• 대한수학회논문집
• /
• 제17권1호
• /
• pp.165-173
• /
• 2002

We first review Ramaswami's find Apostol's identities involving the Zeta function in a rather detailed manner. We then present corrected, or generalized formulas, or a different method of proof for some of them. We also give closed-form evaluation of some series involving the Riemann Zeta function by an integral representation of ζ(s) and Apostol's identities given here.

• 충청수학회지
• /
• 제24권2호
• /
• pp.247-252
• /
• 2011

We study some properties of the Riemann-Stieltjes integrals with respect to the Riesz-$N\acute{a}gy$-$Tak\acute{a}cs$ distribution $H_{a,p}$ and its inverse $H_{p,a}$ on the unit interval satisfying the equation 1 - a = $a^2$ and p = 1 - a. Using the properties of the dual distributions $H_{a,p}$ and $H_{p,a}$, we compare the Riemann-Stieltjes integrals of $H_{a,p}$ over some essential intervals with that of its inverse $H_{p,a}$ over the related intervals.

• 호남수학학술지
• /
• 제36권2호
• /
• pp.263-273
• /
• 2014

Ever since Euler solved the so-called Basler problem of ${\zeta}(2)=\sum_{n=1}^{\infty}1/n^2$, numerous evaluations of ${\zeta}(2n)$ ($n{\in}\mathbb{N}$) as well as ${\zeta}(2)$ have been presented. Very recently, Ritelli [61] used a double integral to evaluate ${\zeta}(2)$. Modifying mainly Ritelli's double integral, here, we aim at evaluating certain interesting alternating series.

• Journal of applied mathematics & informatics
• /
• 제40권1_2호
• /
• pp.173-184
• /
• 2022

We investigated the invariant subspaces of the fractional integral operator in the Sobolev space W^k_p[0, 1] and prove unicellularity of the operator J^α by using the Duhamel product.

• Kyungpook Mathematical Journal
• /
• 제64권1호
• /
• pp.31-46
• /
• 2024

The aim of this paper is to present an approach to improve reverse Minkowski and Hölder-type inequalities using k-weighted fractional integral operators _a⁺𝔍^𝜇_w with respect to a strictly increasing continuous function 𝜇, by introducing two parameters of integrability, p and q. For various choices of 𝜇 we get interesting special cases.

• 비파괴검사학회지
• /
• 제31권6호
• /
• pp.665-670
• /
• 2011

In this paper, a new approach to detect surface cracks from a noisy thermal image in the infrared thermography is presented using an holomorphic characteristic of temperature field in a thin plate under steady-state thermal condition. The holomorphic function for 2-D heat flow field in the plate was derived from Cauchy Riemann conditions to define a contour integral that varies according to the existence and strength of a singularity in the domain of integration. The contour integral at each point of thermal image eliminated the temperature variation due to heat conduction and suppressed the noise, so that its image emphasized and highlighted the singularity such as crack. This feature of holomorphic function was also investigated numerically using a simple thermal field in the thin plate satisfying the Laplace equation. The simulation results showed that the integral image selected and detected the crack embedded artificially in the plate very well in a noisy environment.

• East Asian mathematical journal
• /
• 제31권1호
• /
• pp.41-53
• /
• 2015

A large number of series and integral representations for the Stieltjes constants (or generalized Euler-Mascheroni constants) ${\gamma}_k$ and the generalized Stieltjes constants ${\gamma}_k(a)$ have been investigated. Here we aim at presenting certain integral representations for the generalized Stieltjes constants ${\gamma}_k(a)$ by choosing to use four known integral representations for the generalized zeta function ${\zeta}(s,a)$. As a by-product, our main results are easily seen to specialize to yield those corresponding integral representations for the Stieltjes constants ${\gamma}_k$. Some relevant connections of certain special cases of our results presented here with those in earlier works are also pointed out.

• 대한수학회논문집
• /
• 제38권3호
• /
• pp.755-772
• /
• 2023

Our aim is to establish certain image formulas of the (p, q)-extended modified Bessel function of the second kind M_ν,p,q(z) by employing the Marichev-Saigo-Maeda fractional calculus (integral and differential) operators including their composition formulas and using certain integral transforms involving (p, q)-extended modified Bessel function of the second kind M_ν,p,q(z). Corresponding assertions for the Saigo's, Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) fractional integral and differential operators are deduced. All the results are represented in terms of the Hadamard product of the (p, q)-extended modified Bessel function of the second kind M_ν,p,q(z) and Fox-Wright function _rΨ_s(z).

• 한국지능시스템학회:학술대회논문집
• /
• 한국퍼지및지능시스템학회 2006년도 춘계학술대회 학술발표 논문집 제16권 제1호
• /
• pp.365-369
• /
• 2006

구간치 퍼지집합은 Gorzalczang(1983)과 Turken(1986)에 의해 처음 제의되었다. 이를 토대로 Wang과 Li는 구간치 퍼지수에 관한 연산으로 일반화 하여 연구하였다. 최근에 홍(2002)는 왕과 리의 이론을 리만적분에 의해 구간치 퍼지집합상의 거리측도에 관한 연구를 하였다. 우리는 일반측도와 관련된 리만적분 대신에 퍼지측도와 관련된 쇼케이적분을 이용한 구간치 퍼지수 상의 쇼케이 거리측도를 연구하였다(2005). 본 논문에서는 퍼지수에서 퍼지수로의 쇼케이 거리측도를 정의하고 이와 관련된 성질들을 조사하였다.

• 한국지능시스템학회논문지
• /
• 제15권7호
• /
• pp.789-793
• /
• 2005

구간치 퍼지집합은 Gorzalczan응(1983)과 Turken(1986)에 의해 처음 제의되었다. 이를 토대로 Wang과 Li는 구간치 퍼지수에 관한 연산으로 일반화하여 연구하였다. 최근에 홍(2002)는 왕과 리의 이론을 기만적분에 의해 구간치 퍼지집합상의 거리측도에 관한 연구를 하였다. 본 논문에서 우리는 일반측도와 관련된 리만적분 대신에 퍼지측도와 관련된 쇼케이적분을 이용한 구간치 퍼지수 상의 쇼케이 거리측도를 정의하고 이와 관련된 성질들을 조사하였다.