## 4 Practical Examples
### Example II  
The second example is also related to the fluid flow in pipes. [PERSON] analyzed the accelerated flow in the pipe [25] by assuming a constant bulk velocity during the accelerating period. This led to the Voltera convolutional integral of the first kind that needed to be solved to find the pressure gradient time dependence. Using the Whitaker method [26], he was able to solve this problem. The coefficient of the exponential function of the pressure gradient solution was found from the roots of equations of the following type:  
\[\sum_{j=1}^{\infty}\frac{4}{x+\xi_{j}^{2}}=1\ \Rightarrow\ \sum_{j=1}^{\infty} \frac{1}{x+\xi_{j}^{2}}=\frac{1}{4} \tag{50}\]  
An analysis of this summation shows that these roots are only present for \(x\leq 0\). [PERSON] found the first roots to be as follows: \(x_{1}=0;\ x_{2}=-26.374;\ x_{3}=-70.849;\ x_{4}=-135.023;\ x_{5}=-218.922\). [PERSON] did not notice that these roots (except for the minus sign) are the roots of the Bessel equation of the second order \(J_{2}(\eta_{j})=0\).  
All solutions were conducted with the minus sign; then, these solutions could be written to avoid the minus sign problem in the following form with positive values:  
\[\sum_{j=1}^{\infty}\frac{1}{\xi_{j}^{2}-x}=\frac{1}{4} \tag{51}\]  
Inserting the zeros of the Bessel function \(J_{2}\big{(}\eta_{j}\big{)}=0\) for \(x\) made the equivalence of Equation (A10) evident:  
\[\sum_{k}^{\infty}=1\ \frac{1}{\beta_{p,k}^{2}-\beta_{p+2,n}^{2}}=\frac{1}{4(1+p)} \tag{52}\]In the [PERSON] case \(p=0\), the final solution of A10 was exactly the same as the one derived by Vogelphl  
\[\sum_{k}^{\infty}=1\ \frac{1}{\overline{\dot{\beta}}_{0,k}^{2}-\dot{\beta}_{2,n}^{2} }=\frac{1}{4(1+0)}=\frac{1}{4} \tag{53}\]  
The above-presented examples show that such functions (series based on zeros of Bessel functions) can be found and are useful in real physical problems. Other practical examples found in the references are related to the following: (a) the problem of finding the eigenfunctions and eigenvalues of the free Schrodinger equation inside this cylinder [10]; (b) relativistic quantum mechanics (identity for the static polarizability) where transitions between the corresponding quantum states are related to the Thomas-Reiche-Kuhn sum rule [11].  
Presented in this paper, the theory may contribute to the development of many practical issues found in applied science fields, such as (a) the electronics theory of passive n-parts (chapter 6 of [27]); (b) the analysis of time-domain responses of electrical networks and the synthesis of these circuits (amplifiers, etc.) with specified time domain characteristics [28; 29; 30]; (c) in communication circuits with an emitter current, the output voltages and average transconductance ratio of the modified Bessel function play the major role [31]; (d) analytical solutions for the transient and steady-state responses of linear time-invariant networks [32]; (e) finding the mean charge in graphene nanodots [33]; (f) dynamic analogy in the framework of a linear theory of viscoelasticity [34]; (g) electrical analogs of mechanical models [35]; (h) in mathematical fields, approximate expressions for Bessel functions zero, using a similar technique based on nonlinear equations for the zeros [36; 37], the zeros of the equations arising in spectral problems [13; 38], the identities between similar summation formulas [12], and important bounds [39; 40] are found. The Calogero type series (some of which can be found in the upper part of Table A1) have been recently very useful in finding an inverse Laplace transform of a complicated frequency domain function [41].