< 1 >
Integral of the sinc function
With an integral of the sinc function you can calculate the value for the number π = 3.1415… as
Explanation
You can convert the improper integral of the sinus cardinalis to
Evaluating the integrals in the last sum by numerical integration we get
k 0 1.85194 1 .43379 2 .25661 3 .18260 ∆ ∆2 ∆3 ∆4 4 .14180 −2587 5 .11593 799 −1788 −321 6 .09805 478 153 −1310 −168 7 .08495 310 −1000 8 .07495
The sum to k = 3 is
1.85194 − .43379 + .25661 − .18260 = 1.49216
Applying the Euler transform to the remainder we obtain
We obtain the value of the integral as
1.4962 + .07862 = 1.57078
as compared with 1.57080.
HistoryThe Swiss mathematician Leonhard Euler (1707 - 1783) developed this numerical integration. |