Solutions by infinite series and Bessel functions

2015 ◽  
pp. 199-223
Author(s):  
Shair Ahmad ◽  
Antonio Ambrosetti
Keyword(s):  
Infinite Series ◽  
Bessel Functions

scholarly journals A Recursive Formula for the Evaluation of Earth Return Impedance on Buried Cables

2015 ◽  
Vol 35 (3) ◽  
pp. 34-43
Author(s):  
Reynaldo Iracheta
Keyword(s):  
Infinite Series ◽  
Bessel Functions ◽  
Integration Method ◽  
Recursive Formula ◽  
Definite Integral ◽  
Mathematical Software ◽  
Font Size ◽  
Buried Cables ◽  
The Time Domain ◽  
Numerical Laplace Transform

<p class="Abstractandkeywordscontent"><span style="font-size: small;"><span style="font-family: Century Gothic;">This paper presents an alternative solution based on infinite series for the accurate and efficient evaluation of cable earth return impedances. This method uses Wedepohl and Wilcox’s transformation to decompose Pollaczek’s integral in a set of Bessel functions and a definite integral. The main feature of Bessel functions is that they are easy to compute in modern mathematical software tools such as Matlab. The main contributions of this paper are the approximation of the definite integral by an infinite series, since it does not have analytical solution; and its numerical solution by means of a recursive formula. The accuracy and efficiency of this recursive formula is compared against the numerical integration method for a broad range of frequencies and cable  configurations. Finally, the proposed method is used as a subroutine for cable parameter calculation in the inverse Numerical Laplace Transform (NLT) to obtain accurate transient responses in the time domain.</span></span></p>

Summation of Series over Bourget Functions

2008 ◽  
Vol 51 (4) ◽  
pp. 627-636
Author(s):  
Mirjana V. Vidanović ◽  
Slobodan B. Tričković ◽  
Miomir S. Stanković
Keyword(s):  
Closed Form ◽  
Infinite Series ◽  
Bessel Functions ◽  
Integer Order ◽  
Summation Of Series ◽  
Finite Sums ◽  
Riemann Ζ Function

AbstractIn this paper we derive formulas for summation of series involving J. Bourget's generalization of Bessel functions of integer order, as well as the analogous generalizations by H. M. Srivastava. These series are expressed in terms of the Riemann ζ function and Dirichlet functions η, λ, β, and can be brought into closed form in certain cases, which means that the infinite series are represented by finite sums.

scholarly journals Arithmetic Means for a Class of Functions and the Modified Bessel Functions of the First Kind

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 60 ◽  
Author(s):  
Feng Qi ◽  
Shao-Wen Yao ◽  
Bai-Ni Guo
Keyword(s):  
Infinite Series ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Residue Theorem ◽  
Arithmetic Means ◽  
Complex Functions ◽  
Class Of Functions

In the paper, by virtue of the residue theorem in the theory of complex functions, the authors establish several identities between arithmetic means for a class of functions and the modified Bessel functions of the first kind, present several identities between arithmetic means for a class of functions and infinite series, and find several series expressions for the modified Bessel functions of the first kind.

scholarly journals A note on the summation of an infinite series involving a hypergeometric function

1974 ◽  
Vol 10 (2) ◽  
pp. 305-309 ◽  
Author(s):  
A.G. Williamson
Keyword(s):  
Hypergeometric Function ◽  
Infinite Series ◽  
Bessel Functions

The sum of an infinite series involving the 4F3 function is found by considering an infinite series whose terms involve the product of two Bessel functions of the first kind. Furthermore the infinite series involving the 4F3 function can be utilized to find an approximation to the 4F3 function of unit argument, for particular values of the parameters.

scholarly journals On some infinite series involving the zeros of Bessel functions of the first kind

1960 ◽  
Vol 4 (3) ◽  
pp. 144-156 ◽  
Author(s):  
Ian N. Sneddon
Keyword(s):  
Positive Integer ◽  
Bessel Function ◽  
Rational Function ◽  
Recurrence Relation ◽  
Infinite Series ◽  
Bessel Functions ◽  
Zeros Of Bessel Functions ◽  
Image Position

In this paper we shall be concerned with the derivation of simple expressions for the sums of some infinite series involving the zeros of Bessel functions of the first kind. For instance, if we denote by γv, n (n = l, 2, 3,…) the positive zeros of Jv(z), then, in certain physical applications, we are interested in finding the values of the sumsandwhere m is a positive integer. In § 4 of this paper we shall derive a simple recurrence relation for S2m,v which enables the value of any sum to be calculated as a rational function of the order vof the Bessel function. Similar results are given in § 5 for the sum T2m,v.

scholarly journals Topological Derivative for Imaging of Thin Electromagnetic Inhomogeneity: Least Condition of Incident Directions

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Won-Kwang Park
Keyword(s):  
Numerical Simulations ◽  
Theoretical Investigation ◽  
Infinite Series ◽  
Bessel Functions ◽  
Noisy Data ◽  
Mathematical Structure ◽  
Topological Derivative ◽  
Small Thickness ◽  
Integer Order ◽  
Data Support

It is well-known that using topological derivative is an effective noniterative technique for imaging of crack-like electromagnetic inhomogeneity with small thickness when small number of incident directions are applied. However, there is no theoretical investigation about the configuration of the range of incident directions. In this paper, we carefully explore the mathematical structure of topological derivative imaging functional by establishing a relationship with an infinite series of Bessel functions of integer order of the first kind. Based on this, we identify the condition of the range of incident directions and it is highly depending on the shape of unknown defect. Results of numerical simulations with noisy data support our identification.

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