The Work of Niels Henrik Abel Christian Houzel 1 Functional Equations 2 Integral Transforms and Definite Integrals 3 Algebraic Equations 4 Hyperelliptic Integrals 5 Abel Theorem 6 Elliptic functions 7 Development of the Theory of Transformation of Elliptic Functions 8 Further Development of the Theory of Elliptic Functions and Abelian Integrals 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 pure mathematics rather than in mathematical physics. It is possible to classify these subjects in the following way: 1. solution of algebraic equations by radicals; 2. new transcendental functions, in particular elliptic integrals, elliptic functions, abelian integrals; 3. functional equations; 4. integral transforms; 5. theory of series treated in a rigourous way. 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 first published papers are related to subjects 3 and 4, we will begin our study with functional equations and the integral transforms. 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, naar en Egenskab af disse Funktioner er udtrykt ved en Ligning mellom to Variable (Œuvres, t. I, p. 1–10), Abel considers a very general type of functional equation: V(x, y, ϕα, fβ, Fγ,... ,ϕ′α, f ′β, F′γ,...) 0, where ϕ, f, F,... are unknown functions in one variable and α, β, γ, . are= known functions of the two independent variables x, y. His method consists in successive eliminations of the unknown ϕ, f, F,... between the given equation V 0 and the equations obtained by differentiating this equation with α constant, then= with β constant, etc. If, for instance α const, there is a relation between x and y, and y may be considered as a function of= x and the constant value of α; if n is the highest order of derivative of ϕ present in V, it is possible to eliminate ϕα and its derivatives by differentiating V n 1 times with α constant. We then eliminate fβ and its derivative, and so on, until we+ arrive at a differential equation with only one unknown function of one variable. Naturally, all the functions, known and unknown, are tacitly supposed indefinitely differentiable. Abel applies this to the particular case ϕα 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 log xy log x log y corresponds to the case where α(x, y) xy, β(x, y) x, γ(x, y) =y + = = = and f(x, y, t, u) t u; differentiating with xy const, we get 0 xϕ′x yϕ′ y, = + c = = − from which, with y const, we get ϕ′x x , where c yϕ′ y. In the same way, the functional equation= for arctangent, = = x y arctan + arctan x arctan y, 1 xy = + − x y corresponds to α(x, y) 1 +xy, β(x, y) x, γ(x, y) y and f(x, y, t, u) t u; = − = 2 = 2 = + differentiating with α constant gives 0 (1 x )ϕ′x (1 y )ϕ′ y, whence ϕ′x c if c (1 y2)ϕ y. = + − + = 1 x2 = + ′ + When β(x, y) x, γ(x, y) y and f(x, y, t, u) t u, we get first = = = · ∂α ∂α ϕy ϕ′x ϕx ϕ′ y 0, · ∂y − · ∂x = whence ϕ′x as a known function of x if y is supposed constant. For α(x, y) x y, ϕx = + this gives ϕ′x c ϕ′ y , so log ϕx cx(for ϕ(0) 1) and ϕx ecx; for ϕx = = ϕy = = = α(x, y) xy, ϕ′x c , so log ϕx c log x (ϕ(1) 1) and ϕx xc. = ϕx = x = = = All these examples were classical as is the next one, coming from mechanics. The law of composition of two equal forces making an angle 2x leads to the functional equation The Work of Niels Henrik Abel 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 ϕ′x ϕy ϕx ϕ′ y 2ϕ′(x y); another differentiation, with+ x= y const, gives ϕ x· ϕy− ϕx · ϕ y = 0. If y −is regarded as − = ′′ · − · ′′ = constant, this gives ϕ′′x cϕx 0 and ϕx α cos(βx γ) with α, β and γ constant. + = = + π From (1), one sees that α 2 and γ 0 and the problem imposes ϕ 2 0, so β 1 and ϕ(x) 2 cos x.= = = =Here is another= case of application of Abel’s general method: the equation has the form ψα F(x, y,ϕx,ϕ′x,..., fy, f ′ y,...), where α is a given function of x and y and ϕ,=f,ψ are unknown functions. By differentiating with α constant, one gets a relation between x,ϕx,ϕ′x,... and y, fy, f ′ y,..., whence two differential equations, with respect to ϕ and to f , considering successively y and x as constant; 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 ϕ x, so that α(x, y) x y, the + = · ′ + · ′ = + differentiation with α constant gives ϕx f ′′ y fy ϕ′′x 0, and ϕx a sin(bx c), fy a sin(by c) then ψ(x y) · aa b−sin(b·(x =y) c c =) so that ψα+ = ′ + + = ′ + + + ′ = aa′b sin(bα c c′). In the case+ of+ψ(x y) f(xy) ϕ(x y), one gets 0 f (xy)(y x) 2ϕ (x y). + = + − = ′ − + ′ − Abel takes xy c as constant and writes ϕ α kα, where α x y and k f ′(c) , = ′ = = − = 2 k 2 2ϕ′c so ϕα k′ 2 α ; then he takes x y c constant and writes f ′β c′ c , so fβ c= +c β. Finally − = = = = ′′ + ′ k 2 ψ(x y) c′′ c′xy k′ (x y) + = + + + 2 − k 2 k 2 2 or ψα c′′ c′x(α x) k′ 2 (2x α) c′′ 2 α k′ xα(c′ 2k) (2k c′)x = + − + + − = + + + − k +2 − and we see that the condition c′ 2k is necessary; ψα k′ c′′ 2 α . The third example is ϕ(x y=) ϕx fy fx ϕy,= which+ gives+ + = · + · 0 ϕ′x fy ϕx f ′ y f ′x ϕy fx ϕ′ y (2) = · − · + · − · ; if one supposes that f(0) 1 and ϕ(0) 0, one gets 0 ϕ′x ϕx c fx c′ by making y 0(c f (0)=and c ϕ (=0)); so fx kϕ=x k −ϕ x and,· + substituting· = = ′ ′ =− ′ = + ′ ′ this value in (2) and making y constant: ϕ′′x aϕ′x bϕx 0 etc. Abel returned to the study of functional equations+ + in the= paper “Recherche des fonctions de deux quantites´ variables independantes´ x et y, telles que f(x, y), qui ont la propriet´ e´ que f(z, f(x, y)) est une fonction symetrique´ de z, xety”, published in German in the first volume of Crelle’s Journal in 1826 (Œuvres, t. I, p. 61–65). The condition of the title characterises a composition law which is associative and commutative; it may be written as 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 and to z and= multiplying= the results, one= gets 24 C. Houzel ∂r ∂v ∂s ∂r ∂v ∂s . (4) ∂x ∂y ∂z = ∂y ∂z ∂x ∂v ∂v But, by the definition of v, the quotient of ∂y by ∂z is a function ϕy when z is ∂s ∂s regarded as constant; in the same manner, ϕx is the quotient of ∂x by ∂z , so (4) ∂r ∂r becomes ∂x ϕy ∂y ϕx and this gives r as an arbitrary function ψ of Φ(x) Φ(y), where Φ is a primitive= of ϕ. So f(x, y) ψ(Φ(x) Φ(y)); putting this expression+ in (3) and making Φz Φy 0 and= Φx p,+ one gets Φψp p c, where c Φψ(0), and then Φ=f(x, y) = Φ(x) Φ(=y) c or = + = = + + Ψ f(x, y) Ψ(x) Ψ(y) (5) = + where Ψ(x) Φ(x) c. In other words, Abel finds that f is conjugate to the addition law by the function= Ψ+: he has determined the one-parameter groups. The second volume of Crelle’s Journal (1827) (Œuvres, t. I, p. 389–398) contains another paper of Abel on a functional equation: ϕx ϕy ψ(xfy yfx) ψ(r), (6) + = + = where r xfy yfx; this equation includes, as particular cases, the laws of addition for= log (+fy 1 y, ϕx ψx log x) and for arcsin ( fy 1 y2, = 2 = = = − ϕx ψx arcsin x).