A Note on the Summation of the Incomplete Gamma Function

We examine the improved infinite sum of the incomplete gamma function for large values of the parameters involved. We also evaluate the infinite sum and equivalent Hurwitz-Lerch zeta function at special values and produce a table of results for easy reading. Almost all Hurwitz-Lerch zeta functions have an asymmetrical zero distribution.

An exact integral-to-sum relation for products of Bessel functions

A useful identity relating the infinite sum of two Bessel functions to their infinite integral was discovered in Dominici et al. (Dominici et al. 2012 Proc. R. Soc. A 468 , 2667–2681). Here, we extend this result to products of N Bessel functions, and show it can be straightforwardly proven using the Abel-Plana theorem, or the Poisson summation formula. For N = 2, the proof is much simpler than that of Dominici et al. and significantly enlarges the range of validity.

Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function

We apply our simultaneous contour integral method to an infinite sum in Prudnikov et al. and use it to derive the infinite sum of the Incomplete gamma function in terms of the Hurwitz zeta function. We then evaluate this formula to derive new series in terms of special functions and fundamental constants. All the results in this work are new.

About force of interaction of a point charge with a dielectric ball

The decision of a problem on force of interaction of a point charge with a dielectric ball which is represented in the form of the infinite sum is resulted. It is shown that in a case when dielectric permeability of a ball aspires in infinity, force of interaction aspires to force of interaction of a point charge with the conducting isolated ball which contains only one composed. n the basis of the received results the simple approached formula for calculation of force of interaction of a point charge and the dielectric ball, containing too only one composed is offered. Using the received formula, the problem about force of interaction of a point charge and the same charged dielectric ball is solved. For various values of parameters the neutral curves dividing scopes of forces of pushing away and an attraction are constructed.

Be okay with small steps, as the harmonic series grows without bound

“Be okay with small steps, as the harmonic series grows without bound” considers the question: if each term in the infinite sum represented a day or a fraction of a day you live, would you live forever? That is, does the harmonic series—an infinite sum in which you start with a 1 and then continue adding the reciprocals of increasingly positive integers—grow without bound? The discussion is illustrated with hand-drawn sketches. Mathematics students and enthusiasts are encouraged to appreciate that small steps in mathematical and life pursuits may accumulate, leading to progress over time. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Divergent Series and Its Assigned Value in a Hyperreal Context

This letter discusses the deep connection between the infinite sum of natural numbers and the value -1/12. Aside of more widely known facts, we consider a nontrivial way in which we show the veracity of this connection; more precisely this concerns the BGN method \citep{bgn} applied on the so-called damped oscillated Abel summed variant of the series. Moreover, we have found a generalization of this method which `correctly' assigns finite values to other divergent series. We conclude with some questions concerning whether and how we can analytically relate our hyperreal terms to frame the method in a more justifiable and applicable context.

Analytical solution to an LQG homing problem in two dimensions

An analytical solution is found to the problem of maximising the time spent in the first quadrant by the two-dimensional diffusion process (X(t), Y(t)), where Y(t) is a controlled Brownian motion and X(t) is proportional to its integral. Moreover, we force the process to exit the first quadrant through the y-axis. This type of problem is known as LQG homing and is very difficult to solve explicitly, especially in two or more dimensions. Here the partial differential equation satisfied by a transformation of the value function is solved by making use of the method of separation of variables. The exact solution is expressed as an infinite sum of Airy functions.

Zeta function and some of its properties

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 z(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.