Trigonometric Series, Polynomial: Definitions

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Trigonometric Series & Polynomial

A trigonometric series is a series of cosines and sines of multiple angles. This powerful way to represent and study functions has the form (Kechris, 2020):


Where:

• a_n ∈ ℂ (the complex plane),
• x ∈ ℝ (the reals).

The nth partial sum of the trigonometric series is called the trigonometric polynomial.

The trigonometric series turns into the Fourier series if a_n and b_n (the coefficients of the trigonometric series) have the form:


Where f is an integrable function.

One application of trigonometric sequences and series is to approximate π. For example, an accelerated version of Vieta’s formula


can approximate π to 300,000 decimal places (Kreminski, 2008).

Theory of Trigonometric Series

The theory of trigonometric series spawned several other important theoretical branches including construction of the Riemann integral and Lebesgue integrals plus several generalizations of the theory, including (Hazelwinkel, 1993):

• Abstract harmonic analysis,
• Almost-periodic functions,
• Fourier integral,
• General orthogonal series.

In addition, the theory of trigonometric series was a starting point for the development of set theory.

Trigonometric Function Series

Although the term “trigonometric series” usually refers to the formula at the top of this article, it may also refer to the expansion of trigonometric functions into their power series. For example: (Nave, 2020).

• sine(x) = x – (x³/3!) + (x⁵/5!) – (x⁷/7!)…
• cos(x) = 1 – (x²/2!) + (x⁴/4!) – (x⁶/6!)…
• cos(x) = x + (x³/3!) + (2x⁵/15!) _ (17x⁷/315!)…

Where ! is a factorial.

These expansions can be used to approximate tiny values of x. For example, when the angle x is in radians, small values for sin(x) = x, small cos(x) = 1 and small tan(x) = x.

References

Hazelwinkel, M. (1993). Encyclopaedia of Mathematics: Stochastic Approximation — Zygmund Class of Functions. Springer.
Kechris, A. (2020). Set Theory and Uniqueness for Trigonometric Series. Retrieved November 15, 2020 from: http://www.math.caltech.edu/~kechris/papers/uniqueness.pdf
Kreminski, R. (2008). π to Thousands of Digits from Vieta’s Formula, Mathematics
Magazine 81, No. 3, June, 2008, 201-207.
Nave, R. (2020). Trig Function Series. Retrieved November 15, 2020.
Zygmund, A. (2002). Trig. Series. Cambridge University Press.

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