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Mark W. Coffey

Publications and source records attributed to Mark W. Coffey.

At least 37 records · Page 2Linked to original sources
arXiv2012

Expected exit times of Brownian motion from planar domains: Complements to a paper of Markowsky

We supplement a very recent paper of G. Markowsky concerned with the expected exit times of Brownian motion from planar domains. Besides the use of conformal mapping, we apply results from potential theory. We treat the case of a wedge-shaped region exactly, subsuming an example of Markowsky. A number of other results are presented, including for a half disc, $n$-grams, and a variety of other regions.

math-ph↗
arXiv2012

Series representation of the Riemann zeta function and other results: Complements to a paper of Crandall

We supplement a very recent paper of R. Crandall concerned with the multiprecision computation of several important special functions and numbers. We show an alternative series representation for the Riemann and Hurwitz zeta functions providing analytic continuation through out the whole complex plane. Additionally we demonstrate some series representations for the initial Stieltjes constants appearing in the Laurent expansion of the Hurwitz zeta function. A particular point of elaboration in these developments is the hypergeometric form and its equivalents for certain derivatives of the incomplete Gamma function. Finally, we evaluate certain integrals including $\int_{\tiny{Re} s=c} {{ζ(s)} \over s} ds$ and $\int_{\tiny{Re} s=c} {{η(s)} \over s} ds$, with $ζ$ the Riemann zeta function and $η$ its alternating form.

math-ph↗
arXiv2012

Certain logarithmic integrals, including solution of Monthly problem #tbd, zeta values, and expressions for the Stieltjes constants

We solve problem x proposed by O. Oloa, AMM xxx 2012 {\bf 119?} (to appear), p. yyy for certain definite logarithmic integrals. A number of generating functions are developed with certain coefficients $p_n$, and some extensions are presented. The explicit relation of $p_n$ to Nörlund numbers $B_n^{(n)}$ is discussed. Certain inequalities are conjectured for the $\{p_n\}$ sequence of coefficients, including its convexity, and an upper bound is demonstrated. It is shown that $p_n$ values may be used to express the Stieltjes constants for the Hurwitz and Riemann zeta functions, as well as values of these zeta functions at integer argument. Other summations with the $p_n$ coefficients are presented.

math-ph↗
arXiv2011

Fractional part integral representation for derivatives of a function related to ln Gamma(x+1)

For $0\neq x>-1$ let $$Δ(x)={{\ln Γ(x+1)} \over x}.$$ Recently Adell and Alzer proved the complete monotonicity of $Δ'$ on $(-1,\infty)$ by giving an integral representation of $(-1)^n Δ^{(n+1)}(x)$ in terms of the Hurwitz zeta function $ζ(s,a)$. We reprove this integral representation in different ways, and then re-express it in terms of fractional part integrals. Special cases then have explicit evaluations. Other relations for $Δ^{(n+1)}(x)$ are presented, including its leading asymptotic form as $x \to \infty$.

math-ph↗
arXiv2011

Sums of alternating products of Riemann zeta values and solution of a Monthly problem

We solve problem 11585 proposed by B. Burdick, AMM June-July 2011 {\bf 118} (6), p. 558 for the sum of certain products of Riemann zeta function values. We further point out an alternating sum analog, and then present and prove different alternating sum analogs. In addition, we present summation by parts and other results for the Hurwitz and Riemann zeta functions and for the digamma and trigamma functions.

math-ph↗
arXiv2011

Series representations of the Riemann and Hurwitz zeta functions and series and integral representations of the first Stieltjes constant

We develop series representations for the Hurwitz and Riemann zeta functions in terms of generalized Bernoulli numbers (Nörlund polynomials), that give the analytic continuation of these functions to the entire complex plane. Special cases yield series representations of a wide variety of special functions and numbers, including log Gamma, the digamma, and polygamma functions. A further byproduct is that $ζ(n)$ values emerge as nonlinear Euler sums in terms of generalized harmonic numbers. We additionally obtain series and integral representations of the first Stieltjes constant $γ_1(a)$. The presentation unifies some earlier results.

math-ph↗
arXiv2011

Hypergeometric summation representations of the Stieltjes constants

The Stieltjes constants $γ_k$ appear in the regular part of the Laurent expansion of the Riemman and Hurwitz zeta functions. We demonstrate that these coefficients may be written as certain summations over mathematical constants and specialized hypergeometric functions $_pF_{p+1}$. This family of results generalizes a representation of the Euler constant in terms of a summation over values of the trigonometric integrals Si or Ci. The series representations are suitable for acceleration. As byproducts, we evaluate certain sine-logarithm integrals and present the leading asymptotic form of the particular $_pF_{p+1}$ functions.

math-ph↗
arXiv2011

Double series expression for the Stieltjes constants

We present expressions in terms of a double infinite series for the Stieltjes constants $γ_k(a)$. These constants appear in the regular part of the Laurent expansion for the Hurwitz zeta function. We show that the case $γ_k(1)=γ$ corresponds to a series representation for the Riemann zeta function given much earlier by Brun. As a byproduct, we obtain a parameterized double series representation of the Hurwitz zeta function.

math-ph↗
arXiv2011

Evaluation of some second moment and other integrals for the Riemann, Hurwitz, and Lerch zeta functions

Several second moment and other integral evaluations for the Riemann zeta function $ζ(s)$, Hurwitz zeta function $ζ(s,a)$, and Lerch zeta function $Φ(z,s,a)$ are presented. Additional corollaries that are obtained include previously known special cases for the Riemann zeta function $ζ(s)=ζ(s,1)=Φ(1,s,1)$. An example special case is: $$\int_R {{|ζ(1/2+it)|^2} \over {t^2+1/4}}dt=2π[\ln(2π)-γ],$$ with $γ$ the Euler constant. The asymptotic forms of certain fractional part integrals, with and without logarithmic factors in the integrand, are presented. Extensions and other approaches are mentioned.

math-ph↗
arXiv2010

Integral and series representations of the digamma and polygamma functions

We obtain a variety of series and integral representations of the digamma function $ψ(a)$. These in turn provide representations of the evaluations $ψ(p/q)$ at rational argument and for the polygamma function $ψ^{(j)}$. The approach is through a limit definition of the zeroth Stieltjes constant $γ_0(a)=-ψ(a)$. Several other results are obtained, including product representations for $\exp[γ_0(a)]$ and for the Gamma function $Γ(a)$. In addition, we present series representations in terms of trigonometric integrals Ci and Si for $ψ(a)$ and the Euler constant $γ=-ψ(1)$.

math-ph↗
arXiv2010

Parameterized summation relations for the Stieltjes constants

The Stieltjes constants $γ_k(a)$ appear in the regular part of the Laurent expansion of the Hurwitz zeta function about its only polar singularity at $s=1$. We present multi-parameter summation relations for these constants that result from identities for the Hurwitz zeta function. We also present multi-parameter summation relations for functions $A_k(x)$ that may be expressed as sums over the Stieltjes constants. Integral representations, especially including Mellin transforms, play an important role. As a byproduct, reciprocity and other summatory relations for polygamma functions and Bernoulli polynomials may be obtained.

math-ph↗
arXiv2010

Integral representations of functions and Addison-type series for mathematical constants

We generalize techniques of Addison to a vastly larger context. We obtain integral representations in terms of the first periodic Bernoulli polynomial for a number of important special functions including the Lerch zeta, polylogarithm, Dirichlet $L$- and Clausen functions. These results then enable a variety of Addison-type series representations of functions. Moreover, we obtain integral and Addison-type series for a variety of mathematical constants.

math-ph↗
arXiv2010

Some definite logarithmic integrals from Euler sums, and other integration results

We present explicit expressions for multi-fold logarithmic integrals that are equivalent to sums over polygamma functions at integer argument. Such relations find application in perturbative quantum field theory, quantum chemistry, analytic number theory, and elsewhere. The analysis includes the use of properties of a variety of special functions.

math-ph↗
arXiv2009

Addison-type series representation for the Stieltjes constants

The Stieltjes constants $γ_k(a)$ appear in the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $ζ(s,a)$ about its only pole at $s=1$. We generalize a technique of Addison for the Euler constant $γ=γ_0(1)$ to show its application to finding series representations for these constants. Other generalizations of representations of $γ$ are given.

math-ph↗
arXiv2009

On certain sums over the nontrivial zeta zeros

We study coefficients $b_n$ that are expressible as sums over the Li/Keiper constants $λ_j$. We present a number of relations for and representations of $b_n$. These include the expression of $b_n$ as a sum over nontrivial zeros of the Riemann zeta function, as well as integral representations. Conditional on the Riemann hypothesis, we provide the asymptotic form of $b_n \sim 2^{-n-2}\ln n$.

math-ph↗
arXiv2009

Greenberger-Horne-Zeilinger state protocols for fully connected qubit networks

We generalize the recently proposed Greenberger-Horne-Zeilinger (GHZ) tripartite protocol [A. Galiautdinov, J. M. Martinis, Phys. Rev. A 78, 010305(R) (2008)] to fully connected networks of weakly coupled qubits interacting by way of anisotropic Heisenberg exchange g(XX+YY)+g1*ZZ. Our model adopted here differs from the more familiar Ising-Heisenberg chain in that here every qubit interacts with every other qubit in the circuit. The assumption of identical couplings on all qubit pairs allows an elegant proof of the protocol for arbitrary N. In order to further make contact with experiment, we study fidelity degradation due to coupling imperfections by numerically simulating the N=3 and N=4 cases. Our simulations indicate that the best fidelity at unequal couplings is achieved when (a) the system is initially prepared in the uniform superposition state (similarly to how it is done in the ideal case), and (b) the entangling time and the final rotations on each of the qubits are appropriately adjusted.

quant-ph↗
arXiv2009

Series representations for the Stieltjes constants

The Stieltjes constants γ_k(a) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(s,a) about s=1. We present series representations of these constants of interest to theoretical and computational analytic number theory. A particular result gives an addition formula for the Stieltjes constants. As a byproduct, expressions for derivatives of all orders of the Stieltjes coefficients are given. Many other results are obtained, including instances of an exponentially fast converging series representation for γ_k=γ_k(1). Some extensions are briefly described, as well as the relevance to expansions of Dirichlet L functions.

math-ph↗