On the function

1963 ◽  
Vol 59 (4) ◽  
pp. 735-737
Author(s):  
A. S. Meligy ◽  
E. M. EL Gazzy
Keyword(s):  
Positive Integer ◽  
General Formula ◽  
Bessel Functions ◽  
Exponential Integral ◽  
Euler’S Constant ◽  
Image Position ◽  
Euler's Constant

In a previous paper (3) one of us reported an expansion for the exponential integralin terms of Bessel functions. In this note, we shall obtain the more general formulawhere n is any positive integer, γ is Euler's constant andIt reduces to that in (3) when n = 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.

scholarly journals Some Finite Analogues of the Poisson Summation Formula

1961 ◽  
Vol 12 (3) ◽  
pp. 133-138 ◽  
Author(s):  
L. Carlitz
Keyword(s):  
Summation Formula ◽  
Poisson Summation Formula ◽  
Euler’S Constant ◽  
Image Position ◽  
Positive Integers ◽  
Euler's Constant ◽  
Poisson Summation

1. Guinand (2) has obtained finite identities of the typewhere m, n, N are positive integers and eitherorwhere γ is Euler's constant and the notation ∑′ indicates that when x is integral the term r = x is multiplied by ½. Clearly there is no loss of generality in taking N = 1 in (1.1).

scholarly journals A note on Mathieu functions

1957 ◽  
Vol 3 (3) ◽  
pp. 132-134 ◽  
Author(s):  
M. Bell
Keyword(s):  
Positive Integer ◽  
General Formula ◽  
Algebraic Equation ◽  
Explicit Construction ◽  
High Order ◽  
Mathieu Functions ◽  
Unstable Zone ◽  
Leading Term ◽  
Image Position ◽  
Integral Order

The Mathieu functions of integral order [1] are the solutions with period π or 2π of the equationThe eigenvalues associated with the functions ceN and seN, where N is a positive integer, denoted by aN and bN respectively, reduce toaN = bN = N2when q is zero. The quantities aN and bN can be expanded in powers of q, but the explicit construction of high order coefficients is very tedious. In some applications the quantity of most interest is aN – bN, which may be called the “width of the unstable zone“. It is the object of this note to derive a general formula for the leading term in the expansion of this quantity, namelySuppose first that N is an odd integer. Then there is an expansionwhereThese functions π satisfyandOn Substituting (3) in (1), one obtains the algebraic equationwhereExplicitly,{11} = q{lm} = 0 otherwise.

A Note on Some Integrals Involving Bessel Functions

1929 ◽  
Vol 25 (2) ◽  
pp. 130-131
Author(s):  
C. Fox
Keyword(s):  
Positive Integer ◽  
Bessel Functions ◽  
Image Position

The object of this note is to prove the following results, all of which hold when |a| < 1.(2) If r is any positive integer other than zero, the

scholarly journals An Elementary Derivation of the Exponential Limit and of Euler's Constant

1935 ◽  
Vol 29 ◽  
pp. xxi-xxiv ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Area Under The Curve ◽  
Euler’S Constant ◽  
Elementary Derivation ◽  
Fundamental Properties ◽  
Image Position ◽  
The Right ◽  
Euler's Constant

By defining a logarithm aswe may visualise the function as the area under the curve, measured to the right from the zero value at the ordinate AB. The fundamental properties follow at once from (1): for if u = av, then

scholarly journals On Series for calculating Euler's Constant, and the Constant in Stirling's Theorem

1909 ◽  
Vol 28 ◽  
pp. 48-59
Author(s):  
K. J. Sanjana
Keyword(s):  
Euler’S Constant ◽  
Elementary Methods ◽  
Image Position ◽  
Euler's Constant ◽  
Made In

Let γn denote the value ofwhere n is a definite integer; and let γ denote the limit ofwhen the integer k is indefinitely increased. It is known that the expansion of γn – γ in ascending powers of 1/n iswhere B1, B3, B5… are the numbers of Bernoulli. The series (3) is, however, divergent, as B2r+1 not only increases indefinitely with r, but bears† an infinite ratio to B2r–1 in this case. It is proposed to find by elementary methods the expansion of γn – γ up to the term in nr and to estimate the error (of order l/nr+1) made in omitting further terms of series (3). I shall take the case of r = 9, but the process is quite general.

scholarly journals RATIONAL APPROXIMATIONS TO VALUES OF BELL POLYNOMIALS AT POINTS INVOLVING EULER’S CONSTANT AND ZETA VALUES

2012 ◽  
Vol 92 (1) ◽  
pp. 71-98
Author(s):  
KH. HESSAMI PILEHROOD ◽  
T. HESSAMI PILEHROOD
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Bell Polynomials ◽  
Rational Approximations ◽  
Euler’S Constant ◽  
The Riemann Zeta Function ◽  
Zeta Values ◽  
Riemann Zeta ◽  
Image Position ◽  
Euler's Constant

AbstractIn this paper we present new explicit simultaneous rational approximations which converge subexponentially to the values of the Bell polynomials at the points where m=1,2,…,a, a∈ℕ, γ is Euler’s constant and ζ is the Riemann zeta function.

scholarly journals On Dirichlet's divisor problem

1931 ◽  
Vol 134 (823) ◽  
pp. 192-202 ◽  
Keyword(s):  
Positive Integer ◽  
Divisor Problem ◽  
Euler’S Constant ◽  
Number Of Divisors ◽  
Prime Factors ◽  
Euler's Constant

1. Let d ( n ) denote the number of divisors of the positive integer n , so that, if n = p 1 a 1 . . . p r ar is the canonical expression of n in prime factors, d ( n ) = (1 + a 1 ) . . . (1 + a r ), and let d ( x ) = 0 if x is not an iteger; then if (1. 1) D ( x ) = Σ' n ≤ x d ( n ) = Σ n ≤ x d ( n ) ─ ½ d ( x ), and (1. 2) Δ ( x ) = D ( x ) ─ x log x ─ (2C ─ 1) x ─ ¼, where C is Euler's constant, it was proved by Dirichlet in 1849 that (1. 21) Δ ( x ) = O (√ x ),

scholarly journals On some infinite series involving the zeros of Bessel functions of the first kind

1960 ◽  
Vol 4 (3) ◽  
pp. 144-156 ◽  
Author(s):  
Ian N. Sneddon
Keyword(s):  
Positive Integer ◽  
Bessel Function ◽  
Rational Function ◽  
Recurrence Relation ◽  
Infinite Series ◽  
Bessel Functions ◽  
Zeros Of Bessel Functions ◽  
Image Position

In this paper we shall be concerned with the derivation of simple expressions for the sums of some infinite series involving the zeros of Bessel functions of the first kind. For instance, if we denote by γv, n (n = l, 2, 3,…) the positive zeros of Jv(z), then, in certain physical applications, we are interested in finding the values of the sumsandwhere m is a positive integer. In § 4 of this paper we shall derive a simple recurrence relation for S2m,v which enables the value of any sum to be calculated as a rational function of the order vof the Bessel function. Similar results are given in § 5 for the sum T2m,v.

scholarly journals On certain new connections between Legendre and Bessel Functions

1935 ◽  
Vol 4 (3) ◽  
pp. 111-111
Author(s):  
S. C. Mitra
Keyword(s):  
Positive Integer ◽  
Bessel Functions ◽  
Image Position

Let n be a positive integer. Then we know that, if m>– 1,Consider the integralwhich is equal to

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