Estimates of Zeros of a Polynomial

1962 ◽  
Vol 58 (2) ◽  
pp. 229-234 ◽  
Author(s):  
L. Mirsky
Keyword(s):  
Integral Formula ◽  
Transcendental Numbers ◽  
Image Position ◽  
Complex Coefficients

Throughout this note we shall consider a fixed polynomial with complex coefficients and of degree n ≥ 2. Its zeros will be denoted by ξ1, ξ2, …, ξn where the numbering is such that Making use of Jensen's integral formula, Mahler (4) showed that, for l ≥ k < n, A slightly weaker result had been established by Feldman in an earlier publication (2). Mahler's inequality (1) is of importance in the study of transcendental numbers, and our first object is to sharpen his bound by proving the following result.

Analytic functions of finite dimensional linear transformations

1959 ◽  
Vol 55 (1) ◽  
pp. 51-61 ◽  
Author(s):  
S. N. Afriat
Keyword(s):  
Power Series ◽  
Integral Formula ◽  
Characteristic Root ◽  
Interpolation Formula ◽  
General Function ◽  
Linear Transformations ◽  
Multiple Characteristic ◽  
Characteristic Roots ◽  
Image Position ◽  
Necessary And Sufficient

Since the first introduction of the concept of a matrix, questions about functions of matrices have had the attention of many writers, starting with Cayley(i) in 1858, and Laguerre(2) in 1867. In 1883, Sylvester(3) defined a general function φ(a) of a matrix a with simple characteristic roots, by use of Lagrange's interpolation formula, and Buchheim (4), in 1886, extended his definition to the case of multiple characteristic roots. Then Weyr(5) showed in 1887 that, for a matrix a with characteristic roots lying inside the circle of convergence of a power series φ(ζ), the power series φ(a) is convergent; and in 1900 Poincaré (6) obtained the formulaefor the sum, where C is a circle lying in and concentric with the circle of convergence, and containing all the characteristic roots in its ulterior, such a formula having effectively been suggested by Frobenius(7) in 1896 for defining a general function of a matrix. Phillips (8), in 1919, discovered the analogue, for power series in matrices, of Taylor's theorem. In 1926 Hensel(9) completed the result of Weyr by showing that a necessary and sufficient condition for the convergence of φ(a) is the convergence of the derived series φ(r)(α) (0 ≼ r < mα; α) at each characteristic root α of a, of order r at most the multiplicity mα of α. In 1928 Giorgi(10) gave a definition, depending on the classical canonical decomposition of a matrix, which is equivalent to the contour integral formula, and Fantappie (11) developed the theory of this formula, and obtained the expressionfor the characteristic projectors.

scholarly journals An infinite integral formula

1939 ◽  
Vol 6 (2) ◽  
pp. 75-77
Author(s):  
C. G. Lambe
Keyword(s):  
Integral Formula ◽  
The Other ◽  
Infinite Integral ◽  
Image Position

§ 1. The object of this note is to discuss the formulathe integral being supposed convergent for certain ranges of values of x and z. The contour is such that the poles of Γ(– s)lie to its right and the other poles of the integrand to its left. It will be seen that all the Pincherle-Mellin-Barnes integrals are particular cases of this formula.

On the product of n linear forms

1953 ◽  
Vol 49 (2) ◽  
pp. 190-193 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Complex Conjugate ◽  
Linear Forms ◽  
Image Position ◽  
Complex Coefficients ◽  
Homogeneous Linear

Let L1, …, Ln be n homogeneous linear forms in n variables u1, …, un, with non-zero determinant Δ. Suppose that L1, …, Lr have real coefficients, that Lr+1, …, Lr+s have complex coefficients, and that the form Lr+s+j is the complex conjugate of the form Lr+j for j = 1, …, s, where r + 2s = n. Letfor integral u1, …, un, not all zero. For any n numbers α1, …, αn of the same ‘type’ as the forms L1, …, Ln (that is, α1, …, αr real, αr+1, …, αr+s complex, αr+s+j = ᾱr+j), let

On convergence and quasiconformality of complex planar spline interpolants

1986 ◽  
Vol 99 (2) ◽  
pp. 347-356 ◽  
Author(s):  
H. P. Dikshit ◽  
A. Ojha
Keyword(s):  
Finite Elements ◽  
Integral Formula ◽  
Holomorphic Functions ◽  
The Other ◽  
Piecewise Function ◽  
Cauchy’S Integral Formula ◽  
Recent Origin ◽  
Image Position ◽  
Spline Interpolants ◽  
Identity Theorem

There appear to be two main approaches for developing complex splines. One of these, which has been in use for quite some time, consists in defining splines on the boundary of a given region which are then extended into the interior by Cauchy's integral formula (see e.g. [1]). The other approach, which is of a more recent origin, is motivated in spirit by the theory of finite elements (see e.g. [10], p. 320) and is contained in [8] and [9]. Observing that the foregoing extension into the interior is not easy to execute numerically, certain continuous piecewise non-holomorphic functions, called complex planar splines have been studied in [8] and [9]. The choice of non-holomorphic functions is justified, since if we take the pieces to be holomorphic functions like polynomials, then by the well known identity theorem ([5], p. 132, theorem 60) the continuity of such a piecewise function implies that all the pieces represent just one holomorphic function. Thus, we shall consider polynomials in z and its conjugate z¯ of the formwhich are generally non-holomorphic functions. The numberwill be called the degree of q. For simplicity we also write q(z) for q(z, z¯).

Linear Combinations of Univalent Functions with Complex Coefficients

1971 ◽  
Vol 23 (4) ◽  
pp. 712-717 ◽  
Author(s):  
Robert K. Stump
Keyword(s):  
Linear Combination ◽  
Convex Domain ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Linear Combinations ◽  
Radius Of Starlikeness ◽  
Complex Constant ◽  
Complex Functions ◽  
Image Position ◽  
Complex Coefficients

Let U be the class of all normalized analytic functionswhere z ∈ E = {z : |z| < 1} and ƒ is univalent in E. Let K denote the sub-class of U consisting of those members that map E onto a convex domain. MacGregor [2] showed that if ƒ1 ∈ K and ƒ2 ∈ K and if1then F ∉ K when λ is real and 0 < λ < 1, and the radius of univalency and starlikeness for F is .In this paper, we examine the expression (1) when ƒ1 ∈ K, ƒ2 ∈ K and λ is a complex constant and find the radius of starlikeness for such a linear combination of complex functions with complex coefficients.

scholarly journals An Integral Formula For Qn (cos θ)

1945 ◽  
Vol 7 (2) ◽  
pp. 81-82
Author(s):  
E. T. Copson
Keyword(s):  
General Solution ◽  
Present Note ◽  
Integral Formula ◽  
Simple Expression ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Particular Solution ◽  
Image Position

The functionis, as is well-known, a general solution of Laplace's equation of degree −1 in (x, y, z). In 1926* I proved that the particular solution r−1Q0 (z/r) cannot be represented in this form whereas the solution r−1Q0 (y/r) can. In the present note I find a very simple expression for the latter solution in the form (1.1), and I deduce from it an apparently new integral formula for Qn (cos θ).

scholarly journals An inequality for a pair of polynomials that are relatively prime

1964 ◽  
Vol 4 (4) ◽  
pp. 418-420 ◽  
Author(s):  
K. Mahler
Keyword(s):  
Arbitrary Complex ◽  
Image Position ◽  
Complex Coefficients

Let f(x) and g(x) be two polynomials with arbitrary complex coefficients that are relatively prime. Hence the maximum is positive for all complex x. Since m(x) is continuous and tends to infinity with |x|, the quantity E(f, g) = min m(x) is therefore also positive.

A Symmetric Proof of the Riemann-Roch Theorem, and a New Form of the Unit Theorem

1952 ◽  
Vol 4 ◽  
pp. 136-148
Author(s):  
S. Beatty ◽  
N. D. Lane
Keyword(s):  
Algebraic Equation ◽  
Rational Functions ◽  
Independent Variable ◽  
New Form ◽  
Roch Theorem ◽  
Image Position ◽  
Complex Coefficients

Let F(z, u) denote1where F1(z),… , Fn(z) are rational functions of z with complex coefficients. We shall speak of F (z, u) = 0 as the fundamental algebraic equation and shall adopt z as the independent variable and u as the dependent, except in § 4, where we use x and y instead of them, and where it is understood that x and y are connected birationally with z and u.

On lower estimates for linear forms involving certain transcendental numbers

1976 ◽  
Vol 14 (2) ◽  
pp. 161-179 ◽  
Author(s):  
Keijo Väänänen
Keyword(s):  
Rational Numbers ◽  
Linear Forms ◽  
Transcendental Numbers ◽  
Image Position

Letwhere λ is rational and not an integer. The author investigates lower estimates for example forwhere the αi are distinct rational numbers not 0, and where x1, …, xk, are integers and

Substitution Groups of Formal Power Series

1954 ◽  
Vol 6 ◽  
pp. 325-340 ◽  
Author(s):  
S. A. Jennings
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Point Of View ◽  
Group Operation ◽  
Several Variables ◽  
Special Cases ◽  
Image Position ◽  
Formal Power ◽  
Complex Coefficients

In this paper we are concerned with the group of formal power series of the form,the coefficients being elements of a commutative ring R and the group operation being substitution. Little seems to be known of the properties of groups of this type, except in special cases, although groups of formal power series in several variables with complex coefficients have been investigated from a different point of view by Bochner and Martin (1, chap. I) and Gotô (2).

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