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# i

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Modified Bessel function of the first kind of integer order.
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Contents

C++
Excel

## Overview

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

The differential equation
$z^2&space;\frac{d^2y}{dz^2}&space;+&space;z&space;\frac{dy}{dz}&space;-&space;(z^2&space;+&space;v^2)y&space;=&space;0$
where v is a real constant, is called the modified Bessel's equation, with the solution known as the modified Bessel function, with solutions: $\inline&space;&space;I_v(z)$ and $\inline&space;I_{-v}(z)$, and $\inline&space;K_v(z)$ where

$I_{v}(z)&space;=&space;\frac{z}{2}^v&space;\sum_{k=0}^{\infty}&space;\frac{\left&space;(&space;\frac{1}{4}z^2&space;\right&space;)^k}{k!&space;\Gamma(v+k+1)}$

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

A second solution, which is independent of $\inline&space;&space;I_{v}(z)$, is known as the modified Bessel function of the kind Maths/Special/Bessel/K/K

## References:

http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html

## I

 doubleI( double x int v )[inline]
Returns modified Bessel function of the first kinds for any integer order (v)

The function is defined as $\inline&space;&space;I_1(x)&space;=&space;-i&space;H_1(ix)$

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

## Accuracy:

<pre> Relative error: arithmetic domain # trials peak rms DEC 0, 30 3400 1.2e-16 2.3e-17 IEEE 0, 30 30000 1.9e-15 2.1e-16</pre>

## References:

Cephes Math Library Release 2.8: June, 2000

### Parameters

 x value to be transformed. v order of bessel function.

### 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.

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## I

 doubleI( double x double v )
Returns modified Bessel function of the first kinds for any order (v)

The function is defined as $\inline&space;&space;I_1(x)&space;=&space;-i&space;J_1(ix)$

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

## Accuracy:

<pre> Relative error: arithmetic domain # trials peak rms DEC 0, 30 3400 1.2e-16 2.3e-17 IEEE 0, 30 30000 1.9e-15 2.1e-16</pre>

## Example:

#include <codecogs/maths/special/bessel/i/i.h>
#include <stdio.h>

int main()
{
using namespace Maths::Special::Bessel::I;
printf("\n  x      v=0      v=1      v=2      v=3      v=4      v=5");
for(double x=0; x<6; x++)
{
printf("\nx=%.1lf",x);
for(int v=0;v<=5;v++)
printf(" %8.6lf", I(x,v));
}
return 0;
}

## Output:

x      v=0      v=1      v=2      v=3      v=4      v=5
x=0.0 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000
x=1.0 1.266066 0.565159 0.135748 0.022168 0.002737 0.000271
x=2.0 2.279585 1.590637 0.688948 0.212740 0.050729 0.009826
x=3.0 4.880793 3.953370 2.245212 0.959754 0.325705 0.091206
x=4.0 11.301922 9.759465 6.422189 3.337276 1.416276 0.504724
x=5.0 27.239872 24.335642 17.505615 10.331150 5.108235 2.157975

## References:

Cephes Math Library Release 2.8: June, 2000

### Parameters

 x input argument. v order of bessel function.

### 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.