A note on integral functions

1932 ◽  
Vol 28 (3) ◽  
pp. 262-265 ◽  
Author(s):  
R. E. A. C. Paley
Keyword(s):  
Finite Order ◽  
Infinite Order ◽  
Integral Function ◽  
Zero Order ◽  
Image Position

1. Let f(z) denote an integral function of finite order ρ. We writeIt has been shown thatwhere hρ is a constant which depends only on ρ. We are naturally led to enquire whether some equation of the form (1.1) may be true with lim sup replaced by lim inf. In this note we show that the reverse is true. We construct an integral function of zero order for whichThe proof may easily be modified to construct a function of any finite order or of infinite order for which (1.2) is satisfied.

scholarly journals ON PROPERTIES OF FINITE-ORDER MEROMORPHIC SOLUTIONS OF A CERTAIN DIFFERENCE PAINLEVÉ I EQUATION

2011 ◽  
Vol 85 (3) ◽  
pp. 463-475 ◽  
Author(s):  
MEI-RU CHEN ◽  
ZONG-XUAN CHEN
Keyword(s):  
Finite Order ◽  
Meromorphic Solutions ◽  
Rational Solutions ◽  
Polynomial Solutions ◽  
The Difference ◽  
Image Position

AbstractIn this paper, we investigate properties of finite-order transcendental meromorphic solutions, rational solutions and polynomial solutions of the difference Painlevé I equation where a, b and c are constants, ∣a∣+∣b∣≠0.

Congruence Relations Between the Traces of Matrix Powers

1949 ◽  
Vol 1 (3) ◽  
pp. 303-304 ◽  
Author(s):  
J. S Frame
Keyword(s):  
Finite Order ◽  
Rational Integer ◽  
Roots Of Unity ◽  
Finite Degree ◽  
Characteristic Roots ◽  
Congruence Relations ◽  
Image Position

Let A be a matrix of finite order n and finite degree d, whose characteristic roots are certain nth roots of unity a1, a2…, ad. We wish to prove a congruence (6) between the traces (tr) of certain powers of A, which is suggested by two somewhat simpler congruences (1) and (3). First, if tr (A) is a rational integer, it is easy to establish the familiar congruenceeven though tr(Ap) may not itself be rational.

scholarly journals On the application of the operational calculus to the expansion of a function in a series of Legendre's functions of the second kind

1942 ◽  
Vol 7 (1) ◽  
pp. 1-2
Author(s):  
D. P. Banerjee
Keyword(s):  
Analytic Function ◽  
Present Note ◽  
Operational Calculus ◽  
Integral Function ◽  
Legendre Functions ◽  
Image Position

In the present note we shall obtain the expansion in a series of Legendre functions of the second kind of an integral function φ (ω) represented by Laplace's integralwhere f (x) is an analytic function of x, regular in the circlewhere an are constants and qn (ω) = in+1Qn (iω).

An extension of a theorem of Kulakoff

1938 ◽  
Vol 34 (3) ◽  
pp. 316-320
Author(s):  
T. E. Easterfield
Keyword(s):  
Finite Order ◽  
Dihedral Group ◽  
Mixed Group ◽  
Quaternion Group ◽  
Sylow Subgroups ◽  
Cyclic Groups ◽  
Image Position

It has been shown by Kulakoff that if G is a group, not cyclic, of order pl, p being an odd prime, the number of subgroups of G of order pk, for 0 < k < l, is congruent to 1 + p (mod p2); and by Hall that if G is any group of finite order whose Sylow subgroups of G of order pk, p being odd, are not cyclic, then, for 0 < k < l, the number of subgroups of G of order pk is congruent to 1 + p (mod p2). No results were given for the case p = 2. In the present paper it is shown that analogous results hold for the case p = 2, but that the role of the cyclic groups is played by groups of four exceptional types: the cyclic groups themselves, and three non-Abelian types. These groups are defined as follows:(1) The dihedral group, of order 2k, generated by A and B, where(2) The quaternion group, of order 2k, generated by A and B, where(3) The "mixed" group, of order 2k, generated by A and B, where

scholarly journals On a family of logarithmic and exponential integrals occurring in probability and reliability theory

1994 ◽  
Vol 35 (4) ◽  
pp. 469-478 ◽  
Author(s):  
M. Aslam Chaudhry
Keyword(s):  
Closed Form ◽  
Recurrence Relation ◽  
Reliability Theory ◽  
Integral Function ◽  
Integral Representations ◽  
Exponential Integral ◽  
Error Functions ◽  
Special Cases ◽  
Image Position ◽  
Form Evaluation

AbstractWe define an integral function Iμ(α, x; a, b) for non-negative integral values of μ byIt is proved that Iμ(α, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals. New integral representations of the exponential integral and complementary error functions are found as special cases.

On the question of whether f″+ e−zf′ + B(z)f = 0 can admit a solution f ≢ 0 of finite order

1986 ◽  
Vol 102 (1-2) ◽  
pp. 9-17 ◽  
Author(s):  
Gary G. Gundersen
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Infinite Order ◽  
Independent Interest ◽  
Partial Result ◽  
Transcendental Entire Function ◽  
Degree Polynomial ◽  
Odd Degree ◽  
Polynomial Of Odd Degree

SynopsisWe show that if B(z) is either (i) a transcendental entire function with order (B)≠1, or (ii) a polynomial of odd degree, then every solution f≠0 to the equation f″ + e−zf′ + B(z)f = 0 has infinite order. We obtain a partial result in the case when B(z) is an even degree polynomial. Our method of proof and lemmas for case (i) of the above result have independent interest.

scholarly journals A note on the finite Fourier transform

1959 ◽  
Vol 1 (1) ◽  
pp. 95-98
Author(s):  
James L. Griffith
Keyword(s):  
Fourier Transform ◽  
Real Axis ◽  
Exponential Type ◽  
Integral Function ◽  
Finite Fourier Transform ◽  
The Real ◽  
Almost Everywhere ◽  
Type C ◽  
Image Position

1. One of the best known theorems on the finite Fourier transform is:—The integral function F(z) is of the exponential type C and belongs to L2 on the real axis, if and only if, there exists an f(x) belonging to L2 (—C, C) such that ( Additionally, if f(x) vanishes almost everywhere in a neighbourhood of C and also in a neighbourhood of —C, then F(z) is of an exponential type lower than C.

On some infinite integrals involving logarithmic exponential and powers

1992 ◽  
Vol 122 (1-2) ◽  
pp. 11-15 ◽  
Author(s):  
M. Aslam Chaudhry ◽  
Munir Ahmad
Keyword(s):  
Recurrence Relation ◽  
Integral Function ◽  
Special Cases ◽  
Infinite Integral ◽  
Image Position

SynopsisIn this paper we define an integral function Iμ(α; a, b) for non-negative integral values of μ byIt is proved that the function Iμ(α; a, b) satisfies a functional recurrence-relation which is then exploited to evaluate the infinite integralSome special cases of the result are also discussed.

On the Asymptotic Periods of Integral Functions

1935 ◽  
Vol 31 (4) ◽  
pp. 543-554 ◽  
Author(s):  
Sheila Scott
Keyword(s):  
Integral Function ◽  
Image Position ◽  
Single Sequence

A period of a function f(z) is defined to be a number ω (≠ 0) such thatis identically zero; and it can be shown that an integral function may either have no periods or else a single sequence kλ (k = ± 1, ± 2, …).

On methods of summability based on integral functions

1959 ◽  
Vol 55 (1) ◽  
pp. 23-30 ◽  
Author(s):  
D. Borwein
Keyword(s):  
Integral Function ◽  
Radius Of Convergence ◽  
Complex Numbers ◽  
Arbitrary Complex ◽  
Image Position

Suppose throughout thatand thatis an integral function. Suppose also that l, sn(n = 0,1,…) are arbitrary complex numbers and denote by ρ(ps) the radius of convergence of the series

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