The Work of Niels Henrik Abel

157 Pages·2007·3.57 MB·English

The Work of Niels Henrik Abel
ChristianHouzel
1 FunctionalEquations
2 IntegralTransformsandDefiniteIntegrals
3 AlgebraicEquations
4 HyperellipticIntegrals
5 AbelTheorem
6 Ellipticfunctions
7 DevelopmentoftheTheoryofTransformation
ofEllipticFunctions
8 FurtherDevelopmentoftheTheoryofEllipticFunctions
andAbelianIntegrals
9 Series
10 Conclusion
References
During his short life, N.-H. Abel devoted himself to several topics characteristic
of the mathematics of his time. We note that, after an unsuccessful investigation
of the influence of the Moon on the motion of a pendulum, he chose subjects in
puremathematicsratherthaninmathematicalphysics.Itispossibletoclassifythese
subjectsinthefollowingway:
1. solutionofalgebraicequationsbyradicals;
2. newtranscendental functions,inparticular ellipticintegrals, ellipticfunctions,
abelianintegrals;
3. functionalequations;
4. integraltransforms;
5. theoryofseriestreatedinarigourousway.
The first two topics are the most important and the best known, but we shall
see that there are close links between all the subjects in Abel’s treatment. As the
firstpublishedpapersarerelatedtosubjects3and4,wewillbeginourstudywith
functionalequationsandtheintegraltransforms.
22 C.Houzel
1 Functional Equations
In the year 1823, Abel published two norwegian papers in the first issue of Ma-
gasinet for Naturvidenskaberne, a journal edited in Christiania by Ch. Hansteen.
In the first one, titled Almindelig Methode til at finde Funktioner af een variabel
Størrelse,naarenEgenskabafdisseFunktionererudtryktvedenLigningmellom
toVariable(Œuvres,t.I,p.1–10),Abelconsidersaverygeneraltypeoffunctional
equation: V(x,y,ϕα, fβ,Fγ,... ,ϕα, f β,F γ,...) 0, where ϕ, f,F,... are
′ ′ ′
=
unknownfunctionsinonevariableandα,β,γ,... areknownfunctionsofthetwo
independent variables x,y. His method consists in successive eliminations of the
unknownϕ, f,F,... betweenthegivenequationV 0andtheequationsobtained
=
by differentiating this equation with α constant, then with β constant, etc. If, for
instanceα const,thereisarelationbetweenx and y,and ymaybeconsideredas
=
afunctionofx andtheconstantvalueofα;ifn isthehighestorderofderivativeof
ϕpresentinV,itispossibletoeliminateϕαanditsderivativesbydifferentiatingV
n 1timeswithαconstant.Wetheneliminate fβanditsderivative,andsoon,until
+
wearriveatadifferentialequationwithonlyoneunknownfunctionofonevariable.
Naturally, all the functions, knownand unknown, are tacitly supposed indefinitely
differentiable.
Abelappliesthistotheparticularcaseϕα f(x,y,ϕβ,ϕγ),where f,α,β and
=
γ are given functions and ϕ is unkown; he gets a first order differential equation
with respect to ϕ. For instance, the functional equation of the logarithm logxy
=
logx logycorrespondstothecasewhereα(x,y) xy,β(x,y) x,γ(x,y) y
+ = = =
and f(x,y,t,u) t u;differentiatingwithxy const,weget0 xϕx yϕ y,
′ ′
fromwhich,with=y +const,wegetϕx c,whe=rec yϕ y.Inthe=samew−ay,the
= ′ = x = ′
functionalequationforarctangent,
x y
arctan + arctanx arctany,
1 xy = +
−
correspondstoα(x,y) = 1x+xyy,β(x,y) = x,γ(x,y) = y and f(x,y,t,u) = t+u;
differentiatingwithαconsta−ntgives0 (1 x2)ϕx (1 y2)ϕ y,whenceϕx
′ ′ ′
c ifc (1 y2)ϕ y. = + − + =
1 x2 = + ′
+ Whenβ(x,y) x,γ(x,y) yand f(x,y,t,u) t u,wegetfirst
= = = ·
∂α ∂α
ϕy ϕx ϕx ϕ y 0,
′ ′
· ∂y − · ∂x =
whence ϕ′x asaknownfunctionofxifyissupposedconstant.Forα(x,y) x y,
ϕx = +
this gives ϕ′x c ϕ′y, so logϕx cx(for ϕ(0) 1) and ϕx ecx; for
ϕx = = ϕy = = =
α(x,y) xy, ϕ′x c,sologϕx clogx (ϕ(1) 1)andϕx xc.
= ϕx = x = = =
Alltheseexampleswereclassicalasisthenextone,comingfrommechanics.The
lawofcompositionoftwoequalforcesmakinganangle2x leadstothefunctional
equation
TheWorkofNielsHenrikAbel 23
ϕx ϕy ϕ(x y) ϕ(x y) (1)
· = + + − ;
where ϕx is the ratio of the resultant force to one of the two equal forces. Differ-
entiating with y x const, one gets ϕx ϕy ϕx ϕ y 2ϕ(x y); another
′ ′ ′
+ = · − · = −
differentiation,withx y const,givesϕ x ϕy ϕx ϕ y 0.Ifyisregardedas
′′ ′′
− = · − · =
constant,thisgivesϕ x cϕx 0andϕx αcos(βx γ)withα,βandγ constant.
′′
From(1),oneseesthat+α 2=andγ 0=andthepro+blemimposesϕ π 0,so
= = 2 =
β 1andϕ(x) 2cosx.
= = (cid:2) (cid:3)
HereisanothercaseofapplicationofAbel’sgeneralmethod:theequationhas
theformψα F(x,y,ϕx,ϕx,... , fy, f y,...),whereαisagivenfunctionofx
′ ′
=
and y and ϕ, f,ψ are unknown functions. By differentiating with α constant, one
gets arelation between x,ϕx,ϕx,... and y, fy, f y,..., whencetwodifferential
′ ′
equations,withrespecttoϕandto f,consideringsuccessively yandx asconstant;
if ϕ and f are determined, it is easy to determine ψ by the functional equation.
In particular, if ψ(x y) ϕx f y fy ϕx, so that α(x,y) x y, the
′ ′
+ = · + · = +
differentiationwithαconstantgivesϕx f y fy ϕ x 0,andϕx asin(bx c),
′′ ′′
· − · = = +
fy a sin(by c) then ψ(x y) aabsin(b(x y) c c) so that ψα
′ ′ ′
= + + = + + + =
aabsin(bα c c).
′ ′
+ +
Inthecaseofψ(x y) f(xy) ϕ(x y),onegets0 f (xy)(y x) 2ϕ(x y).
′ ′
+ = + − = − + −
Abeltakesxy casconstantandwritesϕα kα,whereα x yandk f′(c),
= ′ = = − = 2
soϕα k kα2;thenhetakesx y cconstantandwrites f β c 2ϕ′c,so
= ′+ 2 − = ′ = ′ = c
fβ c cβ.Finally
′′ ′
= +
k
ψ(x y) c cxy k (x y)2
′′ ′ ′
+ = + + + 2 −
orψα c cx(α x) k k(2x α)2 c kα2 k xα(c 2k) (2k c)x2
= ′′+ ′ − + ′+2 − = ′′+2 + ′+ ′− + − ′
andweseethattheconditionc 2kisnecessary;ψα k c kα2.
′ = = ′+ ′′+ 2
Thethirdexampleisϕ(x y) ϕx fy fx ϕy,whichgives
+ = · + ·
0 ϕx fy ϕx f y f x ϕy fx ϕ y (2)
′ ′ ′ ′
= · − · + · − · ;
ifonesupposesthat f(0) 1andϕ(0) 0,onegets0 ϕx ϕx c fx c by
′ ′
= = = − · + ·
making y 0(c f (0)andc ϕ(0));so fx kϕx kϕx and,substituting
′ ′ ′ ′ ′
= = =− = +
thisvaluein(2)andmaking yconstant:ϕ x aϕx bϕx 0etc.
′′ ′
+ + =
Abelreturnedtothestudyoffunctionalequationsinthepaper“Recherchedes
fonctions de deux quantite´s variables inde´pendantes x et y, telles que f(x,y), qui
ontlaproprie´te´que f(z, f(x,y))estunefonctionsyme´triquedez,xety”,published
inGermaninthefirstvolumeofCrelle’sJournalin1826(Œuvres,t.I,p.61–65).
Theconditionofthetitlecharacterisesacompositionlawwhichisassociativeand
commutative;itmaybewrittenas f(x,y) f(y,x), f(z, f(x,y)) f(x, f(y,z))
= = =
f(y, f(z,x))or
f(z,r) f(x,v) f(y,s) (3)
= =
if f(x,y) r, f(y,z) v and f(z,x) s.Differentiating with respect to x,to y
= = =
andtozandmultiplyingtheresults,onegets
24 C.Houzel
∂r ∂v∂s ∂r ∂v ∂s
. (4)
∂x∂y∂z = ∂y∂z ∂x
But, by the definition of v, the quotient of ∂v by ∂v is a function ϕy when z is
∂y ∂z
regarded as constant; in the same manner, ϕx is the quotient of ∂s by ∂s, so (4)
∂x ∂z
becomes ∂rϕy ∂rϕx andthisgivesr asanarbitraryfunctionψ ofΦ(x) Φ(y),
∂x = ∂y +
whereΦ isaprimitiveofϕ.So f(x,y) ψ(Φ(x) Φ(y));puttingthisexpression
= +
in (3) and making Φz Φy 0 and Φx p, one gets Φψp p c, where
= = = = +
c Φψ(0),andthenΦf(x,y) Φ(x) Φ(y) cor
= = + +
Ψf(x,y) Ψ(x) Ψ(y) (5)
= +
whereΨ(x) Φ(x) c.Inotherwords,Abelfindsthat f isconjugatetotheaddition
= +
lawbythefunctionΨ:hehasdeterminedtheone-parametergroups.
ThesecondvolumeofCrelle’sJournal(1827)(Œuvres,t.I,p.389–398)contains
anotherpaperofAbelonafunctionalequation:
ϕx ϕy ψ(xfy yfx) ψ(r), (6)
+ = + =
where r xfy yfx; this equation includes, as particular cases, the laws of
addition f=or log (+fy 1y, ϕx ψx logx) and for arcsin (fy 1 y2,
= 2 = = = −
ϕx ψx arcsinx).Onehasϕx ψr ∂r,ϕ y ψr ∂r,soϕx ∂r ϕ y ∂r
= = ′ = ′ · ∂x ′ = ′ · ∂y ′ · ∂y =(cid:4) ′ · ∂x
or
ϕ y (fy yf x) ϕx (fx xf y), (7)
′ ′ ′ ′
· + = · +
whence,for y 0,
=
aα ϕx (fx αx) 0, (8)
′ ′
− · + =
where a ϕ(0), α f(0) and α f (0), a differential equation which deter-
′ ′ ′
= = =
mines ϕ if f is known. Substituting in (7), one gets (fx αx)(fy yf x)
′ ′
+ + =
(fy α y)(fx xf y)or
′ ′
+ +
1 1
(α fy fy f y α yf y) (α fx fx f x αxf x) m,
′ ′ ′ ′ ′ ′ ′ ′
y − · − = x − · − =
necessarilyconstant.So
f x (fx αx) (mx α fx) 0, (9)
′ ′ ′
· + + − =
which determines f; as this differential equation is homogenenous, it is easily
integretedbyputting fx = xz,intheformlogc−logx = 12log(z2−n2)+2αn′ logzz−nn,
wherem n2andcisaconstantofintegration.Onegets +
=−
c2n (fx nx)n+α′(fx nx)n−α′,
= − +
TheWorkofNielsHenrikAbel 25
withc α,thenϕby(8)and(6)isverifiedifψx ϕ x ϕ(0).Abelexplicitly
= = α +
treatsthecaseinwhichn α 1: fx α 1x,thenϕx aαlog(α x) k
= ′ = 2 = + 2 (cid:2) (cid:3) = + +
andψx 2k aαlog(α2 x).
= + +
Therelationα2n (fx nx)n+α′(fx nx)n−α′,whichdetermines f,allowsusto
= − +
express fx nx,andthenxand fx,intermsof fx nx v;turningbackto(8),this
givesϕx −aα log(cnx cfx).Whenn 0,the+relatio=nwhichdetermines f takes
= n+α′ fx + =
theformeα′x fx andwehaveϕx aαlogcα aαx ,ψx 2aαlogcα ax .
= α = α′ + fx = α′ + f(αx)
Theequation (6(cid:5))sig(cid:6)nifies that αf xfy+αyfx = fx · fy and Abel verifies that it is
satisfied. Another particular case i(cid:5)s that in(cid:6)which α′ . When m is finite, (9)
= ∞
reducestoxf x fx 0,sothat fx cx;whenmisinfiniteandequals pα,(9)
′ ′
− = = −
becomesxfx px fx 0and fx pxlogcx.Inthislastcase,onegetsby(7)
− − = =
yϕ y xϕx 0,whencexϕx kconstantandϕx klogmx(anewm)andthen
′ ′ ′
− = = =
ψ(pvlogc2v) klogm2v.
=
AmemoirleftunpublishedbyAbelisdevotedtotheequationϕx 1 ϕ(fx),
+ =
where f isgivenandϕunknown(Œuvres,t.II,p.36–39,mem.VI).Abelintroduces
afunctionψ suchthat fψy ψ(y 1);onemaytakeψ arbitrarilyontheinterval
= +
0,1 and define ψ on 0, by ψ(y n) fn(ψy) (and on ,0 by
[ ] [ +∞[ + = ] − ∞ ]
ψ(y n) f n(ψy) if f is bijective). For x ψy, the functional equation
−
− = =
becomes 1 ϕψy ϕψ(y 1), so that ϕψy y χy where χ is any periodic
+ = + = +
functionof ywithperiod1.Denotingtheinversefunctionofψ by`ψ,Abelgets
ϕx `ψx χ(`ψx).
= +
Asanexample,hetakes fx xn andψy any,sothat`ψx loglogx−logloga and
= = = logn
loglogx logloga loglogx logloga
ϕx − χ − ,
= logn + logn
(cid:7) (cid:8)
forinstanceϕx loglogx ifχ 0anda e.
= logn = =
Abel treats in a similar manner the general equation F(x,ϕ(fx),ϕ(ψx)) 0,
=
where F, f andψ aregivenfunctionsandϕ isunknown.Supposingthat fx y
t
=
andψx y or y ψ(‘fy),onehas F(‘fy,u ,u ) 0,whereu ϕy;
t 1 t 1 t t t t 1 t t
this diffe=renc+e equat+ion=has a solution u θt and ϕz + θ(=‘y). For instan=ce the
t t
equation(ϕx)2 ϕ(2x) 2leadsto(u )2 =u 2and=,ifu a 1,thisgives
ut = a2t−1 + a2=t1−1;onth+eotherhand ytt+1== 2t+y1t,+sothat yt =1c=·2t−+1 a(cconstant)
and2t 1 x.Finally,ϕx bx b x (b a1/c).Aswesee,thistypeofequations
− = c = + − =
is treated with a method different from the preceeding one, by reduction to finite
differenceequation.
Anothertypeoffunctionalequationisrelatedtothedilogarithm
x2 x3 xn
ψx x ... ... ,
= + 22 + 32 + + n2 +
whichAbelstudiesintheposthumousmemoirXIV(Œuvres,t.II,p.189–193)after
Legendre’s Exercices de Calcul inte´gral. The study is based on the summation of
26 C.Houzel
theseries(for x 1)intheformofanintegral
| |≤
dx
ψx log(1 x) (10)
=− x −
(cid:9)0
andAbelreproducesseveralfunctionalequationsgivenbyLegendre,asforexample
π2
ψx ψ(1 x) logx log(1 x).
+ − = 6 − · −
Butheaddsaremarkablenewproperty:
x y y x
ψ ψ ψ (11)
1 x · 1 y = 1 x + 1 y
(cid:7) − − (cid:8) (cid:7) − (cid:8) (cid:7) − (cid:8)
ψy ψx log(1 y)log(1 x)
− − − − −
for(x,y)intheinteriordomainofthefigure
Inordertoprove(11),Abelsubstitutes a y forx in(10):
1 a · 1 y
− −
a y dy dy 1 a y
ψ log − −
1 a · 1 y =− y + 1 y (1 a)(1 y)
(cid:7) − − (cid:8) (cid:9) (cid:7) − (cid:8) − −
dy y dy
log 1 log(1 y)
=− y − 1 a + y −
(cid:9) (cid:7) − (cid:8) (cid:9)
dy a dy
log 1 log(1 a)
− 1 y − 1 y + 1 y −
(cid:9) − (cid:7) − (cid:8) (cid:9) −
y dy a
ψ ψy log 1
= 1 a − − 1 y − 1 y
(cid:7) − (cid:8) (cid:9) − (cid:7) − (cid:8)
log(1 a)log(1 y),
− − −
TheWorkofNielsHenrikAbel 27
wheretheremainingintegraliscomputedbytakingz a asvariable:
= 1 y
−
dy a dz a
log 1 log(1 z) ψz ψ const.
1 y − 1 y = z − =− =− 1 y +
(cid:9) − (cid:7) − (cid:8) (cid:9) (cid:7) − (cid:8)
Theconstantofintegrationisdeterminedbytaking y 0andisfoundtobeψa.
=
Abelwasthefirstmathematiciantogiveageneraland(almost)rigourousproof
ofNewton’sfamousbinomialformula
m(m 1) m(m 1)(m 2)
(1 x)m 1 mx − x2 − − x3 ... (12)
+ = + + 2 + 2 3 +
·
He published his demonstration in the first volume of Crelle’s Journal (1826,
Recherches sur la se´rie 1 mx m(m 1)x2 m(m 1)(m 2)x3 ..., Œuvres, t. I,
+ + 2− + −23 − +
p.218–250).HeusesanideaofEuler,alreadyexploit·edbyLagrangeandCauchy:
writingϕ(m)thesecondmemberof(12),oneprovesthat
ϕ(m n) ϕ(m)ϕ(n), (13)
+ =
sothatϕ(m) Am (1 x)m form rationalaswasobservedbyEuler.Lagrange
= = +
extendedthisprooftoeveryvalueofmadmittingthatϕisananalyticfunctionofm.
Cauchyusedananalogousstrategyformrealand x <1usingthecontinuityofϕ,
| |
forwhichhisproofwasunfortunatelyincomplete.Abelconsidersthemostgeneral
case,with x andm complex,with x < 1or x 1andRem > 1(if x 1,
| | | | = − = −
oneneedsRem >0).
For m k ki, ϕ(m) f(k,k)(cosψ(k,k) isinψ(k,k)), with f, ψ
′ ′ ′ ′
= + = +
continuous functions of k, k real. The continuity is almost established by Abel
′
in his theorem V, but this theorem is not entirely correct. The concept of uniform
convergencedidnotexistatthattimeanditwasnoteasytogiveageneraltheorem
for the continuity of the sum of a series of continuous functions. The functional
equation(13)becomes
f(k ℓ,k ℓ) f(k,k)f(ℓ,ℓ) (14)
′ ′ ′ ′
+ + = ;
ψ(k ℓ,k ℓ) 2mπ ψ(k,k) ψ(ℓ,ℓ),
′ ′ ′ ′
+ + = + +
where m is an integer, which must be constant because of the continuity of ψ. In
afirststep,Abeltreatsthefunctionalequationforψ;puttingθk ψ(k,k ℓ)
′ ′
= + =
2mπ ψ(k,k) ψ(0,ℓ)hegets
′ ′
+ +
θk θℓ a θ(k ℓ), (15)
+ = + +
witha 2mπ ψ(0,k) ψ(0,ℓ),whence
′ ′
= + +
θk ck a, (16)
= +
wherecisafunctionofk,ℓ.Indeed,takingℓ k,2k,... ,ρkin(15)andadding
′ ′
=
theresults,Abelgetsρθk (ρ 1)a θ(ρk)andθρ ρ(θ(1) a) afork 1,
= − + = − + =
28 C.Houzel
ρ anaturalinteger;then,fork µ (µ,ρ N,ρ 0),ρθ µ (ρ 1)a θµ
= ρ ∈ (cid:6)= ρ = − +
andθ µ cµ a,withc θ(1) a.Thisformulaisex(cid:5)ten(cid:6)dedtothenegative
ρ = ρ + = −
values(cid:5)of(cid:6)kusingθ( k) 2a θkand,bycontinuity,toeveryrealvalueofk.So
− = −
ψ(k,k ℓ) ck 2mπ ψ(0,k) ψ(0,ℓ), (17)
′ ′ ′ ′
+ = + + +
wherec θ(k,ℓ),afunctionofk andℓ.Fork 0,thisgives
′ ′ ′ ′
= =
ψ(0,k ℓ) 2mπ ψ(0,k) ψ(0,ℓ),
′ ′ ′ ′
+ = + +
afunctionalequationwhichmaybetreatedas(15)andwhichhasthesolution
ψ(0,k) βk 2mπ,
′ ′ ′
= −
withanarbitraryconstantβ;then(17)becomes
′
ψ(k,k ℓ) θ(k,ℓ) k β(k ℓ) 2mπ,
′ ′ ′ ′ ′ ′ ′
+ = · + + −
alsoequalto2mπ ψ(k,k) ψ(0,ℓ) ψ(k,k) βℓ by(14),sothatψ(k,k)
′ ′ ′ ′ ′ ′
+ + = + =
Fk k βk 2mπ,with Fk θ(k,ℓ)independentofℓ and F(k ℓ) Fk
′ ′ ′ ′ ′ ′ ′ ′ ′ ′
· + − = + = =
F(0) βaconstant.Finally
=
ψ(k,k) βk βk 2mπ. (18)
′ ′ ′
= + −
To treat the functional equation (14) for f, Abel writes f(k,k′) eF(k,k′) and
=
F(k ℓ,k ℓ) F(k,k) F(ℓ,ℓ), a functional equation analog to that for ψ
′ ′ ′ ′
+ + = +
with m 0, so its solution is of the form F(k,k) δk δk, with two arbitrary
′ ′ ′
= = +
constantδ,δ.Finally
′
ϕ(k k′i) eδk+δ′k′(cos(βk β′k′) isin(βk β′k′)) (19)
+ = + + +
anditremainstodeterminetheconstantsβ,β,δandδ.
′ ′
Fork 1andk 0,ϕ(1) 1 x 1 αcosφ iαsinφ,whereα x <1
′
= = = + = + + =| |
andφ argx;thisgiveseδcosβ 1 αcosφandeδsinβ αsinφ,sothat
= = + =
αsinφ
eδ (1 2αcosφ α2)12 and tanβ ,β s µπ, (20)
= + + = 1 αcosφ = +
+
with π s π andµ Z.Now,fork 0andanyk,let p fαandq θα
−2 ≤ ≤ 2 ∈ ′ = = =
designate the real and the imaginary part of the series ϕ(k), which are continuous
functionsofαafterAbel’stheoremIV(whichiscorrect);onehas
fα eδkcoskscoskµπ eδksinkssinkµπ,
= −
θα eδksinkscoskµπ εδκcoskssinkµπ
= +
andcoskµπ e δk(fα cosks θα sinks),sinkµπ e δk(θα cosks fα sinks),
− −
= · + · = · − ·
independent of α by continuity. For α 0, eδ 1 and s 0 after (19) whereas
= = =
fα 1andθα 0,sokµπ 0and
= = =
TheWorkofNielsHenrikAbel 29
fα (1 2αcosφ α2)2k cosks, θα (1 2αcosφ α2)2k sinks (21)
= + + = + + ;
thisisCauchy’sresultfor fα iθα 1 x k(cosks isinks) (1 x)k.
+ =| + | + = +
Abel now considers the case in which m in is purely imaginary; then the
=
series (12) is convergent for any value of n by d’Alembert’s rule (which is Abel’s
theoremII)andAbelstatesitscontinuityasafunctionofnasaconsequenceofhis
theoremV.Hewritestherealandimaginarypartsoftheseriesintheform
p 1 λ αcosθ ... λ αµcosθ ...
1 1 µ µ
= + + + +
and q λ αsinθ ... λ αµsinθ ... ,
1 1 µ µ
= + + +
whereλ δ δ ...δ ,θ µφ γ γ ... γ and
µ 1 2 µ µ 1 2 µ
= = + + + +
ni µ 1
− + δ (cosγ isinγ ).
µ µ µ
µ = +
From(19)heknowsthat p eδ′ncosβ′nandq eδ′nsinβ′n;inordertodetermine
= =
δ′resp.β′,hetakesthelimitsof eδ′nconsβ′n−1 resp. eδ′nsninβ′n forn =0.Asδµ → µµ−1
andγ π(µ 2;forµ 1,γ π),hegets λµ 1 andγ µ(φ π) π
µ → ≥ = 1 = 2 n → µ µ → + − 2
so
1 1
β αcosφ α2cos2φ aα3cos3φ ... ,
′
= − 2 + 3 −
1 1
δ αsinφ α2sin2φ α3sin3φ ...
′
=− + 2 − 3 +
Now,computinginthesamemannerthelimits,fork =0,of fαk−1 and θkα,onegets
from(21)
1 1
δ αcosφ α2cos2φ α3cos3φ ... (22)
= − 2 + 3 −
1 1
and β αsinφ α2sin2φ α3sin3φ ... ,
= − 2 + 3 −
sothatβ δandδ β.Thesum(19)oftheseries(12)form k ki is
′ ′ ′
= =− = +
eδk−βk′(cos(βk δk′) isin(βk δk′))
+ + +
withβandδasin(20).LetusinterpretAbel’sresult:writingδ iβ log(1 x),
+ = +
onegets
mlog(1 x) (k ik)(δ iβ) kδ kβ i(kβ kδ),
′ ′ ′
+ = + + = − + +
sothatϕ(m) (1 x)m.
= +
Comparing(20)and(22),Abelgets
1 1 1
log(1 2αcosφ α2) αcosφ α2cos2φ α3cos3φ ...
2 + + = − 2 + 3 −
30 C.Houzel
and
αsinφ 1 1
arctan αsinφ α2sin2φ α3sin3φ ... (23)
1 αcosφ = − 2 + 3 − ;
+
bymakingαtendtoward 1,1log(2 2cosφ) cosφ 1cos2φ 1cos3φ ...
± 2 ± =± −2 ±3 −
and 1φ sinφ 1sin2φ 1sin3φ ... for π < φ < π. If φ π and
2 = − 2 + 3 − − = 2
1 α 1in(23),onegetsGregory’sseriesarctanα α 1α3 1α5 ...
− ≤ ≤ = − 3 + 5 −
Takingx itanφandm realinthebinomialseries,Abel’sfinds
=
m(m 1) m(m 1)(m 2)(m 3)
cosmφ (cosφ)m 1 − (tanφ)2 − − − (tanφ)4 ... ,
= − 1 2 + 1 2 3 4 −
(cid:7) · · · · (cid:8)
m(m 1)(m 2)
sinmφ (cosφ)m mtanφ − − (tanφ)3 ...
= − 1 2 3 +
(cid:7) · · (cid:8)
for π φ π (forφ π,m mustbe> 1).
−4 ≤ ≤ 4 =±4 −
Now,taking x 1andm > 1,hefindsastherealpartof
| |= −
(1 x)m(cosα isinα)
+ − :
m m(m 1)
cosα cos(α φ) − cos(α 2φ) ...
+ 1 − + 1 2 − +
·
mφ
m
(2 2cosφ)2 cos α mρπ
= + − 2 +
(cid:7) (cid:8)
where ρ is an integer such that φ 2ρπ π (with the restriction m > 0 in
case of equality). The substitution|s −φ 2x| ≤and α mx,mx π,m x π or
= = + 2 + 2
m x π π giveAbelvariousformulae,forinstance
+ 2 − 2 (cid:2) (cid:3)
(cid:2) (cid:3) m m(m 1)
(2cosx)mcos2mρπ cosmx cos(m 2)x − cos(m 4)x ...
= + 1 − + 1 2 − +
·
m m(m 1)
(2cosx)msin2mρπ sinmx sin(m 2)x − sin(m 4)x ...
= + 1 − + 1 2 − +
·
for2ρπ π x 2ρπ π.Abelwasthefirsttoproverigourouslysuchformulae
− 2 ≤ ≤ + 2
form noninteger;inalettertohisfriendHolmboe(16January1826,Œuvres,t.II,
p. 256), he states his result and alludes to the unsuccessful attempts of Poisson,
Poinsot,PlanaandCrelle.
Other examples of functional equations in Abel’s work may be mentioned, as
thefamousAbeltheorem(see§5),whichmaybeinterpretedinthisway.Inaletter
toCrelle(9August1826,Œuvres,t.II,p.267),Abelstateshistheoremforgenus2
inaveryexplicitmanner:heconsidersthehyperellipticintegralϕ(x) (α+βx)dx
= √P(x)
where Pisapolynomialofdegree6;thenAbel’stheoremisthefunctionalequation
(cid:10)
ϕ(x ) ϕ(x ) ϕ(x )=C (ϕ(y ) ϕ(y )),wherex ,x andx areindependant
1 2 3 1 2 1 2 3
+ + − +
variables,C isaconstantand y , y aretherootsoftheequation
1 2