相关论文: A product representation of pi

In this Note, we start off with the primary representation of e and from there present an elementary short proof for the Wallis formula for $\pi$.

This note states and proves a representation theorem for regular quantity functions, based on the theory of quantity spaces, thereby giving a new perspective on dimensional analysis and the classical $\pi$ theorem.

This note comprises a negative resolution of the Efficient Market Hypothesis.

This short note contains elementary evaluations of some Euler sums.

In this short note, we give basic enumerative results on colored integer partitions.

A family of original formulae for computing number PI and its proof are presented. An algorithm is proposed to validate the results of this new algorithm.

The regularized product of the Fibonacci numbers is evaluated.

This note describes a way of obtaining e that differs from the standard one. It could be used as an alternate way of showing how the value of e is obtained. No attempt is made to show the existence of the limit in the definition of e that…

In this note we provide a simple formula of general term of recurrent sequence.

A simple application of the semipositivity.

In this note we prove an inequality involving primes and the product of consecutive primes.

In this short note, we have defined a new "nested square root" function which generates usual Pi number for $x=2$. We have given some useful identities and asymptotic formulas of the Pi-function.

We obtain simple proofs of certain inequalites for bivariate means.

This note contains a short proof of the functional equation for the zeta function.

In this article, we give another visual proof of $\pi^e < e^\pi$.

In our previous publication we have shown a method for calculating series of even powers of $\pi$ based on the product representation of the $sinc$ function. We refer the readers to [1] for more details. In this work we apply the method to…

In this article, we derive, using Fourier series and multiple derivative of the function $\pi/\sin(\pi x)$, series representations for positive powers of $\pi$. We also show that the Euler-Wallis product can be easily obtained from the same…

Lecture notes for an introductory course in elementary particles.

In this note, we will give a short proof of an identity for cubic partitions.

This note presents an interesting counterexample to a basic covering problem.