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Maths › Special › Bessel › Y ›

Y

Bessel function of second kind of integer order.

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Y
Y
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Interface

C++

Excel

Y

doubleY(	double	`x`
	int	`v`	)

A Bessel function of the second kind

is sometimes also denoted using

and is a solution to the differential equation

$x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + (x^2 - v^2)y = 0$

(1)

with

and $I_{-v}(x)$ being the other solutions. See Maths/Special/Bessel/I/I

This solution has a regular singularity at 0 and an irregular singularity at $\infty$ .

1/bessely-969.gif — Bessel function of the 2nd kind

The function is evaluated by forward recurrence on n, starting with values computed by the routines y0() and y1().

If n = 0 or 1 the routine for Maths/Special/Bessel/Y/Y0 or Maths/Special/Bessel/Y/Y1 is called directly.

Accuracy:

Absolute error, except relative when y > 1: <pre> domain # trials peak rms 0, 30 2200 2.9e-16 5.3e-17 0, 30 30000 3.4e-15 4.3e-16 </pre>

Error Messages:

<pre> message condition value returned singularity x = 0 MAXNUM overflow MAXNUM </pre>

References:

Cephes Math Library Release 2.8: June, 2000

Example 1

#include <stdio.h>
#include <codecogs/maths/special/bessel/y/y.h>
 
int main()
{
  printf("\nY\n  x      v=0      v=1      v=2      v=3      v=4      v=5");
  for(double x=1; x<7; x++)
  {
    printf("\nx=%.1lf",x);
    for(int v=0;v<=5;v++)
      printf(" %8.6lf", Maths::Special::Bessel::Y::Y(x,v));
  }
  return 0;
}

Output:

Y
x      v=0      v=1      v=2      v=3      v=4      v=5
x=1.0 0.088257 -0.781213 -1.650683 -5.821518 -33.278423 -260.405867
x=2.0 0.510376 -0.107032 -0.617408 -1.127784 -2.765943 -9.935989
x=3.0 0.376850 0.324674 -0.160400 -0.538542 -0.916683 -1.905946
x=4.0 -0.016941 0.397926 0.215904 -0.182022 -0.488937 -0.795851
x=5.0 -0.308518 0.147863 0.367663 0.146267 -0.192142 -0.453695
x=6.0 -0.288195 -0.175010 0.229858 0.328249 0.098391 -0.197061

Parameters

x	value to be transformed
v	the order

Authors

Stephen L. Moshier. Copyright 1984, 1987, 2000,
Documentation by Nick Owens

Source Code

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Y

doubleY(	double	`x`
	double	`v`	)

Returns the Bessel function of the second kind with real order v.

If the order v is an interger then the integer form of Maths/Special/Bessel/Y/Y is used, otherwise the solution is constructed from a combination of Maths/Special/Bessel/J/J and

Maths/Special/Bessel/K/K:

$Y_v(x) = \frac{ \cos( \pi v ) J_v(x) - K_{-v}(x) }{\sin(\pi v)}$

(2)

Warning

This function has not been tested to be correct or accurate. It maks use of the Bessel K (bessel/k), which currently does not accept a real order. When Bessel K is updated, it is expected this function will then work appropriately. Use with care.