# Sinc and the Basel problem

We have stressed a lot the sinc functionsinc(x) = sin(x)/xThe fact that the definition of sinc(x) can be extended to x=0, where the value is 1 was called the

**fundamental theorem of trigonmetry**(FTT). The reason for this name [2] is that it leads to derivative formulas sin'(x) = cos(x) and cos'(x) = - sin(x). It is fundamental to the calculus aspects of trigonometry. Pedagogically, the sinc function is important [3] because it is a prototype for so many parts of single variable calculus. The sinc function is important in many other parts of mathematics too. Quite recently, it appears in fancy areas like random matrix theory. Euler for example showed the following amazing product formula

sinc(pi x) = (1-xwhich follows from the fact that sin(pi x) has the integers as roots and the fact that sin(pi 0) = 1 thanks to the FTT. In the next line, I cite [1], where Havil refers to Euler, who used sinc(x) = 1-x^{2}/1^{2}) (1-x^{2}/2^{2}) (1-x^{2}/3^{2}) ...

^{2}/3! + x

^{4}/5! -... and where I compare coefficients of sinc(pi x) instead of sin(x): Havlin:

*This astonishing piece of ingenuity is now part of the theory of infinite product and through that theory is made rigorous. Now he [Euler] equated the coefficients on both sides*

-pi^{2}/3! = -1-1/2^{2}-1/3^{2}- ...

Credit: Story of Maths |
Euler so solved the Basel problem and got a formula for the sum of the reciprocals of the squares. This number
is called zeta(2), where zeta is the famous Zeta function
zeta(x) = sumcertainly one of the most important functions in mathematics. |

**References:**

- [1] Julian Havil, Gamma, Exploring Euler's Constant. Princeton univeristy press, 2003, page 39
- [2] Otto Bretscher, Lecture notes calculus, Summer 2009
- [3] Peter Garfield, personal communication
- [4] Movie credit: NOVA: The Story of Maths