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Fourier's formula for pi
The mathematician Joseph Fourier developed for π = 3,1415… the formula
Explanation
We start with the sinc function and write the series
The zeroes of this function are integer multiples of π, so the points x = n · π for n = ±1, ±2, ±3, .... The Weierstrass factorization theorem provides for this
what you can convert to
and thus also to
Now we're going to apply Newton's identities. We only work with the coefficients of x2 and calculate
In the original series we see that the coefficient of x2 there is equal to
thus
so that
HistoryThe first calculation was given in 1735 by Leonhard Euler, and has since been known as the Basel problem. The series is the zeta function |