jeanflop/post-ocr-correction huggingface dataset:
en commun, comme une extension de la théorie des invariants d'une forme algébrique binaire à la forme transcendante ou fonction. Nous noterons la variable indépendante par Z, et la fonction (2) ty, = f(Z) = ao + al(Z, z) + at(Z, z)2 + ... sa expansion dans le voisinage de Z = z. Ceux fonctions des coefficients aj, aj, a2, ... qui sont laissées invariantes par toutes transformations de groupe des trois paramètres sont appelées invariants. Car ces invariants peuvent être regardées comme fonctions de dw, d2w, ... et leurs valeurs changent avec z, nous les appelons plus spécialement différentielles
valeurs changent avec z, nous les appelons plus spécialement différentielles invariants. Plusieurs systèmes complets de tels invariants sont obtenus en Art. 1. En Art. 2 nous introduisons la notion d'invariant intégral. Le plus important invariant intégral est où est le Schwarzien dérivé de w par rapport à z. Le plus important invariant différentiel est Il se trouve que, à partir de notre point de vue, une fonction analytique est essentiellement déterminée lorsque la relation entre o et z est donnée. Nous l'appelons l'équation intrinsèque de la fonction H = f(z). La détermination effective d'une fonction dont l'équation intrinsèque est donnée,
La détermination effective d'une fonction dont l'équation intrinsèque est donnée, requiert l'intégration d'une équation différentielle du second ordre, et peut être réduite à la solution d'une équation intégrale avec un noyau skew-symétrique. En Art. 4 nous introduisons les fonctions rationnelles osculatrices de différentes orders, une notion très proche de certaines investigations dues à Frobenius et Fade. Les relations qui existent entre les pôles de ces fonctions, et aussi entre leurs zéros, sont très simples et élégantes et semblent apparaître ici pour la première fois. La fonction logarithmique osculatrice, introduite en Art. 5, sert à compléter cette théorie en une manière
Art. 5, sert à compléter cette théorie en une manière essentielle, et donne lieu à de nouveaux résultats simples et géométriques. La fonction logarithmique osculatrice joue un rôle essentiel parce que chaque fonction logarithmique a une équation intrinsèque de la forme o = constant. Les pôles et zéros des fonctions rationnelles osculatrices et les singularités de la fonction logarithmique osculatrice sont des fonctions de la position du point z à lequel l'osculation se fait. Si z est soumis à une transformation linéaire, chacun de ces points est transformé par la même transformation linéaire, et nous le nous servons parler de
même transformation linéaire, et nous le nous servons parler de ces points comme des cogré vitaux. Des classes spéciales de fonctions peuvent être définies par des relations prescrites entre certains des cogré vitaux. Nombreuses illustrations de cette méthode sont travaillées en détail en Arts. 9, 10 et 11. Un des résultats les plus intéressants obtenus de la sorte est une nouvelle propriété de certaines fonctions elliptiques. Soit la fonction rationnelle quadratique qui oscule w = F(Z) au point Z = z et let q1, q2 soient ses pôles. Soit T l'harmonique conjugué de z, le point de contact, par rapport
l'harmonique conjugué de z, le point de contact, par rapport à q1, q2. Nous appelons T le satellite quadratique de z. Nous posons la question; quelles fonctions sont celles pour lesquelles T est un point fixe? Il est évident que ce sera le cas lorsque F(Z) est une fonction quadratique avec des pôles coincidants. Mais il existe une autre cas lorsque T est un point fixe; namely, lorsque w = F(Z) peut être obtenu par une transformation linéaire d'une fonction Weierstrass -fonction pour laquelle l'invariant g2 a la valeur zéro, de sorte que le rapport croissant des points de branchement
de sorte que le rapport croissant des points de branchement de la surface de Riemann soit équivalent-harmonique. Les expressions générales des cogré vitaux permettent d'écrire down plusieurs nouvelles formules qui sont valides pour toute fonction analytique. Chacune de ces formules donne une expression pour w = f(z) dans la forme d'un integral dont l'intégrand dépend de un ou plusieurs des cogré vitaux. Ces formules admettent l'itération et sont très utiles dans un grand nombre d'investigations. Des expressions similaires ont également été trouvées pour l'invariant intégral g2, le plus important parmi ceux qui dépendent des singularités de la fonction logarithmique osculatrice.
ceux qui dépendent des singularités de la fonction logarithmique osculatrice. Ces formules pour g2 sont développées en Art. 7. Mais nous donnons, en Art. 7, une définition indépendante de g2 qui est de plus grande importance, parce qu'elle est formulée entièrement en termes de notions qui restent invariables sous les transformations linéaires, et par le moyen d'un processus de limite qui est l'analogon multiplicatif du processus utilisé pour définir un integral défini. En Art. 8 nous introduisons l'invariant intégral ρ comme une nouvelle variable indépendante. Ainsi de w = f(z), nous avons Si nous introduisons comme variable indépendante, w devient
nous avons Si nous introduisons comme variable indépendante, w devient une fonction de zn et nous pouvons mettre Nous pouvons maintenant introduire zn comme variable indépendante et continuer de la sorte. Les relations qui existent entre les dérivés Schwarziani w, z, ..., HJ, Zk( prennent la forme de fractions continues de très simple et remarquable forme. Un certain nombre de nouveaux problèmes se présentent à partir de ce fait; en quels cas les fractions continues se termineront? lorsque elles seront périodiques? si elles ne terminent pas, et k est autorisé à grandir au delà des limites; converge-ront-elles? Nous avons actuellement
à grandir au delà des limites; converge-ront-elles? Nous avons actuellement résolu quelques-uns des plus simples de ces problèmes; mais ici, comme ailleurs dans ce mémoire, les nouveaux problèmes sont trop nombreux et trop à longues portées pour qu'une solution immédiate à tous, ou même à beaucoup d'entre eux, soit une entreprise possible. En Art. 12 nous élargonnons notre groupe en considérant des transformations linéaires indépendantes des variables dépendantes et indépendantes. Les correspondants combinaisons invariables sont appelées hyper invariants. L'invariant intégral ρ conserve sa propriété invariante même sous le groupe élargi. La plus simple différence hyperinvariante, dénotée par I, est de
La plus simple différence hyperinvariante, dénotée par I, est de l'ordre du cinquième. La relation entre I et ρ, appelée l'équation hyperintrinsèque, définit une fonction w = f (z) excepté pour les transformations linéaires des deux variables. The simplest hyperintrinsic equation, namely I = const., defines the power functions where a, b, c, d, e, f, g, h, x, y, z, and r are constants. The problem immediately presents itself to determine a power function of this form which shall have the closest possible contact with a given analytic function at a given point. This problem is solved completely in
at a given point. This problem is solved completely in Art. 13, and leads again to interesting geometric relations between the singularities of the osculating power function and the singularities of other osculants which have been introduced before. We have presented in this introduction only a few of the most striking features of this new theory, just enough to indicate the general point of view. And even in the body of the paper we have purposely refrained at many points from developing the theory more in detail, because it was our desire to obtain merely a first general outlook over
our desire to obtain merely a first general outlook over this new territory. There remains much to be done. 1. — The differential invariants of a function of a complex variable. Let w = f(z) be a function of the complex variable z defined, in the neighborhood of the Origin, by means of its Taylor expansion (I) w = f (z) = a0 + a1 z + a2 z2 + ... + a_k zk + ... If we transform the independent variable by putting where a, b, c, d are arbitrary constants, w becomes a function f(z) of z and
arbitrary constants, w becomes a function f(z) of z and to properties of the function f(z) in the neighborhood of z = 0 will correspond properties of the transformed function f(z) in the neighborhood of z = Let us assume d = 0 and put z = K. We may then write (2) in the form If d = 0, the transformation (2) may also be reduced to the same form (3) provided that z in that case be interpreted to mean As a result of the substitution (3) the series (I) will become a power series in Z, where
series (I) will become a power series in Z, where Z is either equal to z - K or We proceed to determine the coefficients of this new expansion. We find from (3) If we substitute these values for z in (I) and denote by a_k the coefficient of s^1 we find The following special cases of (5) are especially important By an absolute invariant of the function f(z) under the transformations of the group (2) or (3) we mean a function of the coefficients a0, a1, a2, ... which, as a result of the relations (5), is identically equal
as a result of the relations (5), is identically equal to the corresponding function of a0, a1, a2, ... Evidently a0 is such an invariant. In order to find the others we might use the method of infinitesimal transformations. Thus (3) represents an infinitesimal transformation if we put e^z = J + Aot. j - [J - oft. where O't is an infinitesimal. The corresponding infinitesimal transformations of the coefficients are given by (7) da0 = o, da_k = [k a_k] + (k-1) a_{k-1} ot, k > 0. Therefore every absolute invariant must be a solution of the two differential
absolute invariant must be a solution of the two differential equations The first of these two equations is easy to interpret. Let us attribute to a_k the weight k, and let us agree that the weight of a product shall be the sum of the weights of its factors. Then, the first equation (8) merely asserts that every absolute invariant is isobaric of weight zero. We may therefore find the invariants by constructing isobaric functions of a given weight p, in such a way as to also satisfy the second equation of (8), and then divide by a. Isobaric functions
equation of (8), and then divide by a. Isobaric functions of weight different from zero, which also satisfy the second equation of (8), shall be called relative invariants. But it is easier to obtain a complete set of invariants by a different process. Let us assume A0 = 0. We may then according to (6) determine a and [3 in a unique fashion so as to make a1 = 1, a2 = 0, namely by putting We obtain in this way the canonical expansion (of the first kind) (I0) W = f (z) = A0 + Z1 + A3 z3
= f (z) = A0 + Z1 + A3 z3 + A4 z4 + ... where, on account of (5), (6), and (9), Another canonical form (the second kind) is obtained by putting The resulting expansion is characterized by the relations This form is very useful. We shall discuss the convergence of (10) later. At present we only wish to emphasize the fact that this canonical expansion (TO) is unique. That is, to every series of the form (I), in which a1 is not equal to zero, there corresponds a uniquely defined canonical expansion of form (10). From this it
uniquely defined canonical expansion of form (10). From this it follows that the coefficients of the canonical expansion are absolute invariants of f(z). In fact, suppose we consider any function obtainable from f(z) by a transformation of form (2). By means of the transformation would be transformed into its canonical form where B0, B3, etc., are related to b0, b1, b2, ... in the same way as are A0, A1, A2, ... to a0, a1, a2, ... But the expansions (10) and (14) must be identical on account of the uniqueness of the canonical form, so that = A_s, showing
uniqueness of the canonical form, so that = A_s, showing that A0, A3, A4, ... are actually absolute invariants. It is not difficult, moreover, to verify this fact by applying the infinitesimal transformations (7) to the expression (II) for A_k. For we find dA_s = 0, which again proves that A_s is an invariant. The invariants A0, A3, A4, ..., A_s, ... are independent functions of the coefficients a0, a1, a2, ... For, taken in this order, each of the invariants involves a coefficient as which does not appear in any of the earlier invariants. Every absolute invariant is a
any of the earlier invariants. Every absolute invariant is a function of A0, A3, A4, ... For, let I(a0, a1, a2, ...) be an absolute invariant. Then we shall have (15) I(a3, a2, ...) = I(Ga, Ga-1, G-2, ...) whenever a0, ap, a2, ... are connected with a9, an, a2, ... by means of the relations (5). In particular, (15) will be verified if we choose the values (9) for c and P, as a result of which choice, the expansion assumes its canonical form. Therefore we shall have I(Ga, a2, a3, ...) = I(Ao, A1, O, A3, A4, ...)
a2, a3, ...) = I(Ao, A1, O, A3, A4, ...) showing that I is a function of Ao, A1, A2, ... as stated. We may summarize our last three theorems by saying, that the invariants Ao, A1, A2, ... form a complete system of independent absolute invariants whenever a1 is not equal to zero. We shall ordinarily maintain the hypothesis a1 = 1. If a1 is equal to zero we study, in place of the expansion (i), the Taylor expansion of J(z) for some point z = k where k ≠ o. Of course k may always be chosen in
≠ o. Of course k may always be chosen in infinitely many ways so as to ensure that the coefficient of z - k in the new expansion will not be equal to zero, provided that w is not a constant. It is possible, however, to replace the invariants A0, A3, A4, ... by a new system adapted to the case a1 == a2 == ... == dp-1 = 0, av ≠ o, by an obvious extension of the method which we have used for the case a1 ≠ o. Since we have a j = j • a, is
Since we have a j = j • a, is a relative invariant of weight I. Consequently as As is a relative invariant of weight s. According to (I I), the last term of as A, (corresponding to t = s - I) contains as a factor, and no other term in as A-s contains a higher power of a1 in its denominator. Therefore a*-1 As is a relative invariant of weight 2s2, which is an integral rational function of a1, a2, ..., as. Moreover this invariant is a linear function of a1, a2 as. The only coefficients which appear
function of a1, a2 as. The only coefficients which appear raised to a higher power are a1 and a2. For this reason we shall occasionally refer to the invariants s as the linear invariants of the function f(z). It is now easy to see that these linear invariants form a complete system in a more specific sense than that mentioned above. Every absolute invariant, which is a rational function of the coefficients of f(z), will be a rational function of the linear invariants. The invariant A3 is of special importance. We have so that A3 is equal to the value
We have so that A3 is equal to the value which assumes for z = o, if we write so that z represents the Schwarzian derivative of w with respect to z. We shall frequently think of an expansion of the function W = f(z) in the neighborhood of a point not at the origin, where f(z) is analytic, W = f(Z) - ao + adZ + 4(Z - z)2 + ... and we shall then write We may then think of ao, a1, a2, ... as well as w, w', W", ... as functions of z and we shall
w', W", ... as functions of z and we shall have where a specific determination of the square root being chosen in any particular case. Since A3, A4, A5, ... are the coefficients of z, z, ... in the canonical expansion, it is apparent that A4, A5, ... may be obtained from A3 by repeated differentiation with respect to z. But we may exhibit another and more convenient differentiation process which accomplishes the same purpose. If we make any linear transformation of the independent variable, we find and From our knowledge of the fact that A2 is an absolute invariant
knowledge of the fact that A2 is an absolute invariant and from (22), we conclude a familiar formula which may also be verified directly. More generally we make the following remark. If I is a relative invariant of weight p, the transformation (20) will transform I into I where If I is an absolute invariant, we have p = 0, and therefore Thus we may take tv, 0, as a fundamental system of invariants. II. — Integral invariants. All of the invariants obtained so far are functions of the values, w, w', w", which W = f(Z) and its derivatives
w, w', w", which W = f(Z) and its derivatives assume for a given value z of the complex variable Z, the function f(Z) being analytic in the neighborhood of this point. We shall henceforth speak of these invariants as differential invariants. But equation (23) of Art. i enables us to define a new kind of invariant which we shall call an integral invariant of the function. Equation (23) may be written as follows if the square roots are properly determined. Let us select a curve C of finite length in the z plane. This curve may be open or
in the z plane. This curve may be open or closed but it should be so chosen that f(z), z' is analytic in the neighborhood of each of its points. Let C be the curve in the z plane obtained from C by the transformation (20). Then we shall have Thus, the value of the integral remains unchanged if the independent variable and the path of integration are transformed simultaneously by the same linear transformation. We may express this by saying that o is an integral invariant. More specifically we shall speak of o as a simple integral invariant because
shall speak of o as a simple integral invariant because the function under the integral sign depends only upon w, w', w", .... The integrand does not itself involve integration. If I is any absolute differential invariant, the integral will be an integral invariant. Conversely, if w is not a linear function of z, any simple integral invariant may be expressed in this form. The truth of the direct statement is apparent. To prove the converse, let J = F(w, w', w", ...) dz be a simple integral invariant. If w is not a linear function of z, the Schwarzian
w is not a linear function of z, the Schwarzian derivative dw, dz will not be equal to zero identically, and we may introduce to as a new independent variable in place of z. Then J becomes which is invariant for all possible curves C, we must have dJ = dJ. Since we have also dz = dw, we find which shows that log (z, w) is an absolute differential invariant. There exists an integral invariant even simpler than log (z, w), namely dw, and by an argument analogous to that just completed we can show that any other simple
that just completed we can show that any other simple integral invariant may be expressed in the form f dw where I is an absolute differential invariant. In fact this form remains valid even if w is a linear function of z, provided it is not a constant. But the absolute differential invariant of lowest order, excepting w itself, is 0. Therefore log dw is the simple integral invariant of lowest order, excepting only the integral f dw. This latter integral however is related so simply to the values of w at the end points of the path of integration,
w at the end points of the path of integration, that we prefer not to think of it as an integral at all. With this understanding we may claim that log is the simplest integral invariant of w = f(z) for linear transformations of the independent variable. III. — The intrinsic equation. Let w be given as an analytic function of z. We put the square root being properly specified, and the integration being performed over a path C at all of whose points V dw, is analytic. If we regard o as a function of its upper limit z,
regard o as a function of its upper limit z, and eliminate z, we obtain a relation of the form 0 = F(©), where F is an analytic function of o. We shall speak of this, which expresses the relation between the simplest differential and integral invariants of the function w = f(z), as the intrinsic equation of w = f(z), a terminology which corresponds to that used by Cesaro in his intrinsic geometry. The importance of this notion for our theory appears in the following theorem. All functions, which can be obtained from each other by linear transformation of
can be obtained from each other by linear transformation of the independent variable, have the same intrinsic equation. Conversely, if the intrinsic equation is given, there will correspond to it infinitely many functions all of which may be obtained from one of them, w = f(z), by adding a constant and making a linear transformation of z. The truth of the direct theorem is obvious, since 9 and p are absolute invariants for all linear transformations of z. To prove the converse we observe that so that consequently w is determined as a function of a except for an additive
determined as a function of a except for an additive constant. From the definition of o we find: On the other hand we have so that Hence we may replace o by the function of w which results for it from (32), giving dz, dw as a known function of w, say and G (w) will be independent of k. If z is any solution of this equation, the most general solution will be where a, y, z, are arbitrary constants. Thus if w = f(z) is one function which corresponds to the given intrinsic equation, will be the most
corresponds to the given intrinsic equation, will be the most general function of the same kind. The following method of proving the same theorem is more elegant and leads to some further results. Since (32) furnishes an expression for o as a function of cp by means of a simple quadrature, we are naturally led to think of z also as a function of 9. Now Cayley first proved the formula which shows how to transform the independent variable of a Schwarzian. If we put s = w, x = z, o'k = 9, we find dw, dz = dV-,
z, o'k = 9, we find dw, dz = dV-, dz, [ ? - 1 ]. If w is not a linear function w, z; is not equal to zero, and we conclude dw, d9; = 1. We may write and our theorem now follows immediately from the fact that (32) gives the right member of (37) as a known function of 9 which is independent of k, and that all solutions z of (37) are linear functions of any particular solution. But we may actually indicate the analytic nature of this problem a little more closely. If G =
of this problem a little more closely. If G = F(s) is the given intrinsic equation, we find from (32) where I is now a known function of ©. Of course the most general solution of (38) is a linear function of w. Consider the linear homogeneous differential equation and let W1, W2 be two of its linearly independent solutions. Then W . W will be a solution of (38). Conversely if w is given as function of ©, as in the present case, we may find W1, W2. Let ©0 be a value of © for which I is
©0 be a value of © for which I is analytic and let more especially let the constant k in equation (32) be equal to zero. Then there exist in the ©-plane a circle of non-vanishing radius around ©0, such that for all points in this circle whence follows if we put and therefore In order to determine z as a function of ©, we must integrate the differential equation Any solution of this equation may be equated to where \(Z_1, Z_2\) are independent solutions of. We may write this equation in the form and integrate it as though the
equation in the form and integrate it as though the right member were a known function of \(z\), that is, as though (44) were a non-homogeneous linear differential equation whose right member is given. We find in this way that any solution of (44) satisfies the equation where \(W_1\) and \(W_2\) are given by (41), or if we put the arguments into evidence, If \(Z_1\) and \(Z_2\) denote those solutions of (43) which are defined by the same initial conditions (40) which determine \(W_1\) and \(W_2\), \(Z_1\) and \(Z_2\) will satisfy the two integral equations (46) as the most general