A “well-known” series

The Endeavour 2024-05-21

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

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

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.

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