jeanflop/post_ocr_correction2 huggingface dataset:
lower limit of the integral invariant \(o\) be selected accordingly. MINI. — Rational osculants. In order to be able to interpret the invariants already found, and for the purpose of obtaining still others of a more fundamental character, we now introduce a device suggested by differential geometry. In the metric theory of plane curves, for instance, certain osculating curves of a simple character are introduced, such as the tangent, the osculating circle, etc. The properties of these osculants, and their relations to each other constitute the subject matter of differential geometry. But the kind of osculants considered in any part of differential geometry depends essentially upon the group of transformations which, in that branch of geometry, is regarded as fundamental. Thus the osculating circle plays no part in projective differential geometry, since a projective transformation does not, in general, transform a circle into a circle. Our present investigations are in the domain of the theory of functions of a complex variable, and we are interested in those properties of such functions which remain unchanged under any linear transformation of the independent variable. Now obviously a rational function of degree \(n\) will be transformed into another rational function of the same degree by any linear transformation of the independent variable. Again, let us consider two analytic functions \(W_1 = a_0 + a_1 z + a_2 z^2 + ... + a_k z^k + a_{k+1} z^{k-1} + ... W_2 = b_0 + b_1 z + b_2 z^2 + ... + b_k z^k + b_{k+1} z^{k-1} + ... which are so related that \(a_0 = b_0, a_1 = b_1, ... , a_k = b_k = b_{k-1}\). We shall say that the two functions have \(k\)th order contact at the point \(z = 0\). The point \(z = 0\) may be called the point of contact. Let us now subject \(z\) to any linear transformation. Formulae (5) show that we shall have \(W_1 = b_0, W_1 = b_1, ..., W_k = b_k\). Consequently, the order of contact between two functions remains unchanged under linear transformations of the independent variable. Therefore, the following problem clearly belongs
degree by any linear transformation of the independent variable. Again, let us consider two analytic functions \(W_1 = a_0 + a_1 z + a_2 z^2 + ... + a_k z^k + a_{k+1} z^{k-1} + ... W_2 = b_0 + b_1 z + b_2 z^2 + ... + b_k z^k + b_{k+1} z^{k-1} + ... which are so related that \(a_0 = b_0, a_1 = b_1, ... , a_k = b_k = b_{k-1}\). We shall say that the two functions have \(k\)th order contact at the point \(z = 0\). The point \(z = 0\) may be called the point of contact. Let us now subject \(z\) to any linear transformation. Formulae (5) show that we shall have \(W_1 = b_0, W_1 = b_1, ..., W_k = b_k\). Consequently, the order of contact between two functions remains unchanged under linear transformations of the independent variable. Therefore, the following problem clearly belongs to our field. To determine the rational function of degree \(n\) which has contact of the highest possible order with a given analytic function at a given point. We shall henceforth speak of this function as the osculating rational function of degree \(n\). This problem may be solved very easily by explicit formulae. It is included in a more general problem studied by several mathematicians, especially by Frobenius and by Fade, namely, to determine the rational function, whose numerator is of degree \(n\) and whose denominator is of degree \(m\), such that the power series expansion of this fraction at a specified point (say at \(z = 0\)) shall agree with a given power series \(a_0 + a_1 z + a_2 z^2 + ...\) as far as possible, that is, ordinarily to \(rI. - h (m + n - h)\) terms. The fractions obtained in this way may be arranged in a double entry table usually known as Pade's table. We are only interested in those fractions of Pade's table for which \(n = m\), that is, those which would be located normally in the principal diagonal. The reason for this is apparent. To specify the degrees of numerator and
by explicit formulae. It is included in a more general problem studied by several mathematicians, especially by Frobenius and by Fade, namely, to determine the rational function, whose numerator is of degree \(n\) and whose denominator is of degree \(m\), such that the power series expansion of this fraction at a specified point (say at \(z = 0\)) shall agree with a given power series \(a_0 + a_1 z + a_2 z^2 + ...\) as far as possible, that is, ordinarily to \(rI. - h (m + n - h)\) terms. The fractions obtained in this way may be arranged in a double entry table usually known as Pade's table. We are only interested in those fractions of Pade's table for which \(n = m\), that is, those which would be located normally in the principal diagonal. The reason for this is apparent. To specify the degrees of numerator and denominator separately is equivalent to the requirement that a certain number of zeros or poles of the rational function shall be at infinity. But in our present theory the point at infinity is in no way distinguished by special properties. The formulae of Frobenius give us immediately the following results. Given the expansion of a function, analytic in the neighborhood of z = 0, w = C_1q + a_1 Z + a_2 Z^2 + ... + a_n Z^n + ... The osculating rational function of degree n will be where a_1, a_2, ..., a_n, a_n_1, a_n_2, ..., a_n_0 + a_0 z^n (53) T_n = " ' 1 ' * J * * ' 1 ' • ' " 1 a_n_1 a_n_2 ... a_n_i a_n + 1 a_n + 2 ... a_n_0_1 a_n - 1 a_n - 2 '-1 • + a_0 (1) Crelle's Journal, vol. 90, 1881. (2) Annales scientifiques de l'École Normale supérieure, 3e série, 1892, vol. 9. and a_1, a_2, ..., a_n (54) C_n = a_2' a_3^1 a_n + 1, z t . . • 5 . > . . ' I .... #a_n_2 a_n_0_2, . • .-, a_n_n_0, I Moreover this rational function will actually be of
Given the expansion of a function, analytic in the neighborhood of z = 0, w = C_1q + a_1 Z + a_2 Z^2 + ... + a_n Z^n + ... The osculating rational function of degree n will be where a_1, a_2, ..., a_n, a_n_1, a_n_2, ..., a_n_0 + a_0 z^n (53) T_n = " ' 1 ' * J * * ' 1 ' • ' " 1 a_n_1 a_n_2 ... a_n_i a_n + 1 a_n + 2 ... a_n_0_1 a_n - 1 a_n - 2 '-1 • + a_0 (1) Crelle's Journal, vol. 90, 1881. (2) Annales scientifiques de l'École Normale supérieure, 3e série, 1892, vol. 9. and a_1, a_2, ..., a_n (54) C_n = a_2' a_3^1 a_n + 1, z t . . • 5 . > . . ' I .... #a_n_2 a_n_0_2, . • .-, a_n_n_0, I Moreover this rational function will actually be of degree n and in its lowest terms, if and only if the determinant a_1, . • • i (55) C_n = a_2, a_3^1 a_n + 1 * I • i ' • • • a_n_0^1 a(n + 1, • • " a_n_t - 1 is different from zero. The following relation between two successive rational osculants should also be noted One very obvious remark may be made at once. The quantities C_n, defined by (55), are relative invariants of the function w = f(z) under linear transformations of the independent variable. To prove this, let us suppose C_1 = 0, C_2 = 0, C_n_1^1 = 0, C(n = 0. Then each of the osculants is actually of the degree indicated and is not reducible to a lower degree; but according to (56) we shall have In other words, the rational osculating function of degree n, which has 2n-th order contact with w, is not in its lowest terms but reduces (1) FROBENIUS, loco cit., p. 6, equation (16). T_v to f^n. That is, the osculating rational function of degree n - 1 has n - 1 not only contact of order 2(n - 1) with w = f(z) but contact
relation between two successive rational osculants should also be noted One very obvious remark may be made at once. The quantities C_n, defined by (55), are relative invariants of the function w = f(z) under linear transformations of the independent variable. To prove this, let us suppose C_1 = 0, C_2 = 0, C_n_1^1 = 0, C(n = 0. Then each of the osculants is actually of the degree indicated and is not reducible to a lower degree; but according to (56) we shall have In other words, the rational osculating function of degree n, which has 2n-th order contact with w, is not in its lowest terms but reduces (1) FROBENIUS, loco cit., p. 6, equation (16). T_v to f^n. That is, the osculating rational function of degree n - 1 has n - 1 not only contact of order 2(n - 1) with w = f(z) but contact of order at least 2n. We shall say that it hyperosculates the function w=f(z). But hyperosculation is obviously a property invariant under linear transformation. Therefore the condition C_n = 0 is an invariant equation, and C_n must be an invariant, as stated above. Since the sum of the indices in any term of the expanded determinant is n, the weight of c_n is equal to n^2, and the effect of the infinitesimal transformation (7) upon C_n is given by (57) dC_n = OC_n ?, C_n, dC_n. The direct proof of (5) is somewhat complicated and may be omitted. The invariants C_1, C_2, C_3, ... have already made their appearance in Art. 1. Moreover, the conditions C_n = 0, C_n + 1 = 0, C_n + 2 = 0, C_n + 3 = • • • obviously imply that w=f(z) is a rational function of degree n - 1, This also follows from the fact that the coefficients of the power series w(z), in this case, possess a scale of relation. Of course the invariants c_1, c_2, c_3, ... do not form a complete system. For, as we advance from c_n to c_n + 1 two new coefficients, a_n, a_n +
any term of the expanded determinant is n, the weight of c_n is equal to n^2, and the effect of the infinitesimal transformation (7) upon C_n is given by (57) dC_n = OC_n ?, C_n, dC_n. The direct proof of (5) is somewhat complicated and may be omitted. The invariants C_1, C_2, C_3, ... have already made their appearance in Art. 1. Moreover, the conditions C_n = 0, C_n + 1 = 0, C_n + 2 = 0, C_n + 3 = • • • obviously imply that w=f(z) is a rational function of degree n - 1, This also follows from the fact that the coefficients of the power series w(z), in this case, possess a scale of relation. Of course the invariants c_1, c_2, c_3, ... do not form a complete system. For, as we advance from c_n to c_n + 1 two new coefficients, a_n, a_n + 1, are introduced at the same time. It is easy however to form a complete system from these invariants by adding to them the invariants That these quantities are invariants is easily seen as follows: Since c_n is a relative invariant of weight n, it is an absolute invariant. Therefore (See Art. 1) its derivative is a relative invariant. But this derivative is Since a_1 is a relative invariant, the numerator of this fraction is also a relative invariant; but this numerator reduces to da_1, since Of course the weight of d(a_n) is n^2 + 2. We can find a simple general expression for c_n' in terms of a_1, a_2, ..., a_n, a_n + 1. To do this, we observe in the first place that now let us write c_n as follows where the upper indices, which indicate the column to which each element belongs, have been introduced merely for the sake of fixing a notation for the cofactors. We actually have and we shall denote by A_n the cofactor of h_n, so that We find or Thus we have found The sum of the first n terms, in the first column a_2 A_1 + ... + a_n o. The
(See Art. 1) its derivative is a relative invariant. But this derivative is Since a_1 is a relative invariant, the numerator of this fraction is also a relative invariant; but this numerator reduces to da_1, since Of course the weight of d(a_n) is n^2 + 2. We can find a simple general expression for c_n' in terms of a_1, a_2, ..., a_n, a_n + 1. To do this, we observe in the first place that now let us write c_n as follows where the upper indices, which indicate the column to which each element belongs, have been introduced merely for the sake of fixing a notation for the cofactors. We actually have and we shall denote by A_n the cofactor of h_n, so that We find or Thus we have found The sum of the first n terms, in the first column a_2 A_1 + ... + a_n o. The same thing is true of the sum of the first n terms in all of the other columns, except the last. Consequently so that we find finally where the determinant differs from that for c_n only in the last row. The derivative of c_n is obtained by increasing the index of each element in the last row of c_n by a single unit, and then multiplying the whole determinant by 2K. This formula assumes an interesting form if we remember that As a result of formula (61) we may regard the invariants dC_n as being known in terms of the coefficients a_1, a_2, ... in the form of sums of two determinants. The direct proof of (57), which has been omitted, may be based upon a calculation very similar to the one just completed. We proceed now to study the distribution of the poles and zeros of these rational osculants. The osculating linear function is -T_n, where Therefore we find the formulae where p is the pole, and e the zero of the osculating linear function. The expression for the osculating quadratic function depends upon a_1, a_2, a_3, and a_4. If, in this expression, we replace a_2 by an arbitrary
of each element in the last row of c_n by a single unit, and then multiplying the whole determinant by 2K. This formula assumes an interesting form if we remember that As a result of formula (61) we may regard the invariants dC_n as being known in terms of the coefficients a_1, a_2, ... in the form of sums of two determinants. The direct proof of (57), which has been omitted, may be based upon a calculation very similar to the one just completed. We proceed now to study the distribution of the poles and zeros of these rational osculants. The osculating linear function is -T_n, where Therefore we find the formulae where p is the pole, and e the zero of the osculating linear function. The expression for the osculating quadratic function depends upon a_1, a_2, a_3, and a_4. If, in this expression, we replace a_2 by an arbitrary parameter A, we obtain a one-parameter family of quadratic functions each of which will have third order contact with the function w = f(z) at z = 0. We shall speak of these quadratic functions as penosculting quadratics. The osculating quadratic is that penosculting quadratic for which A has the value a_4. The denominator of the general osculating quadratic is Since its coefficients are linear functions of À, there will exist in general two values of À, call them X1, and X2, for which the linear factors of (63) will become identical. These values of X are given by Those osculating quadratics which correspond to À = X1, and to À = X2 will be the only ones whose two poles coincide. We shall call them the singular osculating quadratics. Let p, and p2 be the poles of the singular osculating quadratics. Then p, and p2 will be obtained from the factors of (63) by equating À to X1, and X2 respectively. Thus we find that are the poles of the two singular osculating quadratics. Let us assume at =/= 0, and that the series under consideration has been reduced to its first canonical form (See Art. 1). Then we
value a_4. The denominator of the general osculating quadratic is Since its coefficients are linear functions of À, there will exist in general two values of À, call them X1, and X2, for which the linear factors of (63) will become identical. These values of X are given by Those osculating quadratics which correspond to À = X1, and to À = X2 will be the only ones whose two poles coincide. We shall call them the singular osculating quadratics. Let p, and p2 be the poles of the singular osculating quadratics. Then p, and p2 will be obtained from the factors of (63) by equating À to X1, and X2 respectively. Thus we find that are the poles of the two singular osculating quadratics. Let us assume at =/= 0, and that the series under consideration has been reduced to its first canonical form (See Art. 1). Then we shall have tv = Ao + A1 z + A2 z2 + A3 z3 +, , " where Thus we find p = 0, PI + P2 = 0, showing that the four points o, p, p1, p2 form a harmonic set, that is, the cross-ratio (66) (o, p1, p2) = -1. Now the cross-ratio of four points in the z plane is left unchanged by all linear transformations of z; to a function of z and the linear function which osculates it at z = 0 will correspond the transformed function and its linear osculating function. Consequently we obtain the following theorem. Let z be any point of the plane in the neighborhood of which the function f(z) is analytic and where f'(z) is not equal to zero. Consider the osculating linear function and the two singular osculating quadratics of f(z) which belong to the point z, which shall henceforth be called the point of contact. Then, the point of contact and the pole of the osculating linear function are separated harmonically by the poles of the two singular osculating quadratics. Of course this gives us the corollary that the four points mentioned are on one and the same
= -1. Now the cross-ratio of four points in the z plane is left unchanged by all linear transformations of z; to a function of z and the linear function which osculates it at z = 0 will correspond the transformed function and its linear osculating function. Consequently we obtain the following theorem. Let z be any point of the plane in the neighborhood of which the function f(z) is analytic and where f'(z) is not equal to zero. Consider the osculating linear function and the two singular osculating quadratics of f(z) which belong to the point z, which shall henceforth be called the point of contact. Then, the point of contact and the pole of the osculating linear function are separated harmonically by the poles of the two singular osculating quadratics. Of course this gives us the corollary that the four points mentioned are on one and the same circle, it being understood, as is customary in this theory, that a straight line is to be regarded as a special circle, namely a circle through infinity. In proving this theorem we made use of the canonical form of the expansion. Of course the theorem may be proved without this device, but not without some calculation. It suffices for this purpose to note that an equation which is equivalent to (66). Let us now consider any osculating quadratic, and let us denote its poles by p1 and p2. These poles will be distinct, except when X is equal to either X1 or X2, and they will always be the roots of the quadratic equation, (68) (~k (Z' - otg) -u2 + (a2 -i — ~k i a1 as a2 0, o, which is obtained by equating (63) to zero. According to (65) p1 and p2 are the roots of the quadratic (69) 2a2Z -i(X O. But the harmonic invariant of these two quadratics, namely, (Xa2 - a^)a1 + (a1a3 - X®)a3 + (a2a3 - Xa1)a2 is equal to zero for all values of X. Consequently p1 and p2 are harmonic conjugates with respect to p, and P2. This gives rise
device, but not without some calculation. It suffices for this purpose to note that an equation which is equivalent to (66). Let us now consider any osculating quadratic, and let us denote its poles by p1 and p2. These poles will be distinct, except when X is equal to either X1 or X2, and they will always be the roots of the quadratic equation, (68) (~k (Z' - otg) -u2 + (a2 -i — ~k i a1 as a2 0, o, which is obtained by equating (63) to zero. According to (65) p1 and p2 are the roots of the quadratic (69) 2a2Z -i(X O. But the harmonic invariant of these two quadratics, namely, (Xa2 - a^)a1 + (a1a3 - X®)a3 + (a2a3 - Xa1)a2 is equal to zero for all values of X. Consequently p1 and p2 are harmonic conjugates with respect to p, and P2. This gives rise to the following theorem. The poles of every osculating quadratic, and therefore also the poles of the osculating cubic, are harmonic conjugates of each other with respect to the poles of the two singular osculating quadratics. The same result may also be expressed by saying that the poles of the osculating quadratics are pairs of an involution. Equation (68) will furnish, as its roots, the poles of the osculating quadratic if we put X = a,. Therefore we find where q1 and q2 denote the poles of the osculating quadratic. We now pass to the consideration of osculating and osculating cubics. The osculating cubic is and the osculating cubics are obtained from by writing an arbitrary parameter X in place of a6. Thus the poles of the osculating cubics are the roots of where Since the coefficients of (71) are linear functions of X, the poles of every osculating cubic constitute a triple of a cubic involution. The product U:n where U2 is the denominator of the osculating quadratic, will be the denominator of one osculating cubic, namely that one which corresponds to X = 0. There also exists a osculating cubic, one of whose poles coincides with the pole
osculating quadratics are pairs of an involution. Equation (68) will furnish, as its roots, the poles of the osculating quadratic if we put X = a,. Therefore we find where q1 and q2 denote the poles of the osculating quadratic. We now pass to the consideration of osculating and osculating cubics. The osculating cubic is and the osculating cubics are obtained from by writing an arbitrary parameter X in place of a6. Thus the poles of the osculating cubics are the roots of where Since the coefficients of (71) are linear functions of X, the poles of every osculating cubic constitute a triple of a cubic involution. The product U:n where U2 is the denominator of the osculating quadratic, will be the denominator of one osculating cubic, namely that one which corresponds to X = 0. There also exists a osculating cubic, one of whose poles coincides with the pole of the osculating linear function. In order to determine the essential properties of this latter cubic, let us assume that the given expansion for w = f(z) is in the first canonical form, so that (73) a3 = 1, a2=0, a3=A3, we then find In order that p = a3 may be a root of (71), we must have A = 0, and this condition furnishes the following equation for X: Let r1 and r2 be the remaining two poles of this special osculating cubic. Then r1 and r2 are the roots of the quadratic obtained from (71) by substituting for X the value given by (75). Consequently we have But we find so that From (74) we find and from (77) we find similarly so that According to this equation, r is the harmonic conjugate of the origin, both with respect to q1, q2 and with respect to r1, r2. The point T will appear so frequently in our theorems as to make it desirable to give it a name. We therefore formulate the following definition. The point T, which is the harmonic conjugate of the point of contact with respect to the poles of the osculating quadratic, shall
may be a root of (71), we must have A = 0, and this condition furnishes the following equation for X: Let r1 and r2 be the remaining two poles of this special osculating cubic. Then r1 and r2 are the roots of the quadratic obtained from (71) by substituting for X the value given by (75). Consequently we have But we find so that From (74) we find and from (77) we find similarly so that According to this equation, r is the harmonic conjugate of the origin, both with respect to q1, q2 and with respect to r1, r2. The point T will appear so frequently in our theorems as to make it desirable to give it a name. We therefore formulate the following definition. The point T, which is the harmonic conjugate of the point of contact with respect to the poles of the osculating quadratic, shall be called the quadratic satellite of the point of contact. We may now express the contents of equations (80) as follows. Consider the special osculating cubic one of whose poles coincides with the pole of the osculating linear function, its other two poles will be separated harmonically by the point of contact and the quadratic satellite. Still another formulation of this theorem is the following. The involution determined by the two point pairs (q1, q2) and (l1, l2) has as its double points the point of contact and its quadratic satellite. We have characterized geometrically, the denominators of two of the osculating cubics, namely (z - q1) (z - q2) and (z - p)(z - 11)(z - 12), where the latter may be replaced by (z - 1)(z - 2) if the expansion of f(z) is assumed to be in its canonical form. The poles of any osculating cubic will then be the roots of the equation, which is merely another form for the cubic involution, Let s1, s2, s3 be the roots of this cubic for a given value of p, so that s1, s2, s3 form a triple of the cubic involution. We find IS1 + q1 +