//========================================================================
//========================================================================
//
// NAME: double bessel_2(int n, double arg)
//
// DESC: Calculates the Bessel function of the second kind (Yn).
//
// INPUT:
// int n:: Order of Bessel function
// double arg: Bessel function argument
//
// OUTPUT:
// Yn:: Bessel function of the second kind of order n
//
// NOTES: 1) The Bessel function of the second kind is defined as:
// Yn(x) = (2.0*Jn(x)/pi)*(ln(x/2) + gamma_e)
// + sum{m=0, m=inf}[(((-1)^(m-1))*(hs(m) +
// hs(m+n)) + x^(2*m))/(((2.0^(2*m+n))*m!*(m+n)!)
// - sum{m=0, m=(n-1)}[((n-m-1)!*x^(2*m))/((2.0^(2*m-n))*m!)
//
// 2) Y-n = ((-1)^n)*Yn
//
//========================================================================
//========================================================================
double bessel_2(int n, double arg)
{
// get a tolerance for the "infinite" sum
double tol = 100.0*depsilon();
int m, mmax;
double Jn, Yn, sum1, sum1_prev, sum2;
sum1 = sum1_prev = sum2 = 0.0;
// Make sure that we calculate the positive order Bessel function
int k = abs(n);
// GET THE BESSEL FUNCTION OF THE FIRST KIND
Jn = bessel_1(k, arg);
// !!! my concern with this do loop is that if a certain term contributes 0 then the
// loop may inappropriately exit
mmax = static_cast<int>(2.0*arg) + 1;
m = 0;
// GET TERM SUM 1
do
{
sum1_prev = sum1;
sum1 += pow((0.0-1.0), (m-1))*(h_s(m) + h_s(m+k))*pow(arg, (2*m))/(pow(2.0, (2*m+k))*dfactorial(m)*dfactorial(m+k));
m++;
} while((fabs(sum1 - sum1_prev) > tol) || (m < mmax));
sum1 = pow(arg, k)*sum1/(1.0*PI);
// GET TERM SUM 2
for(m = 0; m <= (k-1); ++m)
{
sum2 = sum2 + dfactorial(k-m-1)*pow(arg, (2*m))/(pow(2.0, (2*m-k))*dfactorial(m));
}
sum2 = pow(arg, (0-k))*sum2/(1.0*PI);
// NOW GET Yn
Yn = (2.0*Jn/(1.0*PI))*(log(arg/2.0) + EULERC) + sum1 - sum2;
// NOW WE WILL MAKE USE OF THE BESSEL FUNCTION RELATION FOR NEGATIVE n:
// Y(n-1) = (-1)^n*Yn
if(n < 0)
{
Yn = pow((0.0-1.0), k)*Yn;
}
return Yn;
}
//========================================================================
//========================================================================
//
// NAME: complex<double> bessel_1_complex(int n, complex<double> arg)
//
// DESC: Calculates the Bessel function of the first kind (Jn) for
// complex arguments.
//
// INPUT:
// int n:: Order of Bessel function
// complex<double> arg: Bessel function argument
//
// OUTPUT:
// Jn:: Bessel function of the first kind of order n
//
// NOTES: 1) The Bessel function of the first kind is defined as:
// Jn(x) = ((x/2.0)^n)*sum{m=0, m=inf}[(((-x^2)/4.0)^m)/(m!*(n+m)!)
//
// 2) J-n = ((-1)^n)*Jn
//
// 3) The infinite series representing the Bessel function
// converges for all arguments given that enough terms are taken:
// k >> |arg|
//
//========================================================================
//========================================================================
complex<double> bessel_1_complex(int n, complex<double> arg)
{
complex<double> Jn(0.0, 0.0);
complex<double> Jn_series(0.0, 0.0);
complex<double> Jn_seriesm1(0.0, 0.0);
double tolerance = 100.0*depsilon();
// Make sure that we calculate the positive order Bessel function
int m = abs(n);
// !!! I don't know if this will be large enough
int k_max_r, k_max_im;
k_max_r = static_cast<int>(real(arg));
k_max_im = static_cast<int>(imag(arg));
/*
int k_max = 10*imax(k_max_r, k_max_im);
for(int k = 0; k <= k_max; k++)
{
Jn_series = Jn_series + pow((0.0 - arg*arg/4.0), k)/(dfactorial(k)*dfactorial(k+m));
}
*/
// !!! my concern with this do loop is that if a certain term contributes 0 then the
// loop may inappropriately exit
int k_max2 = static_cast<int>(2.0*static_cast<double>(imax(k_max_r, k_max_im))) + 1;
int k = 0;
do
{
Jn_seriesm1 = Jn_series;
Jn_series = Jn_series + pow((0.0 - arg*arg/4.0), k)/(dfactorial(k)*dfactorial(k+m));
k = k + 1;
} while ((sqrt(real(Jn_series*conj(Jn_series) - Jn_seriesm1*conj(Jn_seriesm1))) > tolerance) || (k < k_max2));
Jn = pow((arg/2.0), m)*Jn_series;
// NOW WE WILL MAKE USE OF THE BESSEL FUNCTION RELATION FOR NEGATIVE n:
// J(n-1) = (-1)^n*Jn
if(n < 0)
{
Jn = pow((0.0 - 1.0), m)*Jn;
}
return Jn;
}
//========================================================================
//========================================================================
//
// NAME: double bessel_1(int n, double arg)
//
// DESC: Calculates the Bessel function of the first kind (Jn) of order n.
//
// INPUT:
// int n:: Order of Bessel function
// double arg: Bessel function argument
//
// OUTPUT:
// Jn:: Bessel function of the first kind of order n
//
// NOTES: 1) The Bessel function of the first kind is defined as:
// Jn(x) = ((x/2.0)^n)*sum{m=0, m=inf}[(((-x^2)/4.0)^m)/(m!*(n+m)!)
//
// 2) J-n = ((-1)^n)*Jn
//
// 3) The infinite series representing the Bessel function
// converges for all arguments given that enough terms are taken:
// k >> |arg|
//
//========================================================================
//========================================================================
double bessel_1(int n, double arg)
{
double tolerance = 100.0*depsilon();
double Jn, Jn1;
Jn = Jn1 = 0.0;
// make sure we calculate the positive order Bessel function
int m = abs(n);
// !!! my concern with this do loop is that if a certain term contributes 0 then the
// loop may inappropriately exit, and I don't know if this will be large enough
int k_max2 = int(2.0*arg + 1.0);
int k = 0;
do
{
Jn1 = Jn;
Jn += pow(-arg*arg/4.0, k)/(dfactorial(k)*dfactorial(k+m));
k++;
} while ((fabs(Jn - Jn1) > tolerance) || (k < k_max2));
Jn *= pow((arg/2.0), m);
// NOW WE WILL MAKE USE OF THE BESSEL FUNCTION RELATION FOR NEGATIVE n
// J(n-1) = (-1)^n*Jn
if(n < 0)
{
Jn *= pow(-1.0, m);
}
return Jn;
}
double factorial(int n)
{
if(n>=20) return sqrt(TWOPI*(n+1.0))*exp(-(n+1.0))*pow(n+1.0, n)*
(1.0+8.333333333333e-2/(n+1.0)
+3.472222222222e-3/(n+1.0)/(n+1.0)
-2.681327160494e-3/(n+1.0)/(n+1.0)/(n+1.0)
-2.294720936214e-4/(n+1.0)/(n+1.0)/(n+1.0)/(n+1.0)
+7.840392217201e-4/(n+1.0)/(n+1.0)/(n+1.0)/(n+1.0)/(n+1.0)
+6.972813758366e-5/(n+1.0)/(n+1.0)/(n+1.0)/(n+1.0)/(n+1.0)/(n+1.0)
-5.921664373537e-4/(n+1.0)/(n+1.0)/(n+1.0)/(n+1.0)/(n+1.0)/(n+1.0)/(n+1.0)
);
return dfactorial( (double) n);
}
//========================================================================
//========================================================================
//
// NAME: complex<double> bessel_2_complex(int n, complex<double> arg)
//
// DESC: Calculates the Bessel function of the second kind (Yn) for
// complex arguments.
//
// INPUT:
// int n:: Order of Bessel function
// complex<double> arg: Bessel function argument
//
// OUTPUT:
// complex<double> Yn:: Bessel function of the second kind of
// order n
//
// NOTES: 1) The Bessel function of the second kind is defined as:
// Yn(x) = (2.0*Jn(x)/pi)*(ln(x/2) + gamma_e)
// + sum{m=0, m=inf}[(((-1)^(m-1))*(hs(m) +
// hs(m+n)) + x^(2*m))/(((2.0^(2*m+n))*m!*(m+n)!)
// - sum{m=0, m=(n-1)}[((n-m-1)!*x^(2*m))/((2.0^(2*m-n))*m!)
//
// 2) Y-n = ((-1)^n)*Yn
//
//========================================================================
//========================================================================
complex<double> bessel_2_complex(int n, complex<double> arg)
{
complex<double> Yn(0.0, 0.0);
// get a tolerance for the "infinite" sum
double tolerance = 100.0*depsilon();
// make sure we calculate the positive order Bessel function
int k = abs(n);
// GET BESSEL FUNCTIONS OF THE FIRST KIND
complex<double> Jn = bessel_1_complex(k, arg);
// GET TERM SUM 1
complex<double> sum1(0.0,0.0), sum11(0.0,0.0);
/*
for(int i = 0; i <= 20; i++)
{
sum1 = sum1 + pow((0.0-1.0), (i-1))*(h_s(i) + h_s(i+k))*pow(arg, (2*i))/(pow(2.0, (2*i+k))*dfactorial(i)*dfactorial(i+k));
}
*/
// !!! my concern with this do loop is that if a certain term contributes 0 then the
// loop may inappropriately exit
int mm_max = static_cast<int>(2.0*real(arg)) + 1;
int mm = 0;
do
{
sum11 = sum1;
sum1 = sum1 + pow((0.0-1.0), (mm-1))*(h_s(mm) + h_s(mm+k))*pow(arg, (2*mm))/(pow(2.0, (2*mm+k))*dfactorial(mm)*dfactorial(mm+k));
mm = mm + 1;
} while((fabs(real(sqrt(sum1*conj(sum1) - sum11*conj(sum11)))) > tolerance) || (mm < mm_max));
sum1 = pow(arg, k)*sum1/(1.0*PI);
// GET TERM SUM 2
complex<double> sum2(0.0,0.0);
for(int m = 0; m <= (k-1); m++)
{
sum2 = sum2 + dfactorial((k-m-1))*pow(arg, (2*m))/(pow(2.0, (2*m-k))*dfactorial(m));
}
sum2 = pow(arg, (0-k))*sum2/(1.0*PI);
// NOW GET Yn
Yn = (2.0*Jn/(1.0*PI))*(log(arg/2.0) + EULERC) + sum1 - sum2;
// NOW WE WILL MAKE USE OF THE BESSEL FUNCTION RELATION FOR NEGATIVE n:
// Y(n-1) = (-1)^n*Yn
if(n < 0)
{
Yn = pow((0.0-1.0), k)*Yn;
}
return Yn;
}
//========================================================================
//========================================================================
//
// NAME: double dtwofac(int l)
// DESC: Calculates the double factorial (2l-1)!!=1*3*5*...*(2l-1) for
// argument l, which can be defined as:
//
// (2l-1)!!=(2l)!/((2^n)*l!)
//
// NOTES: i. !!! there is a better way to define this with the gamma
// function
//
//
//========================================================================
//========================================================================
double dtwofac(int n)
{
return ( dfactorial(2*n)/( pow(2.0, n)*dfactorial(n) ) );
}
//========================================================================
//========================================================================
//
// NAME: int ifactorial(int n)
// DESC: Calculates the factorial of n and returns the value as an
// integer
//
// INPUT:
// int n == integer
//
// OUTPUT:
// int factorial == n!
//
// NOTES: 1) This will only works to a certain max n (max int)
//
//
//========================================================================
//========================================================================
int ifactorial(int n)
{
return static_cast<int>(dfactorial(n));
}
//==================================================================================================
// Factorials n! and Double factorial n!!
//==================================================================================================
double dfactorial(double dn)
{ if(dn<=1.0) return 1.0; return dn*dfactorial(dn-1.0); }
