A Relation Between Maclaurin Coefficients and Laplace Transform

Year 2022, Volume: 5 Issue: 2, 1 - 6, 24.10.2022

Ufuk Kaya

Abstract

In this paper, we formulate Maclaurin coefficients of a function, not necessarily analytic at point $0$, by using Laplace transform as follows:
$$
f^{\left(n\right)}\left(0\right)=\frac{1}{\left(n+1\right)!}\lim_{r\to+0}\frac{d^{n+1}}{dr^{n+1}}L\left\{f\right\}\left(\frac{1}{r}\right),
$$
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

References

• Abramowitz M., Stegun I.: Handbook of Mathematical Functions. Dover Publications, New York (1965)
• Ahlfors V.L.: Complex Analysis. McGraw-Hill Inc., New York (1979)
• Debnath L., Bhatta D.: Integral Transforms and Their Applications. CRC press, Boca Raton (2014)
• Widder V.D.: The Laplace Transform. Princeton University Press, Princeton (1941)