scholarly journals The Connection between the Basel Problem and a Special Integral

2014 ◽  
Vol 05 (16) ◽  
pp. 2570-2584
Author(s):  
Haifeng Xu ◽  
Jiuru Zhou
Keyword(s):  
Special Integral ◽  
Basel Problem

91.19 The Basel problem and the zeros of

2007 ◽  
Vol 91 (520) ◽  
pp. 120-123 ◽  
Author(s):  
Francis Woodhouse
Keyword(s):  
Basel Problem

Probabilistic Proofs of Euler Identities

2013 ◽  
Vol 50 (04) ◽  
pp. 1206-1212 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Generating Function ◽  
Local Limit Theorem ◽  
Infinite Product ◽  
Random Variables ◽  
Local Limit ◽  
Moment Generating Function ◽  
Basel Problem

Formulae for ζ(2n) andLχ4(2n+ 1) involving Euler and tangent numbers are derived using the hyperbolic secant probability distribution and its moment generating function. In particular, the Basel problem, where ζ(2) = π2/ 6, is considered. Euler's infinite product for the sine is also proved using the distribution of sums of independent hyperbolic secant random variables and a local limit theorem.

scholarly journals SPECIAL SPLINES OF HYPERBOLIC TYPE FOR THE SOLUTIONS OF HEAT AND MASS TRANSFER 3-D PROBLEMS IN POROUS MULTI-LAYERED AXIAL SYMMETRY DOMAIN

2017 ◽  
Vol 22 (4) ◽  
pp. 425-440
Author(s):  
Harijs Kalis ◽  
Andris Buikis ◽  
Aivars Aboltins ◽  
Ilmars Kangro
Keyword(s):  
Differential Equations ◽  
Transfer Problem ◽  
Circular Cross Section ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Rate Of Change ◽  
Wood Block ◽  
Initial Value ◽  
Special Integral ◽  
Initial Boundary Value

In this paper we study the problem of the diffusion of one substance through the pores of a porous multi layered material which may absorb and immobilize some of the diffusing substances with the evolution or absorption of heat. As an example we consider circular cross section wood-block with two layers in the radial direction. We consider the transfer of heat process. We derive the system of two partial differential equations (PDEs) - one expressing the rate of change of concentration of water vapour in the air spaces and the other - the rate of change of temperature in every layer. The approximation of corresponding initial boundary value problem of the system of PDEs is based on the conservative averaging method (CAM) with special integral splines. This procedure allows reduce the 3-D axis-symmetrical transfer problem in multi-layered domain described by a system of PDEs to initial value problem for a system of ordinary differential equations (ODEs) of the first order.

Special integral variation of trees

2006 ◽  
Vol 26 (1) ◽  
pp. 49
Author(s):  
Yi-Zheng Fan ◽  
Yi Wang
Keyword(s):  
Special Integral ◽  
Integral Variation

scholarly journals Basel Problem

2017 ◽  
Vol 25 (2) ◽  
pp. 149-155
Author(s):  
Karol Pąk ◽  
Artur Korniłowicz
Keyword(s):  
Elementary Proof ◽  
Link Type ◽  
Basel Problem

Summary A rigorous elementary proof of the Basel problem [6, 1] $$\sum\nolimits_{n = 1}^\infty {{1 \over {n^2 }} = {{\pi ^2 } \over 6}} $$ is formalized in the Mizar system [3]. This theorem is item #14 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.

scholarly journals Is there a π solution for ∑i^{-i+1}

2021 ◽  
Author(s):  
Alex Nguhi
Keyword(s):  
Basel Problem

The sum∑i^{-i+1} converges very fast as compared with∑i^{-i}, but its value is very close to Euler's solution for the Basel Problem. This begs the question, can there be a solution involving π or other identities ?

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