Zeta function and some of its properties
Abstract
Introduction/purpose: Some properties of the zeta function will be shown as well as its applications in calculus, in particular the “golden nugget formula” for the value of the infinite sum 1 + 2 + 3 + ... Some applications in physics will also be mentioned.
Methods: Complex plane integrations and properties of the Gamma function will be used from the definition of the function to its analytic extension.
Results: From the original definition of the (s) function valid for s > 1 a meromorphic function is obtained on the whole complex plane with a simple pole in s = 1.
Conclusion: The relevance of the zeta function cannot be overstated, ranging from the infinite series to the number theory, regularization in theoretical physics, the Casimir force, and many other fields.
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