A Relation Between Maclaurin Coefficients and Laplace Transform

Authors : Ufuk KAYA

Pages : 1-6

View : 15 | Download : 6

Publication Date : 2022-10-24

Article Type : Research Paper

Abstract :In this paper, we formulate Maclaurin coefficients of a function, not necessarily analytic at point $0$, by using Laplace transform as follows: $$ f^{\leftinsert ignore into journalissuearticles values(n\right);}\leftinsert ignore into journalissuearticles values(0\right);=\frac{1}{\leftinsert ignore into journalissuearticles values(n+1\right);!}\lim_{r\to+0}\frac{d^{n+1}}{dr^{n+1}}L\left\{f\right\}\leftinsert ignore into journalissuearticles values(\frac{1}{r}\right);, $$ where $L$ is the Laplace transform, $r=\frac{1}{s}$, $s$ is the variable of the Laplace transform and $n\in\mathbb{N}\cup\left\{0\right\}$. Also, we apply this formula on some functions. Finally, we give new formulas for Bernoulli numbers via Polygamma function and Hurwitz zeta function.
Keywords : Maclaurin coefficients, Laplace transform, Bernoulli numbers, Polygamma function, Hurwitz zeta function