| "Special Functions" |
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| Results of (x)-order Bessel Function |
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Bessel function using "BesselJ method"
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Bessel function using "BesselY method"
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IMPLEMENTATION
Bessel Function Method
Bessel functions are solutions of the following differential equation:
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x2(d2y / dx2) x(dy/dx) + (x2 - v2)y = 0
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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:
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Jv(x) = Σn=0 (-1)n / n! τ(n + v + 1) (x / 2) 2n + v
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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:
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Yv(x) = Jv(x) cos vΓ - J-v(x) / sin vπ
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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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