SoatDev IT Consulting
SoatDev IT Consulting
  • About us
  • Expertise
  • Services
  • How it works
  • Contact Us
  • News
  • May 21, 2024
  • Rss Fetcher

I was reading an article [1] that refers to “a well-known trigonometric series” that I’d never seen before. This paper cites [2] which gives the series as

begin{align*} frac{sin mphi}{cos phi} &= msinphi - frac{m(m^2-2^2)}{3!}sin^3phi \ &phantom{=} ;+ frac{m(m^2-2^2)(m^2 - 4^2)}{5!}sin^5phi - cdots end{align*}

Note that the right hand side is not a series in φ but rather in sin φ.

Motivation

Why might you know sin φ and want to calculate sin mφ / cos φ? This doesn’t seem like a sufficiently common task for the series to be well-known. The references are over a century old, and maybe the series were useful in hand calculations in a way that isn’t necessary anymore.

However, [1] was using the series for a theoretical derivation, not for calculation; the author was doing some hand-wavy derivation, sticking the difference operator E into a series as if it were a number, a technique known as “umbral calculus.” The name comes from the Latin word umbra for shadow. The name referred to the “shadowy” nature of the technique which wasn’t make rigorous until much later.

Convergence

The series above terminates if m is an even integer. But there are no restrictions on m, and in general the series is infinite.

The series obviously has trouble if cos φ = 0, i.e. when φ = ±π/2, but it converges for all m if −π/2 < φ < π/2.

Tangent

If m = 1, sin mφ / cos φ is simply tan φ. The function tan φ has a complicated power series in φ involving Bernoulli numbers, but it has a simpler power series in sin φ.

References

[1] G. J. Lidstone. Notes on Everett’s Interpolation Formula. 1922

[2] E. W. Hobson. A Treatise on Plane Trigonometry. Fourth Edition, 1918. Page 276.

The post A “well-known” series first appeared on John D. Cook.

Previous Post
Next Post

Recent Posts

  • French B2B neobank Qonto reaches 600,000 customers, files for banking license
  • OpenAI condemns Robinhood’s ‘OpenAI tokens’
  • Wonder Dynamics co-founder Nikola Todorovic joins the AI Stage at TechCrunch Disrupt 2025
  • Lucid sales inch forward as EV maker pushes to ramp Gravity production
  • Substack brings new updates to livestreaming as it increases video push

Categories

  • Industry News
  • Programming
  • RSS Fetched Articles
  • Uncategorized

Archives

  • July 2025
  • June 2025
  • May 2025
  • April 2025
  • February 2025
  • January 2025
  • December 2024
  • November 2024
  • October 2024
  • September 2024
  • August 2024
  • July 2024
  • June 2024
  • May 2024
  • April 2024
  • March 2024
  • February 2024
  • January 2024
  • December 2023
  • November 2023
  • October 2023
  • September 2023
  • August 2023
  • July 2023
  • June 2023
  • May 2023
  • April 2023

Tap into the power of Microservices, MVC Architecture, Cloud, Containers, UML, and Scrum methodologies to bolster your project planning, execution, and application development processes.

Solutions

  • IT Consultation
  • Agile Transformation
  • Software Development
  • DevOps & CI/CD

Regions Covered

  • Montreal
  • New York
  • Paris
  • Mauritius
  • Abidjan
  • Dakar

Subscribe to Newsletter

Join our monthly newsletter subscribers to get the latest news and insights.

© Copyright 2023. All Rights Reserved by Soatdev IT Consulting Inc.