# Documentation

### This is machine translation

Translated by
Mouseover text to see original. Click the button below to return to the English verison of the page.

# besseli

Modified Bessel function of first kind

## Syntax

I = besseli(nu,Z)
I = besseli(nu,Z,1)

## Description

I = besseli(nu,Z) computes the modified Bessel function of the first kind, Iν(z), for each element of the array Z. The order nu need not be an integer, but must be real. The argument Z can be complex. The result is real where Z is positive.

If nu and Z are arrays of the same size, the result is also that size. If either input is a scalar, it is expanded to the other input's size.

I = besseli(nu,Z,1) computes besseli(nu,Z).*exp(-abs(real(Z))).

## Examples

collapse all

Create a column vector of domain values.

z = (0:0.2:1)';

Calculate the function values using besseli with nu = 1.

format long
besseli(1,z)
ans =
0
0.100500834028125
0.204026755733571
0.313704025604922
0.432864802620640
0.565159103992485

Define the domain.

X = 0:0.01:5;

Calculate the first five modified Bessel functions of the first kind.

I = zeros(5,501);
for i = 0:4
I(i+1,:) = besseli(i,X);
end

Plot the results.

plot(X,I,'LineWidth',1.5)
axis([0 5 0 8])
grid on
legend('I_0','I_1','I_2','I_3','I_4','Location','Best')
title('Modified Bessel Functions of the First Kind for v = 0,1,2,3,4')
xlabel('X')
ylabel('I_v(X)')

collapse all

### Bessel’s Equation

The differential equation

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

where ν is a real constant, is called the modified Bessel's equation, and its solutions are known as modified Bessel functions.

Iν(z) and Iν(z) form a fundamental set of solutions of the modified Bessel's equation. Iν(z) is defined by

${I}_{\nu }\left(z\right)={\left(\frac{z}{2}\right)}^{\nu }\sum _{\left(k=0\right)}^{\infty }\frac{{\left(\frac{{z}^{2}}{4}\right)}^{k}}{k!\Gamma \left(\nu +k+1\right)},$

where Γ(a) is the gamma function.

Kν(z) is a second solution, independent of Iν(z). It can be computed using besselk.