Formula for pi

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A formula for π containing an improper integral of the sinc function reads

This allows you to calculate the value π = 3,1415… by numerical integration.

We are going to calculate

calculate with an integration in the complex plane. We can write

and

Substituting into Euler's formula then gives

For the pole z = 0, we have

With the pole outside the enclosure. Let us separate the pole z = 0 by surrounding it with a semicircle (γ) of radius r. We can write

The second integral gives −iπ as z → 0

and the fourth integral, following the same reasoning, gives 0 as R → ∞

Substituting this gives for R → ∞

and so applies

such that

The Swiss mathematician Leonhard Euler (1707 - 1783) calculated the value of π with this formula.