Stdcyl Neumann In C++

In this article, we will discuss the std::cyl_neumann function in C++ with its pseudocode and example.

What is the Neumann Function?

Like the more well-known Bessel functions, the cylindrical Neumann function, symbol Y(x), is one of the solutions to Bessel's differential equation. It is specifically connected to issues with cylindrical waves and can be found in several physical phenomena, such as heat conduction, electromagnetic waves, and fluid movement in geometries with cylindrical shapes.

The cylindrical Neumann function shares many characteristics with other Bessel functions, such as orthogonality qualities, recurrence relations, and asymptotic behavior for large arguments.

The cylindrical Neumann function frequently appears in practical applications while attempting to solve partial differential equations (PDEs) with cylindrical symmetry. For instance, it explains how magnetic fields behave around cylindrical conductors or in cylindrically symmetric waveguides in electromagnetics.

The cylindrical Neumann function isn't included in the C++ Standard Library, but it can be implemented quite easily using pre-existing numerical libraries or by creating original methods based on its mathematical characteristics. Bessel functions can be computed using libraries, such as Boost or through custom implementations. After that, these results can be used to compute the cylindrical Neumann function as required.

Pseudocode:

Define the range of orders and arguments
order_start = 1
order_end = 2
arg_start = 0.0
arg_end = 5.0
arg_step = 1
Loop over orders from order_start to order_end
    Loop over arguments from arg_start to arg_end with step arg_step
	Calculating Bessel function of first kind(J_n(x))
	Calculating Bessel function of the second kind (Y_n(x))
        Print J_n(x) and Y_n(x) for the current order and argument
End loops

Example:

Let us take an example to illustrate the std::cyl_neumann method in C++.

#include<iostream>
#include<cmath>
int main() {
    // Define the range of orders and arguments
    int order_start = 0;
    int order_end = 1;
    double arg_start = 0.0;
    double arg_end = 2.0;
    double arg_step = 1;
    // Loop over orders
    for (int n = order_start; n <= order_end; ++n) {
        // Loop over arguments
for(double x= arg_start; x<=arg_end;x+=arg_step)
	{
//calculate the Bessel function of the first kind using std::cyl_bessel_j
            double Jn = std::cyl_bessel_j(n, x);
            // Calculate the Bessel function of the second kind using std::cyl_neumann
            double Yn = std::cyl_neumann(n, x);
	std::cout<<"J_"<<n<<"("<<x<<")="<<Jn<<std::endl;
	std::cout<<"Y"<<n<<"("<<x<<")="<<Yn<<std::endl;
        }
    }
    return 0;
}

Output: