iH )'] Mn-i,r-2,s-2)]>,

(1)

η.

DE

ΜUL TISΕC TI ONE

FÜNCTIONUM ELLIPTICARUM PRIMI GENERIS

DISSERTÅTIO ACADEMICA.

QUAM

VENIA AMPLISS. FACULT. PHILOS. UPSAL.

Γ. P.

Mao. jacobus jvicolaus granlund

Matheseos Docens Stip. Heg. Carol. Joh.

ET

LEONARDUS ÅKERBLOM

Norrlandi.

IN AUDITOR. GUSTAV. DIE I MART. MDCCCLI.

H. p. M. s.

ι

P. VI.

UPSALIAE

EXCUDEBANT REG. ACAD. TYPOGRAPHI.

MDCCCLI. \

(2)

(3)

41

si 8 sa < 2; terminos

"4)'/i=r-*+0-2

2$ S t/(μ-1,2r-2/j-i>,2s-2^f-2)i/'(«-1,2^+1,2^)

7=e r=7

?,=£-;+<(-1}r

-f[l-(-1)s] S

V'i1",2r-2|i-ö,s-Si)»/i(w-i,2/>+1 ,s-2)

P=r\

+[{^-iΗ

)'·] Q

^-

iH )'] Mn-i,r-2,s-2)]>,

si ras <2.

S· 44·

Eodem modo, Substitute η—1 _pro η in secunda aequationum (52), erit

/?„, =

(Jarcuf'^"'[!+/(«-!, 2,2)«V+/(n-i,

4,

2)«V

+/(η,4,4)αV'+/(»-1,6,2)«V+/(n-l,C,4)« V1

1,β,0)ΛrV6+,/(n-i,8,2)«*c2-\-f'.n-1,8,4)a9c4

-f-/*(n-l,8,6)rtR<?64-/'(w-l?8'8)ft8^',+ +/*(n-l,2r,2)«srea 4-/*(ii-l,2r,4)«2r<?*+y(w-l,2r,6)a2rc?s-f·...

...-ff(n-l,2r,2s)«2V#+...-j-/(n-i,2r,2r)rt2Vr⁺ ],

unde, quoniain

2_ 17_/J\n-t i.lM V"1

(Jnmi)3

^

' .—{^~a^c7\X~\{-\)n'x

^\}[4arcu)'2

2V ^,

scquitur < ,

[4+/(2,2)«V +/,(4,2)«V+/,

(4,4)«4c4 +/, (6,2)« V+/, (6,4)«V +/1(e,6KC'+/1(8,2)«'C+/1(8,i)«V+/,(8,8)A·

•+/,(8,8)+....+/,(2r,2)a"-c'+/,(2r,4)«"-«4+/|(2ise)«=--c6

+...+/l(2r,2s)«"c"+...

+/,(2r,2r)«"e"+

ubi

C

(4)

A (Zr&s)⁼F(n-ißr$s)-f/(n-l,2r,2s)

+

VS i\n-ipr-<2p$s-Z({)f(n-ißpfi(f),

^si ^r>s,

<]=i r=i

/1(2r,2e)=s/(»-i,2r,25), si r=s, si_ve, multiplicatione instituta,

Jarcu.An_^Bn_x=(Jarcuf~*[A]

[14/2(2,2)«V+/2(4,2)«V +/9(4/i)aV4+/,(6,2)«6c2+/;(6,4)a6^+/2(6,6)«6c6+/2(8,2)a8c +A(8,4)«8r 4+/2 (8,6)«V+/2(8,8)«sc8-f... +/2 (2r,2)«2rc2 +/"2(2r,4)«2rc4+/*2(2ι·,6)«2Γ£6+... +/*2(2r,2s)«2rc2s+...

...+/;(2r,2r>2Vr+.

ubi

/2(2r,2s)=/1(2r,2s)-[|-|Hri]^(2r-2'2s-2)? si

r>*>

A (2r,2s) =/, (2r,2e)^, si s= 1»

Substituto deinde ia tertia aequationum (52) n-l pro

η invenitur

C,.,=(cosniT«)^^"l^.1[l+<f(n-l,2,0)«s+().(u-l,4,0)aJ

-4-φ(ιι-Ι,4,2)«4ύ2-|-φ(«-1,6,0)α6-}-φ(η-1,β,2)αββ2

-}-(3ρ(η-1,6,4)«6ο44-φ(»-1ι8>0)Λ8+φ(,ί~15852)α8<)2

-fφ(Μ-1,8,4)«ν+φ(η-158,6)α8ο6+....

+φ(»-1,2ί·,0>2Γ -j-qp(»t-i,2i*,2)a2rc2+9>(il-ij2r,4)«2rc4+qF(n-4,2r,6)a2rc6

+.»

...+φ(ιι-1,2r ,2s)«2rc2e+...

+9(n-l,2r,2r-2)«2rcar-2+

—

j»

unde, quoniam

1 |1'_|\"-t 1\π

cos2arcii'(/l(trcu)2 2,1

=(1-«2)(/4«ιτμ)2 2V

^, sequitar, fore

~acicosarcu.Cn.l.Dn.i⁼

(l-a2)(4(trciiy ^ ) Q/j,(2,2)« -<?2

+/i(4,2)«4c2+/3(4

4)«1c4+/3(6,2)«,!c2+/3(6,4)«V+/(6J6)«6i:

+/3(8,2)«8c

2+/3 (8,4)« 8c4+/*J (s>6)u8£ 6+/5 (8,8)«8C8+....

(5)

43

...

+/3(2r,2)a2rc2+/3(2r,4)a2V+/3(2r,6)a2rc64-...

..,+/3(2r,2^a2Ve+...

+f3 (2r,2r)«2rc2r+ .·.·],

ubi

/*3(2r, 2s)=ip(n-i, 2r-l, 2s~2) r/=5-l p=r-s+q-\

+ S S V(n-1,2r-2jj~1,2s-2q-2)φ(μ-

1»2j»,2

, q=o p=(/+\

sive, multiplicatione instituta,

-tufcosarcu.C,.

(2,2)« V+/, (4,2)« V2

+/,(4,4)«V

+/,(6,2)« V +/,(6,4)« V +/,(6,β)«»«6

+/,(8,2)«V

+/,(8,4)«V +/,(8,6)«8«6 +/,(8,8)«V +

....+/,(2>·,2)α2'«2

+/,(2ί',4)α2Γ«4 +/,(2r,6)«2'c6 +

···· +/1(2r,'2s)«'ri"·+...

+/,(2r,2i-)«!'e" +.

^·

··],

-

ubi

/,(2r,2s)=

—/3(2r,2s) +/3(2i*-2,2s), si r>s,

/,(2i-,2s) ⁼

—/,(2r,2s), si

^»· ⁼«.

Sequitur igitur cx

iis,

quae

jaffl sunt dicta, fore

darcu.An_x

.Bn.x-ac2cosarcu.Cn.iDn.l

⁼

(Jarcu)'i 2 ['!+/=3(2,2)«"^"

+/5(4,2)«V+/i(4,4)«V4+/5(C,2)«e<;2+/5(6,4)«V +/5(e,6)«s«6+/5(8,2)«V+/5(8,4)«V+/;(8,C)rt8c6

; +/5(8,8)«s«»+....+

/5(2r,2)«2'c2+/1(2r,4)«3rc,+/i(2r,6)«"ce

+.··+/,(2i-,2s)«2'c2*+

...·+/, (2i-,2i-;«2'c2'+ ],

ubi

/^ir,2«)=/8(SM») +/,(2r,2»), ?

atque

JarcH.AH_vBn_i-ac*cosarcu.C».vDn.i)M[l+/6(2,2)rtV

(6)

+Λ(*.2)«4β2+/β C4'4)«4 c4+/e(6,2)a6c3+/6 (β,4)«6c4

+/(i(e,6)a6cli+/ti(8,2)«8c3+/t.(8,4)asc»+/6(8)6)aec6

+/6(8,8)a8c»+...4/e(2r,2)a"cI+/e(2r,4K'c4

+/6(2'->6)«!,«g+^{· · ·}

+/6(2r,2«y'c2«+...

_+Λ(2ι·,2 _],

ubi

/6^(2r,2s)=/5(2r,2s)^-Ftn-Zßvßs) q=s-1 p=r-s+q-i

— S S F{ti-% 2r-Zp, 2s-2£)/6(2j>,2£), sir>s,

</=1 p=(j

y6(2r,2«)=/5(2r,2«), ^si ^r=^s.

Sed secunda aequalionum (52) membrum dexlrum aequale membro dextro aequatioois (54) habet7 idque itar

I_I(_/|j'1

ut, neglecto factore communi (z1arcu)? 2 ^, coeilicientes

iisdem dignitatibus quantitatum a, c adjunctae sintaequa-

les; uude sequitur^, fore

f{n,1r,3*) =/6;2r, 2«), y(n,2r,2«·) =/5(2r,2s)^—i?[n-2,2)·,2«)

qr=s-l p=r-s-\-q-1

— S jS

i7,(ii-2,2ir-2/i,2s-2//)/'(ii,2/i,2i/),

si ^r^>^sr /(tt,2r,2s) =/ö(2r,2s)^, si r = sy

sive, substitutionibus factis,

/(»,2r,2e> ⁼t/(»i-i,2r,2i)-f- 1,2»·,2a)

p^r-s+q-1

+ S S

,2r~2/j,2s-2iy)/(n-1,2/7,2<y)

y—i v-<i

—[J Γ']

[/("-l,2r-2,2s-2) F(/i-l,2r~232*-2)

p=s-2 p~r-s+q-1

■

4

S S

^(«-1,2ι·-2/ί-2,2ί-2</-2)/(ϋ-1,2^,2^)2,

£=1 p=?

(7)

4o

q—s-lp—r-sq~l

-^(n-i,2r 1,25-2)- S S

^(ίι-1,2ΐ'-2/>-1,25-2^-2)φ(η^1,2^,2^)

9=o p~(/+1

9=5-1 p—r-s-\-q-2

+^(w-l,2r-5,2s-2)·+ S S ^(w-l,2r-2/i-5,2s-2qf-2) φ(ιι-1,2ρ,2//)

9=0 />=9+1 9=5-1 p—r-s-t(f-l

-jF\n-2,2r,2$)-S S

i^(>i-2,2r-2/j,25-2<jf)y(n,2/?,2/jf),

9=1 p~(f .

quae erit formula recursionis in compufandis functionibus

formaet/,(n,2r,25). Admonemus tarnen, Semper evanescere terminos

9=5-1p~r-s+9-I

i*Xu-l,2r,2s) ^-f S <S

i^(n-l,2r-2/?52s-2</)/(n-l^,2i/)

9=1 ;;=9

9=s-lp=r-s+q~2

4-V'(n-l,2r-2,2s-2)-j-$ g

^(n-l,2r-2p-5,25-2</-2)(f(n-l,2p,2i/)

9=0/>—9+1

- F(«-2,2r,2s)- S F(n~2,2r-2p,2s-2f/)/(n,2p,2<y)^?

9=1 />=9 si r= s5 terminos

9—5-1jff —r-5+9-ί

s S jF"(n-i,2r-2/?,2s-2i/)y(n-1,2p,2^)

9=1 ^>=9

- ß-K-O"'1]

[/(w-i?2r-2,2s-2)

+

F(n-l,2r-2,2s-2)

9=5-2 M=r-5+9-l

-f S S

F(n-l,2r-2p-2,25-2r/-2)/ (Vl,2p,2</)]

9=1 />=9 9=5-1 »=»'-«+9-2

H- S S

^(n-l,2r-2p-5,25-2^-2)qp(n-l,2p,2^)

9=0 p=9+1

9=5-1»=>'-5+9-I

- s S

F(n-2,2r-2p)2s-2^)/(n,2p,2^)7

9=1 />=9 si s=4»

(8)

§. 12.

Eodem prorsus modo, quoniam

(cosarcii)

2~ä(~*)w

1^—

iaveoitur etiam

marcM.^»-i.Cn-i={l-a2t|-^(-l)n-1]}Cco«ami)H(^/')n[l+?(2,0)rta

-j-φ,(4,0)«4 +9l(4,2)aV + φι(6,0)«6 +g>t(6,2)«6£2 -{-<^(6,4)«V1 4~Φι(^Ό)α8 4~Φ1(8,2)α8<72-{-φ1(8,4)α8<74 -fΦι(8,6)α8<?6 +^....+φ,(2ι·,0)α2Γ +^φ4(2r,2)«2V

-fΦ!(2r,4)«2rc4-fΦι(2r,6)«2r£6-f... +Φ4(2r,2s)«2rc2i +

...+φ4(2r,2r-2)«2 Vr'2 +....],

ubi

7^(2}*,2s)=<jp(ii-f,2r,2s)+jF(ti-l,2r,2s)

7=s-l p=r-s+q-1

+ S S F(n-l,2r-2p,2s-2</) <jf(n-I,2p,2</)? sis>o, 7=0 p=q+1

^1(2r,2s)⁼

qp(n-l?2r,2s)/,

si s=o, sive, multiplicatione instituta,

cosarcu, An~i'Cn-l^—

(cosarcu)i

+φ2(4,0)α4+φ,(4,3)«V + φ2(β,0)η6 + φ2(6,2>V.

+<j>2(6,.4)aV+φ2(8,0)α8 +,>2(8,2)aV⁺

φ2(8,4)«ν

+φ2(8,β)ι·ν6 + (-<j>2(2r,0)«at+φ2(2,·,2)α2ν2

+φ,(2r,4)«2'c4+^Ψι(2r,6)a2'cs+ (2/-,2s)a2'c2*

+

...

'

+i2(2r,2r-2)a2'i"·.2

ubi

φ2(2r,2s)=Φί(2r,2s)^-[§^-f(-1)η"1]φ1(2r-22s), sir>s+1, qc2(2r,2s) = φ1(2r,2s), si r = s -}- 1,

<3P2c2r,2s)=91C2r32s)-[f-i(-l)n-1], si rÄ|.

(9)

47

Deinde, quoniam

{cosarcuyiJr^r*)n ¹

^(/\arcu)2⁼^(I-α^2c2)

(cosarcu)l' )H

^">

invcnitur

—a/\arcii.Bn-i'Dn-i ⁼-(*-«

V) (cos«rrM)i*i('^n[g>3(2,0)a2

-fφ3(4,0)α4 +

Φ3(4,2)Λ4Ο2 4 Φ3(6,0)α6 -f 93(6,2)a6ca

4"φ3(6,4)<!V -|-

φ3(8,0)λ8-f φ3(8,2)αV

4

g>3(S,4)a8e4

4-φ3(8,β)«8β6 4-...·4-φ3

(2r,0)a2r 4~ φ3(2γ,2)λ2γγ2

4-φ30,4)2ν 4-φ3(2r,6)a2rc6 4-...

+ v3(8r,2*)a9'ea'

+..

-j-qp3(2r,2r-2)«2rc2r"2

ubi

g>3(2r,2s) =ψ(η-1,2ι*-1,2«) q=s-l p=7'-s+(j-l

4- S S /(ft-l,2r-2/j-2,2s-2</)tp(n-l,2/j+

4?2<y)

7=0 p=q

sive, multiplicatione instituta,

-a/\arcu. -#„_!·Ζ>„-! =

(cosarcM)!"^"1 )"[φ4(2,0)α2

4-Φ4(4,0)γ4 4-

<p4(4,2)a4c2

4-

φ4(6,0)ο6 4~φ(8)2)α6ο2

4*<jp4(6,4)«6c4

4"Φ4(8'0)α8+Φ4(8'2)α8ί;2 4^4<M)a*c4

4-Φ4(85β)"8 c6 4 4-

Φ4(2,,>°)λ2γ 4" Φ4(2r,2)a2rc2

4

φ4?(2f*j4)λ2rc4 4|φ4(2r,6)a2rc6 4-\..4-<p4(2r,2s)a2rc2e 4-,

+

φ4(2Γ52ι·-2)«2ν·24....],

ubi

Φ4(2r,2e)= -Φ3(2r,2s)4^g>3(2r-2,2s-2),

si

s>o, Φ4(2γ,2«) =-<p3(2/',2s), si 5 = 0.

Ex iis, quae jam sunt

dicta, sequitur, fore

(10)

cosarcu.Jtn-\'Cn-\-a^ttrcuiBn.i:Dn-x-{cosarcu)^''^'^

[1+φδ(2,0)α*

4-φ5(4,0)α4 4-9#(4,8)eV+ φ5(6,0)α6 4- Φδ(6,2)αV

4- qp5(6,4)rt6c4 4-φ5<8?°)λ8+φ5(8>2)Λ2+Φ^Μ)«®*4

+Φδ(8,G)a8!c6 4·^{· - * ·}4"Φ5ί^Γ)°)«2Γ4" Ψ5(2/\2)α2Γ£2

4-φ5(2r,4)«2 V4 4-φ5(2r,6>2rc64-· · - 4-Φ5(2Γ,2β)α2^2'4"■· ■ ·

4-<p5(2r,2r-2)a2r£2r*24-^{· · ·}·]>

ubi

g>5(2n,2s)= <p2(2r,2s)4-</>4(2r,2s), atque

^n-i1^n-i (t/^flVCllt £)n-i

£5^) _ —

■"»-a

=(cosarcu)h~

+<ρ6(2,0)α7+φ0(4,0)a4+<p6 (4,2)/i4c*

4-^6(6)0)«64-^!6,2]eV4-f)6(6/i)fl6fH^M«e 4- (/)6(8,2)a8rc2 4-<^ö(8,4)«8c4 4~<pö(8j6)a8c6 4-·^··.

4-<p6(2r,0)a2r4-<ft.(2r,2)a2r£2 4~<jf>6(2r,4)a2rc*

4-<p6(2r,6)a2rc64"^{· · · ·}4" <pti(2r,2s)a2'·*?2* 4-...

4-^(2r,2r-2)a^2r-24-....],

ubi " * ' '

φu(2r,2s)= ^>s(2r,2s)--F(*i~2,2r,2s)

q=s-i p=r-s+q~l '

— S S F(n-252r-2/i,25-2</^6(2/)J2^)) sii>o,

q=o p=q+i

<^ö(2r,2s) =ψo(2r,2s)^, si s= o.

Sed tertia aequationum (32) membrum dextrum aequale

meinbro dextro aequationis (33) habet, idque ita, ut, ne-

glecto factore communi {cosarcu/ϊ~

·Κ~*)",

eaedem dignitates

iH )'] Mn-i,r-2,s-2)]>,

"4)'/i=r-*+0-2

V'i*1"*,2r-2|i-ö,s-Si)»/i(w-i,2/>+1 ,s-2)

)'·] Q

iH )'] Mn-i,r-2,s-2)]>,

(Jarcuf'^"'[!+/(«-!, 2,2)«V+/(n-i,

2)«V

^

^\}[4arcu)'2

(4,4)«4c4 +/, (6,2)« V+/, (6,4)«V +/1(e,6KC'+/1(8,2)«'C+/1(8,i)«V+/,(8,8)A·

•+/,(8,8)+....+/,(2r,2)a"-c'+/,(2r,4)«"-«4+/|(2ise)«=--c6

+/,(2r,2r)«"e"+

VS i\n-ipr-<2p$s-Z({)f(n-ißpfi(f),

Jarcu.An_^Bn_x=(Jarcuf~*[A]

/2(2r,2s)=/1(2r,2s)-[|-|Hri]^(2r-2'2s-2)? si

C,.,=(cosniT«)^^"l^.1[l+<f(n-l,2,0)«s+().(u-l,4,0)aJ

-4-φ(ιι-Ι,4,2)«4ύ2-|-φ(«-1,6,0)α6-}-φ(η-1,β,2)αββ2

+φ(»-1,2ί·,0>2Γ -j-qp(»t-i,2i*,2)a2rc2+9>(il-ij2r,4)«2rc4+qF(n-4,2r,6)a2rc6

+9(n-l,2r,2r-2)«2rcar-2+

j»

=(1-«2)(/4«ιτμ)2 2V

(l-a2)(4(trciiy ^ ) Q/j,(2,2)« -<?2

4)«1c4+/3(6,2)«,!c2+/3(6,4)«V+/(6J6)«6i:

2+/3 (8,4)« 8c4+/*J (s>6)u8£ 6+/5 (8,8)«8C8+....

+/3(2r,2)a2rc2+/3(2r,4)a2V+/3(2r,6)a2rc64-...

+f3 (2r,2r)«2rc2r+ .·.·],

1»2j»,2

(2,2)« V+/, (4,2)« V2

+/,(6,2)« V +/,(6,4)« V +/,(6,β)«»«6

+/,(8,4)«V +/,(8,6)«8«6 +/,(8,8)«V +

+/,(2ί',4)α2Γ«4 +/,(2r,6)«2'c6 +

+/,(2r,2i-)«!'e" +.

··],

—/3(2r,2s) +/3(2i*-2,2s), si r>s,

—/,(2r,2s), si

iis,

jaffl sunt dicta, fore

.Bn.x-ac2cosarcu.Cn.iDn.l

(Jarcu)'i 2 ['!+/=3(2,2)«"^"

+/5(4,2)«V+/i(4,4)«V4+/5(C,2)«e<;2+/5(6,4)«V +/5(e,6)«s«6+/5(8,2)«V+/5(8,4)«V+/;(8,C)rt8c6

/5(2r,2)«2'c2+/1(2r,4)«3rc,+/i(2r,6)«"ce

...·+/, (2i-,2i-;«2'c2'+ ],

/^ir,2«)=/8(SM») +/,(2r,2»), ?

JarcH.AH_vBn_i-ac*cosarcu.C».vDn.i)M[l+/6(2,2)rtV

+/(i(e,6)a6cli+/ti(8,2)«8c3+/t.(8,4)asc»+/6(8)6)aec6

+/6(2r,2«y'c2«+...

i7,(ii-2,2ir-2/i,2s-2//)/'(ii,2/i,2i/),

,2r~2/j,2s-2iy)/(n-1,2/7,2<y)

[/("-l,2r-2,2s-2) F(/i-l,2r~232*-2)

4

^(«-1,2ι·-2/ί-2,2ί-2</-2)/(ϋ-1,2^,2^)2,

^(ίι-1,2ΐ'-2/>-1,25-2^-2)φ(η^1,2^,2^)

i^(>i-2,2r-2/j,25-2<jf)y(n,2/?,2/jf),

i^(n-l,2r-2/?52s-2</)/(n-l^,2i/)

^(n-l,2r-2p-5,25-2</-2)(f(n-l,2p,2i/)

[/(w-i?2r-2,2s-2)

F(n-l,2r-2,2s-2)

F(n-l,2r-2p-2,25-2r/-2)/ (Vl,2p,2</)]

^(n-l,2r-2p-5,25-2^-2)qp(n-l,2p,2^)

F(n-2,2r-2p)2s-2^)/(n,2p,2^)7

2~ä(~*)w

marcM.^»-i.Cn-i={l-a2t|-^(-l)n-1]}Cco«ami)H(^/')n[l+?(2,0)rta

qp(n-l?2r,2s)/,

(cosarcu)i

φ2(8,4)«ν

+

{cosarcuyiJr^r*)n 1

(cosarcu)l' )H

V) (cos«rrM)i*i('^n[g>3(2,0)a2

Φ3(4,2)Λ4Ο2 4 Φ3(6,0)α6 -f 93(6,2)a6ca

φ3(8,0)λ8-f φ3(8,2)αV

g>3(S,4)a8e4

(2r,0)a2r 4~ φ3(2γ,2)λ2γγ2

+ v3(8r,2*)a9'ea'

4?2<y)

(cosarcM)!"^"1 )"[φ4(2,0)α2

<p4(4,2)a4c2

φ4(6,0)ο6 4~φ(8)2)α6ο2

4"Φ4(8'0)α8+Φ4(8'2)α8ί;2 4^4<M)a*c4

Φ4(2,,>°)λ2γ 4" Φ4(2r,2)a2rc2

V'i1",2r-2|i-ö,s-Si)»/i(w-i,2/>+1 ,s-2)

{cosarcuyiJr^r*)n ¹