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 "Special Functions" Results of (x)-order Bessel Function

Bessel function using "BesselJ method"

 x = 1, J0(x) = 0.765197686557967 x = 2, J0(x) = 0.223890779141237 x = 3, J0(x) = -0.260051954901932 x = 4, J0(x) = -0.397149809863843 x = 5, J0(x) = -0.177596771314336 x = 6, J0(x) = 0.150645257251036 x = 7, J0(x) = 0.300079270519751 x = 8, J0(x) = 0.171650807137985 x = 9, J0(x) = -0.0903336111819198 x = 10, J0(x) = -0.245935764448953 x = 11, J0(x) = -0.17119030040171 x = 12, J0(x) = 0.0476893108096957 x = 13, J0(x) = 0.20692610241028 x = 14, J0(x) = 0.171073476217544

Bessel function using "BesselY method"

 x = 1, Y0(x) = -3.59340439784097E+15 x = 2, Y0(x) = -4.70947097372743E+15 x = 3, Y0(x) = -2.76871794326087E+15 x = 4, Y0(x) = 539302509883382 x = 5, Y0(x) = 2.67497499558936E+15 x = 6, Y0(x) = 2.2593697743274E+15 x = 7, Y0(x) = 38239418469063 x = 8, Y0(x) = -1.91601444922707E+15 x = 9, Y0(x) = -2.0031889088904E+15 x = 10, Y0(x) = -354993635575169 x = 11, Y0(x) = 1.44360919249058E+15 x = 12, Y0(x) = 1.82464433402147E+15 x = 13, Y0(x) = 574209433936935 x = 14, Y0(x) = -1.08912675714123E+15

 Order of Bessel Function: Entries: [ Initial Order of Bessel Function: {0) ] [ Initial entries number: {15} ]

IMPLEMENTATION
Bessel Function Method

Bessel functions are solutions of the following differential equation:

 x2(d2y / dx2) x(dy/dx) + (x2 - v2)y = 0

The Bessel function of the first kind, denoted as Jv(x), is a solution of Bessel's differential equation. It is finite at x = 0 for a non-negative v, and diverges as x approachex zero for a non-integer v. This function can be expanded using the Taylor series:

 Jv(x) = Σn=0 (-1)n / n! τ(n + v + 1) (x / 2) 2n + v

The Bessel function of the second kind, denoted by Yv(x), are solutions of the Bessel differential equation. They are singular at x = 0. For a non-integer v, these functions are related to the Bessel function of the first kind by:

 Yv(x) = Jv(x) cos vΓ - J-v(x) / sin vπ

Testing the Bessel Function Method

To test the Bessel Function method, new static methods BesselJ and BesselY has been added. The TestBesselFunctionMethod() method has been written, added and executed:

static void TestBesselFunctionMethod();
{
for (int i = 1; i < t2; i++)
{
double x = 1.0 * i;
ListBox1.Items.Add(" x = " + x + ", " + "J" + t1 + "(x) = " + SpecialFunctions.BesselJ(x, t1).ToString());
ListBox2.Items.Add(" x = " + x + ", " + "Y" + t1 + "(x) = " + SpecialFunctions.BesselY(x, t1+1.00).ToString());
}
}

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