COST (GBP)

5.62

0.50

Maths › Special › Bessel › I ›

i0

viewed 1906 times and licensed 51 times

Modified Bessel function of the first kind, with order zero and exponential scaling.

Controller: CodeCogs

Contents

Interface
I0 Exp
I0
Page Comments

Dependents

Interface

C++

Excel

Overview

These function return solutions to the Modified Bessel Function of the first kind with zero order.

The differential equation

$z^2 \frac{d^2y}{dz^2} + z \frac{dy}{dz} - (z^2)y = 0$

(2)

is called the modified Bessel's equation of zero order, with the solution known as the modified Bessel function,

$I_{0}(z) = \sum_{k=0}^{\infty} \frac{\left ( \frac{1}{4}z^2 \right )^k}{(k!)^2}$

(3)

where $\inline \Gamma(n)$ is the gamma function.

I0 Exp

doubleI0_exp( double x )

Returns exponentially scaled modified Bessel function of the first kind, with order zero.

The function is defined as $\inline I_0(x) \exp(x) = \exp(-|x|) J_0(i x)$ .

Accuracy:

<pre> Relative error: arithmetic domain # trials peak rms IEEE 0,30 30000 5.4e-16 1.2e-16 </pre>

References:

Cephes Math Library Release 2.8: June, 2000

Example 1

#include <codecogs/maths/special/bessel/i/i0.h>
#include <stdio.h>
 
int main()
{
  using namespace Maths::Special::Bessel::I;
 
  for(double x=0; x<6; x+=1)
  {
    double y=I0_exp(x);
    printf("\n I0_exp(%.1lf)=%lf", x,y);
  }
  return 0;
}

Output:

I0_exp(0.0)=1.000000
I0_exp(1.0)=0.465760
I0_exp(2.0)=0.308508
I0_exp(3.0)=0.243000
I0_exp(4.0)=0.207002
I0_exp(5.0)=0.183541

Parameters

x	input argument

Authors

Stephen L. Moshier. Copyright 1984, 1987, 2000
Documentation by Will Bateman (August 2005)

Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.

I0

doubleI0( double x )[inline]

Returns modified Bessel function of order zero of the argument (v=0).

The function is defined as $\inline I_0(x) = J_0(ix)$ .

The range is partitioned into the two intervals [0,8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval.