jeanflop/post_ocr_correction2 huggingface dataset:
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 + s2 + s3 = 0, s1s2 + s1s3 + s2s3 = q1 + q2, and therefore s1s2s3 = -1, and the equations, being independent of u, represent those properties of the triples of our involution, which are common to all of them. Let us determine those triples of the involution which have two coincident points. They correspond to those osculating cubics which have a multiple pole, and which we shall call the singular osculating cubics. For such a triple we may put s1 = d, s2 = 2d + e, s3 = d + 2de, e, s1 = s2 = s3 = d2e. If we substitute these values in the equations, we find whence If the value of e from is substituted in the first equation of , we obtain the biquadratic equation, 2(T1 + r2)d2 + 3(r1 + q2)Crt + "a", - 2 t1, r2 (r1 + r2)^ + rxiq1q2 - 0, which has four double points dk of the cubic involution. To each of these four double points there corresponds, by means of a point eke at its companion, and we obtain in general four singular osculating cubics each of which has one of the points dk as
which have two coincident points. They correspond to those osculating cubics which have a multiple pole, and which we shall call the singular osculating cubics. For such a triple we may put s1 = d, s2 = 2d + e, s3 = d + 2de, e, s1 = s2 = s3 = d2e. If we substitute these values in the equations, we find whence If the value of e from is substituted in the first equation of , we obtain the biquadratic equation, 2(T1 + r2)d2 + 3(r1 + q2)Crt + "a", - 2 t1, r2 (r1 + r2)^ + rxiq1q2 - 0, which has four double points dk of the cubic involution. To each of these four double points there corresponds, by means of a point eke at its companion, and we obtain in general four singular osculating cubics each of which has one of the points dk as a double pole and whose remaining pole, in general distinct from dk, will be eke. Of course, equation might also have been obtained by equating to zero the Jacobian of the pencil of binary forms. We may write as follows Consequently we find The first of these equations is capable of a simple interpretation. Let us put where i, j, k, l represent the indices 1, 2, 3, 4, arranged in any order. Then we shall have, according to, Consequently the four double poles of the four singular osculating cubics have the following property. Select any two of them, dt and dj, and determine the harmonic conjugate, d(j, of the point of contact with respect to them. Proceed in the same way with the other two double poles, dk and d., giving rise to the point dkl. The points dj and d kl will be harmonic conjugates of each other with respect to the point of contact and its quadratic satellite. The four points dt may be arranged in two pairs in six different ways, thus giving rise to six pairs (dij, dkl). According to the theorem just proved, each of these pairs determines, with the point of contact, a
of a simple interpretation. Let us put where i, j, k, l represent the indices 1, 2, 3, 4, arranged in any order. Then we shall have, according to, Consequently the four double poles of the four singular osculating cubics have the following property. Select any two of them, dt and dj, and determine the harmonic conjugate, d(j, of the point of contact with respect to them. Proceed in the same way with the other two double poles, dk and d., giving rise to the point dkl. The points dj and d kl will be harmonic conjugates of each other with respect to the point of contact and its quadratic satellite. The four points dt may be arranged in two pairs in six different ways, thus giving rise to six pairs (dij, dkl). According to the theorem just proved, each of these pairs determines, with the point of contact, a circle passing through the point t. We obtain in this way six circles of the pencil determined by the point of contact and its quadratic satellite. A similar relation may be obtained from. Let us put Then we find from, But in deducing equation we used the canonical expansion, so that the pole of the osculating linear function p was at infinity. Consequently equations indicate that (fij, p) and (dj, dj) are harmonic pairs, that (fkl, p) and (dk, dj) are harmonic, etc. We obtain therefore the following second property of the points dt. Select any two, di and dj, of the four points dj, d2, d3, d.,, and determine the harmonic conjugate, fij, of the pole p of the osculating linear function with respect to them. Proceed in the same way with the remaining two points, dk and dh giving rise to the point fkl. The resulting two points, fij and fkl, will be harmonic conjugates of each other with respect to the pole of the osculating linear function p, and the point r, which is itself the harmonic conjugate of p with respect to r, and y-a. We obtain in this way six circles of the pencil determined
that the pole of the osculating linear function p was at infinity. Consequently equations indicate that (fij, p) and (dj, dj) are harmonic pairs, that (fkl, p) and (dk, dj) are harmonic, etc. We obtain therefore the following second property of the points dt. Select any two, di and dj, of the four points dj, d2, d3, d.,, and determine the harmonic conjugate, fij, of the pole p of the osculating linear function with respect to them. Proceed in the same way with the remaining two points, dk and dh giving rise to the point fkl. The resulting two points, fij and fkl, will be harmonic conjugates of each other with respect to the pole of the osculating linear function p, and the point r, which is itself the harmonic conjugate of p with respect to r, and y-a. We obtain in this way six circles of the pencil determined by p and r. Let us put so that q is the harmonic conjugate of p with respect to q1 and q2, dzjlcl is the harmonic conjugate of p with respect to dij and dkl, and fijlcl is the harmonic conjugate of the point of contact with respect to fij and fkl. We shall have qlq2 = qz, rlri = rr:, If we make use of the coefficient of d2 and the constant term of equation, we find (fijd - ij fijdij + fklr - fijfkl - 3r2 qz (qr)^qr, dijdk + ijkl = qr - r', which may be written dijkl!ijkl = q - r = qiq2, so that dijkl and fijkl are a pair of the involution, of which z = o, z' = oo, and z = qt, z' = q2, constitute two other pairs. Thus the involution of which the point of contact and the pole of the osculating linear function form one pair, and for which the poles of the osculating quadratic form a second pair, also contains the pair dijkl, fijkl. Let us assume that dt and d2 are distinct. Then (z — 1)^(z — α), and (z — β)^(z — γ), will be
and fkl. We shall have qlq2 = qz, rlri = rr:, If we make use of the coefficient of d2 and the constant term of equation, we find (fijd - ij fijdij + fklr - fijfkl - 3r2 qz (qr)^qr, dijdk + ijkl = qr - r', which may be written dijkl!ijkl = q - r = qiq2, so that dijkl and fijkl are a pair of the involution, of which z = o, z' = oo, and z = qt, z' = q2, constitute two other pairs. Thus the involution of which the point of contact and the pole of the osculating linear function form one pair, and for which the poles of the osculating quadratic form a second pair, also contains the pair dijkl, fijkl. Let us assume that dt and d2 are distinct. Then (z — 1)^(z — α), and (z — β)^(z — γ), will be two distinct forms of the cubic involution (71), and this involution may be rewritten in the form (98) (z — δ)(z — ε) = (z — d)^2(z — e^2). Let S1, S2, S3 be the roots of (98) for a given value of v. We shall have (Skδ)^(S_kε) = v(S_k — δ^2) − e^2, (k = 1, 2, 3), and therefore, by division, (99) (δ, δ, sk, si; (ε, ε, sk, si) = 1, k = 1, 2, 3. These relations between the poles, S1, S2, S3, of any penosculting cubic and the poles of the four singular penosculting cubics are given by the double-ratio equations (99). In particular, these relations will hold for the poles of the osculating cubic. A similar relation may be obtained between s1, S2, S3 and the points p, q1, q2, r1, r2, by starting from the form (81) of the cubic involution, if we remember that in our canonical form z = 00 takes the place of z = p. Some of the features of the theory developed so far may be extended at once to the osculating rational functions of higher order. There will always exist one osculating rational function of degree a,
— δ^2) − e^2, (k = 1, 2, 3), and therefore, by division, (99) (δ, δ, sk, si; (ε, ε, sk, si) = 1, k = 1, 2, 3. These relations between the poles, S1, S2, S3, of any penosculting cubic and the poles of the four singular penosculting cubics are given by the double-ratio equations (99). In particular, these relations will hold for the poles of the osculating cubic. A similar relation may be obtained between s1, S2, S3 and the points p, q1, q2, r1, r2, by starting from the form (81) of the cubic involution, if we remember that in our canonical form z = 00 takes the place of z = p. Some of the features of the theory developed so far may be extended at once to the osculating rational functions of higher order. There will always exist one osculating rational function of degree a, one of whose poles is the point of contact, and whose other poles are the same as the poles of the osculating rational function of degree a. There will exist another osculating rational function of degree a, one of whose poles coincides with the pole of the osculating linear function. The poles of all of the other osculating functions of degree a will be determined by an involution of degree a, of which the two special integral rational functions of degree a just mentioned are the base forms. Among these osculating functions there will be 2(a — 1) which are singular, that is, which have at least two coincident poles. Our theorems have all been concerned with properties of the poles of the various osculating functions. But the same relations are also true of their zeros. To prove this statement, it suffices to note that the rational function has contact of order 2a with f(z) at z = 0, so that the expansion of U. f(z) = a0 + a1z + a2z2 + ... begins with the term of order at least 20C + 1. Since a0 = 0, we find that the expansion of will then also begin with
The poles of all of the other osculating functions of degree a will be determined by an involution of degree a, of which the two special integral rational functions of degree a just mentioned are the base forms. Among these osculating functions there will be 2(a — 1) which are singular, that is, which have at least two coincident poles. Our theorems have all been concerned with properties of the poles of the various osculating functions. But the same relations are also true of their zeros. To prove this statement, it suffices to note that the rational function has contact of order 2a with f(z) at z = 0, so that the expansion of U. f(z) = a0 + a1z + a2z2 + ... begins with the term of order at least 20C + 1. Since a0 = 0, we find that the expansion of will then also begin with a term of order at least 2(a + 1). Therefore, the rational osculating functions of are the reciprocals of the corresponding rational osculants of f(z). Consequently, the same relations which we have found between the poles of the rational osculants of f(z) must also hold between their zeros. The same statement also applies to the point-sets for which the various osculants assume a given value k, even if k is different from zero or infinity. We have already pointed out that the rational osculating functions of order 1, 2, 3, ... which we have studied are the functions which normally occupy the positions in the principal diagonal of the Pade table. We have also studied certain classes of osculating functions which have not been considered by Pade, and it appears clearly from our exposition that these osculants are needed for the geometric interpretation. We have not so far considered any of the rational functions of Pade's table which are not on the principal diagonal. But our original objection to these functions, namely the non-invariant character of their definition, may now be disposed of, at least if we use some canonical form for our power series so that the point at
statement also applies to the point-sets for which the various osculants assume a given value k, even if k is different from zero or infinity. We have already pointed out that the rational osculating functions of order 1, 2, 3, ... which we have studied are the functions which normally occupy the positions in the principal diagonal of the Pade table. We have also studied certain classes of osculating functions which have not been considered by Pade, and it appears clearly from our exposition that these osculants are needed for the geometric interpretation. We have not so far considered any of the rational functions of Pade's table which are not on the principal diagonal. But our original objection to these functions, namely the non-invariant character of their definition, may now be disposed of, at least if we use some canonical form for our power series so that the point at infinity becomes the pole of the osculating linear function or some other invariantly defined point. Consequently, we may now regard all of the functions of the Pade table as legitimate objects of our theory, opening up a large field for further investigation. The theorems which have been established on the question of the convergence of a sequence of Pade fractions are of course of special interest. We merely mention that they provide, among other things, criteria as to whether a given power-series does or does not represent a meromorphic function, and that, in the former case, the poles and zeros of the rational osculants approach in definite fashion the poles and zeros of the function f(z) as limits. We close this section with a discussion of the convergence of the canonical expansion which we have used so frequently; we have postponed this discussion, because we shall make use in it of some of the results which have been obtained since the canonical expansion was first introduced. Let us suppose that the original series a0 + a1z + a2z^2 + ... is convergent for 1 &lt; R. We obtained the canonical expansion by putting where p is the pole of the
question of the convergence of a sequence of Pade fractions are of course of special interest. We merely mention that they provide, among other things, criteria as to whether a given power-series does or does not represent a meromorphic function, and that, in the former case, the poles and zeros of the rational osculants approach in definite fashion the poles and zeros of the function f(z) as limits. We close this section with a discussion of the convergence of the canonical expansion which we have used so frequently; we have postponed this discussion, because we shall make use in it of some of the results which have been obtained since the canonical expansion was first introduced. Let us suppose that the original series a0 + a1z + a2z^2 + ... is convergent for 1 &lt; R. We obtained the canonical expansion by putting where p is the pole of the osculating linear function. From our method of deriving the canonical series it is evident that it will be convergent if we have simultaneously whence Thus, the canonical expansion will surely be convergent in a circle of center z = 0 and of radius r, where r is the smaller of the two positive numbers, R and 1/p. V. — The osculating logarithm. We saw in Art. 3 that the absolute differential invariant 9 has a constant value 0 = k for any function of the form Since such a function involves four arbitrary constants, b, c, and d, we may expect to find a logarithmic function which has third order contact with a given analytic function w = f(z) = a0 + a1z + a2z^2 + ... , at the origin. The resulting logarithmic function shall be called the osculating logarithm. We see at once that for the determination of k, we have where, of course, the value of 0 for z = 0 is meant, and where a definite determination should be chosen for the square root. However, a change in this determination has no actual significance since it will be balanced by an interchange in the values of
the two positive numbers, R and 1/p. V. — The osculating logarithm. We saw in Art. 3 that the absolute differential invariant 9 has a constant value 0 = k for any function of the form Since such a function involves four arbitrary constants, b, c, and d, we may expect to find a logarithmic function which has third order contact with a given analytic function w = f(z) = a0 + a1z + a2z^2 + ... , at the origin. The resulting logarithmic function shall be called the osculating logarithm. We see at once that for the determination of k, we have where, of course, the value of 0 for z = 0 is meant, and where a definite determination should be chosen for the square root. However, a change in this determination has no actual significance since it will be balanced by an interchange in the values of a and b, the two singular points of the osculating logarithm. With this value for k, we find from (100), In order that (100) may represent the osculating logarithm, these values of w, dw, and d^2w, must be equal to a0, a1, 2-a2 respectively for z = 0. We obtain the following conditions From the last two equations we conclude by division that is; the singular points of the osculating logarithm are harmonic conjugates of each other with respect to the point of contact and the pole of the osculating linear function. Since we have also found we obtain the formulae whence and finally that branch of the logarithm being selected in (105) such that the expansion of (100) will actually have a0 as its constant term. If the function kv = f(z) is not a linear function, we may use the second canonical form, for which a2 = 0, a3 = a1. We then find so that Moreover, the circles determined by (0, a, b) and (0, Pi, P2) are (for this canonical form) the imaginary and real axes of the z plane (circles through p = cc). Since angles are preserved by linear transformations we have proved the
the following conditions From the last two equations we conclude by division that is; the singular points of the osculating logarithm are harmonic conjugates of each other with respect to the point of contact and the pole of the osculating linear function. Since we have also found we obtain the formulae whence and finally that branch of the logarithm being selected in (105) such that the expansion of (100) will actually have a0 as its constant term. If the function kv = f(z) is not a linear function, we may use the second canonical form, for which a2 = 0, a3 = a1. We then find so that Moreover, the circles determined by (0, a, b) and (0, Pi, P2) are (for this canonical form) the imaginary and real axes of the z plane (circles through p = cc). Since angles are preserved by linear transformations we have proved the following theorem. The cross-ratio determined by the poles of the two singular points of the osculating quadratics and the singularities of the osculating logarithm is equal to a complex cube root of unity. The circle determined by one of these points pairs with the point of contact is orthogonal to the corresponding circle for the second pair, and these circles meet again in the pole of the osculating linear function. Let us consider for a moment the conformal representation of the z — plane upon the w — plane established by means of the osculating logarithm. If z moves on a circle of the pencil through a and -b, the argument of z ab will remain constant; if z moves on a circle of the orthogonal family, the modulus of z ab will remain constant. To the circles of the first family there corresponds in the w — plane the straight lines of a family of parallels, and to the circles of the second family there corresponds the straight lines of the orthogonal family. Through the point of contact (z = 0) there will pass a circle of each family. If we use the same canonical form as above, the
the corresponding circle for the second pair, and these circles meet again in the pole of the osculating linear function. Let us consider for a moment the conformal representation of the z — plane upon the w — plane established by means of the osculating logarithm. If z moves on a circle of the pencil through a and -b, the argument of z ab will remain constant; if z moves on a circle of the orthogonal family, the modulus of z ab will remain constant. To the circles of the first family there corresponds in the w — plane the straight lines of a family of parallels, and to the circles of the second family there corresponds the straight lines of the orthogonal family. Through the point of contact (z = 0) there will pass a circle of each family. If we use the same canonical form as above, the circle through 0, a, and b, is the axis of imaginaries. All circles of the second family have the property that a and b are inverse points with respect to each of them. Since in our canonical form the circle of the second family which passes through z = 0 is the axis of reals. Both of these lines, regarded as circles, pass through p = cc. We obtain the following result. Consider the pencil of circles through the singular points of the osculating logarithm, and the orthogonal family of circles. The two circles of the two families which pass through the point of contact meet again in the pole of the osculating linear function. We may also express this result as follows. Consider an analytic function w = f(z) and the osculating logarithm of f(z) at z = 0. Determine the isothermal system of straight lines in the w — plane which corresponds, by means of the osculating logarithm, to the isothermal system of circles which is determined by the singular points of the osculating logarithm. To this system of straight lines in the w plane there corresponds, by means of w = f(z), an isothermal system of curves