Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Trigonometric Fourier series01:17

Trigonometric Fourier series

173
Fourier series is a foundational mathematical technique that decomposes periodic functions into an infinite series of sinusoidal harmonics. This method enables the representation of complex periodic signals as sums of simple sine and cosine functions, facilitating their analysis and interpretation in various fields, including signal processing, acoustics, and electrical engineering.
The trigonometric Fourier series specifically expresses a periodic function with a defined period T using sine...
173
Exponential Fourier series01:24

Exponential Fourier series

168
In audio signal processing, the exponential Fourier series plays a crucial role in sound synthesis, allowing complex sounds to be broken down into simpler sinusoidal components. This decomposition process is fundamental in analyzing and reconstructing musical notes and other audio signals. The exponential Fourier series expresses periodic signals as the sum of complex exponentials at both positive and negative harmonic frequencies, providing a powerful tool for signal analysis.
Euler's identity...
168
Parseval's Theorem01:18

Parseval's Theorem

405
Parseval's theorem is a fundamental concept in signal processing and harmonic analysis. It asserts that for a periodic function, the average power of the signal over one period equals the sum of the squared magnitudes of all its complex Fourier coefficients. This theorem, named after Marc-Antoine Parseval, provides a powerful tool for analyzing the energy distribution in signals.
Interestingly, Parseval's theorem also holds for the trigonometric form of the Fourier series, which...
405
Determination of Pi Terms01:15

Determination of Pi Terms

216
The Buckingham Pi theorem is a valuable method in dimensional analysis, reducing complex relationships between variables into dimensionless terms. Relevant variables in analyzing the lift force on an airplane wing include lift force, air density, wing area, aircraft velocity, and air viscosity. Expressing each variable in terms of fundamental dimensions — mass, length, and time — provides a consistent foundation for constructing these dimensionless terms.
The theorem indicates that...
216
Euler's Formula for Pin-Ended Columns01:21

Euler's Formula for Pin-Ended Columns

282
In structural engineering, the stability of columns under compressive axial loads is a critical consideration, described as buckling. A typical example involves a column PQ, which is pin-connected at both ends and subjected to a centric axial load F applied at one end, with a reaction force of F' = -F at the other end. Here, it is crucial to understand that when an applied load exceeds the critical load, buckling occurs as the system becomes unstable.
To calculate the critical load,...
282
Convergence of Fourier Series01:21

Convergence of Fourier Series

123
The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
123

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

A note on a generalized double series.

PloS one·2026
Same author

Chebyshev series: Derivation and evaluation.

PloS one·2023
Same author

Extended Wang sum and associated products.

PloS one·2022
Same journal

Thymidylate synthase inhibitory drugs induce p53-dependent pathways differently.

PloS one·2026
Same journal

Top-down and bottom-up attention for joint pattern classification and reconstruction.

PloS one·2026
Same journal

Short- and long-term scaling behavior of blood pressure and pulse arrival time during sleep in healthy controls and patients with obstructive sleep apnea.

PloS one·2026
Same journal

Double DQN-based secrecy energy efficiency and fairness performance in IRS-assisted NOMA systems with friendly jamming.

PloS one·2026
Same journal

10 recommendations for strengthening citizen science for improved societal and ecological outcomes: A co-produced analysis of challenges and opportunities in the 21st century.

PloS one·2026
Same journal

Paying in public: Peer effects, impression management, and willingness to pay on digital payment platforms.

PloS one·2026
See all related articles

Related Experiment Video

Updated: May 23, 2025

A Method of Trigonometric Modelling of Seasonal Variation Demonstrated with Multiple Sclerosis Relapse Data
10:46

A Method of Trigonometric Modelling of Seasonal Variation Demonstrated with Multiple Sclerosis Relapse Data

Published on: December 9, 2015

10.6K

Extended Levett trigonometric series.

Robert Reynolds1

  • 1Department of Mathematics and Statistics, York University, Toronto, ON, Canada, M3J1P3.

Plos One
|May 20, 2025
PubMed
Summary
This summary is machine-generated.

This study derives new closed-form formulas for trigonometric series using the Hurwitz-Lerch zeta function. These formulas connect geometric series, special functions, and fundamental constants for advanced mathematical applications.

More Related Videos

A Millimeter Scale Flexural Testing System for Measuring the Mechanical Properties of Marine Sponge Spicules
11:25

A Millimeter Scale Flexural Testing System for Measuring the Mechanical Properties of Marine Sponge Spicules

Published on: October 11, 2017

8.5K
1,3,5-Triphenylbenzene and Corannulene as Electron Receptors for Lithium Solvated Electron Solutions
06:56

1,3,5-Triphenylbenzene and Corannulene as Electron Receptors for Lithium Solvated Electron Solutions

Published on: October 10, 2016

7.7K

Related Experiment Videos

Last Updated: May 23, 2025

A Method of Trigonometric Modelling of Seasonal Variation Demonstrated with Multiple Sclerosis Relapse Data
10:46

A Method of Trigonometric Modelling of Seasonal Variation Demonstrated with Multiple Sclerosis Relapse Data

Published on: December 9, 2015

10.6K
A Millimeter Scale Flexural Testing System for Measuring the Mechanical Properties of Marine Sponge Spicules
11:25

A Millimeter Scale Flexural Testing System for Measuring the Mechanical Properties of Marine Sponge Spicules

Published on: October 11, 2017

8.5K
1,3,5-Triphenylbenzene and Corannulene as Electron Receptors for Lithium Solvated Electron Solutions
06:56

1,3,5-Triphenylbenzene and Corannulene as Electron Receptors for Lithium Solvated Electron Solutions

Published on: October 10, 2016

7.7K

Area of Science:

  • Mathematical Analysis
  • Number Theory
  • Special Functions

Background:

  • Finite trigonometric series are fundamental in various mathematical and physical domains.
  • The Hurwitz-Lerch zeta function is a significant special function with broad applications.
  • Existing literature lacks closed-form expressions for specific trigonometric series extensions.

Purpose of the Study:

  • To extend two finite trigonometric series.
  • To derive novel closed-form formulae involving the Hurwitz-Lerch zeta function.
  • To explore connections between geometric series, special functions, and fundamental constants.

Main Methods:

  • Analysis of trigonometric series with angles based on geometric series (powers of 3).
  • Application of techniques to derive closed-form expressions.
  • Utilizing the properties of the Hurwitz-Lerch zeta function.

Main Results:

  • Development of closed-form formulae for the extended trigonometric series.
  • Derivation of composite finite and infinite series.
  • Inclusion of special functions, trigonometric functions, and fundamental constants in the results.

Conclusions:

  • The study successfully provides new analytical tools for trigonometric series.
  • The derived formulae offer a unified approach to various mathematical series.
  • The findings contribute to the understanding and application of the Hurwitz-Lerch zeta function.