# besseli

## Purpose

Modified Bessel functions of the first kind.

## Synopsis

````I = besseli(alpha,x)`
`E = besseli(alpha,x,1)`
```

## Description

`I = besseli(alpha,x)` computes modified Bessel functions of the first kind for real, non-negative order `alpha` and argument `x`. If `alpha` is a scalar and `x` is a vector, `I` is a vector the same length as `x`. If `x` is a vector of length `m` and `alpha` is a vector of length `n`, then `I` is an `m`-by-`n` matrix and `I(i,k)` is `besseli(alpha(k), x(i))`. The elements of `x` can be any nonnegative real values, in any order. For `alpha`, the increment between elements must be 1, and all elements must be between 0 and 1000, inclusive.

`E = besseli(alpha,x,1)` computes `besseli(alpha,x).*exp(-x)`.

The relationship between the modified Bessel function of the first kind `I` and the Bessel function of the first kind `J` is

## Algorithm

`besseli` uses three-term backward recurrence for most `x`, and an asymptotic expansion for large `x`.

````bessel`, `besselj`, `besselk`, `bessely`