• Korean Journal of Mathematics
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• v.5 no.1
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• pp.17-27
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• 1997

In this paper we define a torsion function ${\log}\;T(M,{\varphi})(t)$) for $t{\gg}0$ and show that it has an asymptotic expansion $\frac{1}{2}{\chi}(M)t$ as $t{\rightarrow}{\infty}$.

• Communications of the Korean Mathematical Society
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• v.27 no.4
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• pp.721-732
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• 2012

The principal aim of the paper is to investigate new integral expression $${\int}_0^{\infty}x^{s+1}e^{-{\sigma}x^2}L_m^{(\gamma,\delta)}\;({\zeta};{\sigma}x^2)\;L_n^{(\alpha,\beta)}\;({\xi};{\sigma}x^2)\;J_s\;(xy)\;dx$$, where $y$ is a positive real number; $\sigma$, $\zeta$ and $\xi$ are complex numbers with positive real parts; $s$, $\alpha$, $\beta$, $\gamma$ and $\delta$ are complex numbers whose real parts are greater than -1; $J_n(x)$ is Bessel function and $L_n^{(\alpha,\beta)}$ (${\gamma};x$) is generalized Laguerre polynomials. Some integral formulas have been obtained. The Maple implementation has also been examined.

• Korean Journal of Mathematics
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• v.30 no.2
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• pp.199-203
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• 2022

Let f(z) be an entire function of exponential type τ such that ║f║ = 1. Also suppose, in addition, that f(z) ≠ 0 for ℑz > 0 and that $h_f(\frac{\pi}{2})=0$. Then, it was proved by Gardner and Govil [Proc. Amer. Math. Soc., 123(1995), 2757-2761] that for y = ℑz ≤ 0 $${\parallel}D_{\zeta}[f]{\parallel}{\leq}\frac{\tau}{2}({\mid}{\zeta}{\mid}+1)$$, where D_ζ[f] is referred to as polar derivative of entire function f(z) with respect to ζ. In this paper, we prove an inequality in the opposite direction and thereby obtain some known inequalities concerning polynomials and entire functions of exponential type.

• Bulletin of the Korean Mathematical Society
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• v.61 no.3
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• pp.671-697
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• 2024

In this paper, we formally introduce the notion of a general parametric digamma function Ψ(−s; A, a) and we find the Laurent expansion of Ψ(−s; A, a) at the integers and poles. Considering the contour integrations involving Ψ(−s; A, a), we present some new identities for infinite series involving Dirichlet type parametric harmonic numbers by using the method of residue computation. Then applying these formulas obtained, we establish some explicit relations of parametric linear Euler sums and some special functions (e.g. trigonometric functions, digamma functions, Hurwitz zeta functions etc.). Moreover, some illustrative special cases as well as immediate consequences of the main results are also considered.

• East Asian mathematical journal
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• v.10 no.1
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• pp.173-186
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• 1994

• Journal of the Korea Society of Computer and Information
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• v.16 no.10
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• pp.101-109
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• 2011

The Riemann's zeta function $\zeta(s)$ has been known as answer for a number of primes $\pi$(x) less than given number x. In prime number theorem, there are another approximation function $\frac{x}{lnx}$,Li(x), and R(x). The error about $\pi$(x) is R(x) < Li(x) < $\frac{x}{lnx}$. The logarithmic integral function is Li(x) = $\int_{2}^{x}\frac{1}{lnt}dt$ ~ $\frac{x}{lnx}\sum\limits_{k=0}^{\infty}\frac{k!}{(lnx)^k}=\frac{x}{lnx}(1+\frac{1!}{(lnx)^1}+\frac{2!}{(lnx)^2}+\cdots)$. This paper shows that the $\pi$(x) can be represent with finite Li(x), and presents generalized prime counting function $\sqrt{{\alpha}x}{\pm}{\beta}$. Firstly, the $\pi$(x) can be represent to $Li_3(x)=\frac{x}{lnx}(\sum\limits_{t=0}^{{\alpha}}\frac{k!}{(lnx)^k}{\pm}{\beta})$ and $Li_4(x)=\lfloor\frac{x}{lnx}(1+{\alpha}\frac{k!}{(lnx)^k}{\pm}{\beta})}k\geq2$ such that $0{\leq}t{\leq}2k$. Then, $Li_3$(x) is adjusted by $\pi(x){\simeq}Li_3(x)$ with ${\alpha}$ and error compensation value ${\beta}$. As a results, this paper get the $Li_3(x)=Li_4(x)=\pi(x)$ for $x=10^k$. Then, this paper suggests a generalized function $\pi(x)=\sqrt{{\alpha}x}{\pm}{\beta}$. The $\pi(x)=\sqrt{{\alpha}x}{\pm}{\beta}$ function superior than Riemann's zeta function in representation of prime counting.

• Journal of applied mathematics & informatics
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• v.38 no.3_4
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• pp.291-300
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• 2020

This type of polynomial is a generating function that substitutes e^λt for e^t in the denominator of the generating function for the Bernoulli polynomial, but polynomials by using this generating function has interesting properties involving the location of the roots. We define these generation functions and observe the properties of the generation functions.

• Journal of the Korean Mathematical Society
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• v.37 no.3
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• pp.391-410
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• 2000

In the complex case, we construct a q-analogue of the Riemann zeta function q(s) and a q-analogue of the Dirichlet L-function L(s,X), which interpolate the 1-analogue Bernoulli numbers. Using the properties of p-adic integrals and measures, we show that Kummer type congruences for the q-analogue Bernoulli numbers are the generalizations of the usual Kummer congruences for the ordinary Bernoulli numbers. We also construct a q0analogue of the p-adic L-function Lp(s, X;q) which interpolates the q-analogue Bernoulli numbers at non positive integers.

• Proceedings of the KIEE Conference
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• 2002.11c
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• pp.382-386
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• 2002

The damping ratio $\zeta$ of the unit-step response of a second-order discrete system is a function of only the location of the closed-loop poles and is not directly related to the location of the system zero. However, the peak overshoot of the response is the function of both the damping ratio $\zeta$ and an angle $\alpha$, which is the phasor angle of the damped sinusoidal response and is determined by the relative location of the zero with respect to the closed-loop poles. Accordingly, when the closed-loop system poles are fixed, the peak overshoot is considered as a function of the angle $\alpha$ or the system zero location. In this paper the effects of the relative location of the zero on the system performance of a second-order discrete system is studied, and a design method of digital compensator which achieves arbitrary steady-state response with minimum peak overshoot while maintaining the desired system mode and the damping ratio of the unit step response is presented.

• Honam Mathematical Journal
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• v.35 no.3
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• pp.373-379
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• 2013

In contrast with numerous identities involving the binomial coefficients and the Stirling numbers of the first and second kinds, a few identities involving the Eulerian numbers have been known. The objective of this note is to present certain interesting and (presumably) new identities involving the Eulerian numbers by mainly making use of Worpitzky's identity.