The Connection between the Basel Problem and a Special Integral

Haifeng Xu, Jiuru Zhou

School of Mathematical Sciences, Yangzhou University, Jiangsu, China.

DOI: 10.4236/am.2014.516246                

Abstract

By using Fubini theorem or Tonelli theorem, we find that the zeta function value at 2 is equal to a special integral. Furthermore, we find that this special integral is two times of another special integral. By using this fact we give an easy way to calculate the value of the alternating sum of without using the Fourier expansion. Also, we discuss the relationship between Genocchi numbers and Bernoulli numbers and get some results about Bernoulli polynomials.

Keywords

Basel Problem, Zeta Function, Bernoulli Numbers, Bernoulli Polynomials

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	http://en.wikipedia.org/wiki/Basel_problem
[2]	Lesko, J.P. and Smith, W.D. (2003) A Laplace Transform Technique for Evaluating Infinite Series. Mathematics Magazine, 76, 394-398.
[3]	Chapman, R. (2003) Evaluating . http://secamlocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf
[4]	Mei, J.Q. (2011) Calculus. Higher Education Press, Beijing.
[5]	Grafakos, L. (2004) Classical Fourier Analysis. Graduate Texts in Mathematics (Book 249). 2nd Edition, Springer, New York.
[6]	Burkard, E. (2010) Math 209C Homework 1. http://math.ucr.edu/~edwardb/Graduate%20Classes/Math%20209C/209C%20HW1.pdf
[7]	http://en.wikipedia.org/wiki/Fubini's_theorem
[8]	Pólya, G. and Szeg?, G. (2004) Problems and Theorems in Analysis I. Springer, Berlin.
[9]	Lang, S. (2003) Ellipic Functions. Springer-Verlag, Berlin.
[10]	Pólya, G. and Szeg?, G. (2004) Problems and Theorems in Analysis II. Springer, Berlin.
[11]	Stein, E.M. and Shakarchi, R. (2003) Fourier Analysis. Princeton University Press, Princeton.
[12]	Stein, E.M. and Shakarchi, R. (2003) Complex Analysis. Princeton University Press, Princeton.
[13]	http://en.wikipedia.org/wiki/Genocchi_number