On the summation of certain trigonometric series

1945 ◽  
Vol 41 (2) ◽  
pp. 145-160 ◽  
Author(s):  
L. S. Goddard
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Trigonometric Series ◽  
Present Note ◽  
Rational Functions ◽  
Recurrent Relation ◽  
Polygamma Function ◽  
Special Values ◽  
Image Position

In the present note, which is introductory to the following paper, closed expressions, suitable for computational purposes, are found for the sums of the serieswhere α > 1, t = 1, 2, 3, …, and n is a positive integer. In each case a recurrent relation is found giving the values of and for t > 2 in terms of and the series Θκ(α) (κ = 1, 2, …, t), whereWhen κ is even the last series is expressed in closed form in terms of the Bernoullian polynomial φκ(l/α) and, when κ is odd and α is rational, a closed form is found involving the polygamma function Ψ(κ)(z), where The general expressions for and involve Ψ(z) and Ψ′(z) when α is rational, but for special values of α they reduce to a form independent of the Ψ-function. and are independent of n and are expressible as simple rational functions of α.

scholarly journals On the representations of numbers by binary cubic forms

1985 ◽  
Vol 27 ◽  
pp. 95-98 ◽  
Author(s):  
C. Hooley
Keyword(s):  
Positive Integer ◽  
Recent Work ◽  
Present Note ◽  
Cubic Form ◽  
Image Position ◽  
Cubic Forms ◽  
Theory Of Numbers

There are not a few situations in the theory of numbers where it is desirable to have as sharp an estimate as possible for the number r(n) of representations of a positive integer n by an irreducible binary cubic formA variety of approaches are available for this problem but, as they stand, they are all defective in that they introduce unwanted factors into the estimate. For instance, an estimate involving the discriminant of f(x, y) is obtained if we adopt the Lagrange procedure [5] of using congruences of the type f(σ, 1)≡0, mod n, to reduce the problem to one where n=1. Alternatively, following Oppenheim (vid. [2]), Greaves [3], and others, we may appeal to the theory of factorization of ideals, which leads to unwanted logarithmic factors owing to the involvement of algebraic units. Having had need, however, in some recent work on quartic forms [4] for an estimate without such extraneous imperfections, we intend in the present note to prove thatuniformly with respect to the coefficients of f(x, y), where ds(n) denotes the number of ways of expressing n as a product of s factors.

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

scholarly journals On certain infinite integrals involving Struve functions and parabolic cylinder functions

1946 ◽  
Vol 7 (4) ◽  
pp. 171-173 ◽  
Author(s):  
S. C. Mitra
Keyword(s):  
Present Note ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Struve Functions ◽  
Image Position

The object of the present note is to obtain a number of infinite integrals involving Struve functions and parabolic cylinder functions. 1. G. N. Watson(1) has proved thatFrom (1)follows provided that the integral is convergent and term-by-term integration is permissible. A great many interesting particular cases of (2) are easily deducible: the following will be used in this paper.

scholarly journals Value distribution of meromorphic functions concerning rational functions and differences

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mingliang Fang ◽  
Degui Yang ◽  
Dan Liu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Rational Function ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Rational Functions ◽  
Value Distribution ◽  
Exponent Of Convergence ◽  
Nonzero Constant ◽  
Zero Points

AbstractLet c be a nonzero constant and n a positive integer, let f be a transcendental meromorphic function of finite order, and let R be a nonconstant rational function. Under some conditions, we study the relationships between the exponent of convergence of zero points of $f-R$ f − R , its shift $f(z+nc)$ f ( z + n c ) and the differences $\Delta _{c}^{n} f$ Δ c n f .

The Distribution Of Totatives

1955 ◽  
Vol 7 ◽  
pp. 347-357 ◽  
Author(s):  
D. H. Lehmer
Keyword(s):  
Positive Integer ◽  
Prime Factors ◽  
Image Position

This paper is concerned with the numbers which are relatively prime to a given positive integerwhere the p's are the distinct prime factors of n. Since these numbers recur periodically with period n, it suffices to study the ϕ(n) numbers ≤n and relatively prime to n.

scholarly journals An algebraically closed field

1968 ◽  
Vol 9 (2) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Rayner
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Zero Characteristic ◽  
Image Position ◽  
Formal Power ◽  
Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

scholarly journals An Integral involving a Modified Bessel Function of the Second Kind and an E-function

1953 ◽  
Vol 1 (3) ◽  
pp. 119-120 ◽  
Author(s):  
Fouad M. Ragab
Keyword(s):  
Positive Integer ◽  
Bessel Function ◽  
Modified Bessel Function ◽  
Image Position

§ 1. Introductory. The formula to be established iswhere m is a positive integer,and the constants are such that the integral converges.

The Representation of (C, k) Summable Series in Fourier Form

1978 ◽  
Vol 21 (2) ◽  
pp. 149-158 ◽  
Author(s):  
G. E. Cross
Keyword(s):  
Trigonometric Series ◽  
Point Of View ◽  
Perron Integral ◽  
Image Position

Several non-absolutely convergent integrals have been defined which generalize the Perron integral. The most significant of these integrals from the point of view of application to trigonometric series are the Pn- and pn-integrals of R. D. James [10] and [11]. The theorems relating the Pn -integral to trigonometric series state essentially that if the series1.1

scholarly journals Integrals involving products of modified Bessel functions of the second kind

1963 ◽  
Vol 6 (2) ◽  
pp. 70-74 ◽  
Author(s):  
F. M. Ragab
Keyword(s):  
Positive Integer ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Image Position

It is proposed to establish the two following integrals.where n is a positive integer, x is real and positive, μi and ν are complex, and Δ (n; a) represents the set of parameterswhere n is a positive integer and x is real and positive.

Sophocles, ‘Electra’ 1205–10

1973 ◽  
Vol 19 ◽  
pp. 45-46
Author(s):  
R. D. Dawe
Keyword(s):  
Present Note ◽  
Greek Tragedy ◽  
Image Position

The attribution of lines to different speakers in Greek tragedy is a matter on which MSS have notoriously little authority. As for Electra itself, there are at least three places where the name of the heroine has been incorrectly added in some or all MSS. In my Studies in the Text of Sophocles, I, 198, I list these places and suggest that the same error has happened at a fourth place, viz. 1323. The purpose of the present note is to suggest that at El. 1205–10 the same mistake has happened yet again.The situation is that Electra is holding the urn which she falsely believes to contain the ashes of her dead brother, Orestes. But Orestes is alive, and before her at this very moment. He is trying to persuade her to give up the urn. If the text before us had been preserved in a MS devoid of ascriptions to speakers, no one would have been so perverse as to do what all MSS and editors do in fact do, namely attribute the words οὔ φημ᾿ ἐάσειν to Orestes.

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