Stirling’s Formula: Derivation and Applications

Wei Ni; Ruohan Ye; Ruiqi Yu

doi:10.54254/2755-2721/2025.PO24877

Applied and Computational EngineeringOpen access

Stirling’s Formula: Derivation and Applications

Research Article

Open Access

Stirling’s Formula: Derivation and Applications

Wei Ni ^1* Ruohan Ye ², Ruiqi Yu ³

¹ Qingdao No. 2 High School of Shandong Province， 266061, China

² Qingdao No.58 High School of Shandong Province， 266199，China

³ Qingdao No. 2 High School of Shandong Province 266061, China

^*Corresponding author: wei592727@gmail.com

Published on 10 July 2025

ACE Vol.174

ISSN (Print): 2755-273X

ISSN (Online): 2755-2721

ISBN (Print): 978-1-80590-235-5

ISBN (Online): 978-1-80590-236-2

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Abstract

In this paper, we take a closer look at Stirling’s formula, a method used to estimate factorials, particularly when nnn is large. Starting with its historical background, we then give a derivation using the Gamma function and examine how the formula behaves asymptotically. While it's a classic result in mathematics, we also discuss where it shows up in real-world problems—like signal processing, rubber-based materials, and even brain imaging. The way it links with Fourier transforms of the Gamma function shows how useful it is for interpreting patterns like power-law decay and frequency changes in signals.

Keywords:

Stirling’s approximation, Gamma function, asymptotic analysis, Fourier transform, signal processing, viscoelasticity, power-law decay, biomedical applications.

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References

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[2]. Metzler, R., Schick, W., Kilian, H. G., & Nonnenmacher, T.F.(1994). Generalized viscoelastic models: Their fractional equations with solutions. The Journal of Chemical Physics, 103(16), 7180-7186.

[3]. Stankovic, J., & Pajic-Lijakovic, I.(2020).Chewing gum rheology: Power-lawrelaxationandtime-temperaturesuperposition.Food Hydrocolloids, 102, 105591.

[4]. J.G.Proakis and M.Salehi, Digital Communications, 5thed., McGraw-Hill, 2007.

[5]. M.I. Skolnik, Radar Handbook, 3rded., McGraw-Hill, 2008.

[6]. A.Metin and C.Glover, "Frequency Modulated fMRI Signal Analysis, "NeuroImage, vol.150, pp.120-135, 2017.

References

[1]. Bagley, R.L., & Torvik, P.J.(1983).Fractional calculus—a different approach to the analysis of viscoelastically damped structures.AIAA Journal, 21(5), 741-748.

[3]. Stankovic, J., & Pajic-Lijakovic, I.(2020).Chewing gum rheology: Power-lawrelaxationandtime-temperaturesuperposition.Food Hydrocolloids, 102, 105591.

[4]. J.G.Proakis and M.Salehi, Digital Communications, 5thed., McGraw-Hill, 2007.

[5]. M.I. Skolnik, Radar Handbook, 3rded., McGraw-Hill, 2008.

[6]. A.Metin and C.Glover, "Frequency Modulated fMRI Signal Analysis, "NeuroImage, vol.150, pp.120-135, 2017.

Cite this article

Ni,W.;Ye,R.;Yu,R. (2025). Stirling’s Formula: Derivation and Applications. Applied and Computational Engineering,174,46-60.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

About volume

Volume title: Proceedings of CONF-CDS 2025 Symposium: Data Visualization Methods for Evaluatio

ISBN: 978-1-80590-235-5(Print) / 978-1-80590-236-2(Online)

Editor: Marwan Omar, Elisavet Andrikopoulou

Conference website: https://2025.confcds.org/portsmouth.html

Conference date: 30 July 2025

Series: Applied and Computational Engineering

Volume number: Vol.174

ISSN: 2755-2721(Print) / 2755-273X(Online)

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