77]	$\Psi \lambda =\int \left({\frac {1}{\log {\frac {1}{z}}}}-{\frac {z^{\lambda }}{1-z}}\right)\operatorname {d} z=\int \left(-{\frac {1}{\log z}}-{\frac {z^{\lambda }}{1-z}}\right)\operatorname {d} z$

a $z=0$ usque ad $z=1.$

36.

Integralia determinata, per quae supra expressae sunt functiones $\Pi \lambda ,$ $\Pi \lambda .\Psi \lambda ,$ restringere oportuit ad valores ipsius $\lambda$ tales, ut $\lambda +1$ evadat quantitas positiva: haec restrictio ex ipsa deductione demanavit, reveraque facile perspicitur, pro aliis valoribus ipsius $\lambda$ illa integralia semper fieri infinita, etiamsi functiones $\Pi \lambda ,$ $\Pi \lambda .\Psi \lambda$ finitae manere possint. Veritati formula 77 certo eadem conditio subesse debet, ut $\lambda +1$ sit quantitas positiva (alioquin enim integrale certo infinitum evadit, etiamsi functio $\Psi \lambda$ maneat finita): sed deductio formulae primo aspectu generalis nullique restrictioni obnoxia esse videtur. Sed propius attendenti facile patebit, ipsi analysi, per quam formula eruta est, hanc restrictionem iam inesse. Scilicet tacite supposuimus, integrale $\int {\frac {1-x^{\lambda }}{1-x}}\operatorname {d} x$ cui aequale $\int {\frac {kx^{k-1}-kx^{\lambda k+k-1}}{1-x^{k}}}\operatorname {d} x$ substituimus, habere valorem finitum, quae conditio requirit, ut $\lambda +1$ sit quantitas positiva. Ex analysi nostra quidem sequitur, haec duo integralia semper esse aequalia, si hoc extendatur ab $x=0$ usque ad $x=1-\omega ,$ illud ab $x=0$ usque ad $x=(1-\omega )^{k},$ quantumvis parva sit quantitas $\omega ,$ modo non sit $=0{:}$ sed hoc non obstante in casu eo, ubi $\lambda +1$ non est quantitas positiva, duo integralia ab $x=0$ usque ad eundem terminum $x=1-\omega$ extensa neutiquam ad aequalitatem convergunt, sed potius tunc ipsorum differentia, decrescente $\omega$ in infinitum, in infinitum crescet. Hocce exemplum monstrat, quanta circumspicientia opus sit in tractandis quantitatibus infinitis, quae in ratiociniis analyticis nostro iudicio eatenus tantum sunt admittendae, quatenus ad theoriam limitum reduci possunt.

37.

Statuendo in formula 77, $z=e^{-u},$ patet, illam etiam ita exhiberi posse

	$\Psi \lambda =-\int \left({\frac {e^{-u}}{u}}-{\frac {e^{-u\lambda -u}}{1-e^{-u}}}\right)\operatorname {d} u,$ ab $u=\infty$ usque ad $u=0,$ i.e.
[78]	$\Psi \lambda =\int \left({\frac {e^{-u}}{u}}-{\frac {e^{-2u}}{e^{u}-1}}\right)\operatorname {d} u,$ ab $u=0$ usque ad $u=\infty$

(Perinde valor ipsius $\Pi \lambda$ in art. 28 allatus, mutatur statuendo $e^{-y}=0,$ in sequentem

$\Pi \lambda =\int \left(\log {\frac {1}{v}}\right)^{\lambda }dv,$ a $v=0$ usque ad $v=1$)

Porro patet e formula 77, esse

[79]	$\Psi \lambda -\Psi \mu =\int {\frac {z^{\mu }-z^{\lambda }}{1-z}}\operatorname {d} z,$ a $z=0$ usque ad $z=1$

ubi praeter $\lambda +1$ etiam $\mu +1$ debet esse quantitas positiva.

Statuendo in eadem formula 77, $z=u^{\alpha },$ designante $\alpha$ quantitatem positivam, fit

$\Psi \lambda =\int \left(-{\frac {u^{\alpha -1}}{\log u}}-{\frac {au^{\alpha \lambda +\alpha -1}}{1-u^{\alpha }}}\right)\operatorname {d} u,$ ab $u=0$ usque ad $u=1$

et quum perinde statui possit, pro valore positivo ipsius $\beta ,$

$\Psi \lambda =\int \left(-{\frac {u^{\beta -1}}{\log u}}-{\frac {\beta u^{\beta \lambda +\beta -1}}{1-u^{\beta }}}\right)\operatorname {d} u$

patet, fieri

$0=\int \left({\frac {u^{\alpha -1}-u^{\beta -1}}{\log u}}+{\frac {\alpha u^{\alpha \lambda +\alpha -1}}{1-u^{\alpha }}}-{\frac {\beta u^{\beta \lambda +\beta -1}}{1-u^{\beta }}}\right)\operatorname {d} u$

sive

$\int {\frac {u^{\alpha -1}-u^{\beta -1}}{\log u}}\operatorname {d} u=\int \left({\frac {\beta u^{\beta \lambda +\beta -1}}{1-u^{\beta }}}-{\frac {\alpha u^{\alpha \lambda +\alpha -1}}{1-u^{\alpha }}}\right)\operatorname {d} u$

integralibus semper ab $u=0$ usque ad $u=1$ extensis. Sed ponendo $\lambda =0,$ integrale posterius indefinite assignari potest; est scilicet $=\log {\tfrac {1-u^{\alpha }}{1-u^{\beta }}},$ si evanescere debet pro $u=0;$ quare quum pro $u=1$ statuere oporteat ${\tfrac {1-u^{\alpha }}{1-u^{\beta }}}={\tfrac {\alpha }{\beta }},$ erit integrale $\log {\tfrac {\alpha }{\beta }}=\int {\frac {u^{\alpha -1}-u^{\beta -1}}{\log u}}\operatorname {d} u,$ ab $u=0$ usque ad $u=1,$ quod theorema olim ab ill. Euler per alias methodos erutum est.

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1. Haec eliminatio, si pro quantitatibus ipsis logarithmos introducimus, aequationibus tantammodo linearibus applicanda erit.
2. Quum hic valor inde a figura vigesima discrepet ab eo quem computavit clar. Mascheroni in Adnotat. ad Euleri Calculum Integr., adhortatus sum Fridericum Bernhardum Gothofredum Nicolai, iuvenem in calculo indefessum, at computum illum repeteret ulteriusque extenderet. Invenit itaque per calculum duplicem, scilicet descendens tum a $z=50$ tum a $z=100,$
   $\Psi 0=-0,5772156649\;\;0153286060\;\;6512090082\;\;4024310421$
   Eidem calculatori exercitatissimo etiam debetur tabulae ad finem huius Sectionis annexae pars altera, exhibens valores functionis $\Psi z$ ad 18 figuras (quarum ultima haud certa), pro omnibus valoribus ipsius $z$ a $0$ usque ad $1$ per singulas partes centesimas. Ceterum methodi, per quas utraque tabula constructa est, innituntur partim theorematibus quae hic traduntur, partim calculi artificiis singularibus, quae alia occasione proferemus.