# Documentation

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# `besselI`

Modified Bessel functions of the first kind

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## Syntax

```besselI(`v`, `z`)
```

## Description

`besselI(v, z)` represents the modified Bessel functions of the first kind:

.

The Bessel functions are defined for complex arguments v and z.

A floating-point value is returned if either of the arguments is a floating-point number and the other argument is numerical. For most exact arguments the Bessel functions return an unevaluated function call. Special values at index v = 0 and/or argument z = 0 are implemented. Explicit symbolic expressions are returned, when the index v is a half integer. See Example 2.

For nonnegative integer indices v some of the Bessel functions have a branch cut along the negative real axis. A jump occurs when crossing this cut. See Example 3.

## Environment Interactions

When called with floating-point arguments, these functions are sensitive to the environment variable `DIGITS` which determines the numerical working precision.

## Examples

### Example 1

Unevaluated calls are returned for exact or symbolic arguments:

`besselI(2, 1 + I), besselI(0, x), besselI(v, x)`

Floating point values are returned for floating-point arguments:

`besselI(2, 5.0), besselI(3.2 + I, 10000.0)`

### Example 2

Bessel functions can be expressed in terms of elementary functions if the index is an odd integer multiple of :

`besselI(1/2, x), besselI(3/2, x)`

`besselI(7/2, x), besselI(-7/2, x)`

### Example 3

The negative real axis is a branch cut of the Bessel functions for non-integer indices v. A jump occurs when crossing this cut:

```besselI(-3/4, -1.2), besselI(-3/4, -1.2 + I/10^10), besselI(-3/4, -1.2 - I/10^10)```

### Example 4

The functions `diff`, `float`, `limit`, and `series` handle expressions involving the Bessel functions:

`diff(besselI(0, x), x, x), float(ln(3 + besselI(17, sqrt(PI))))`

`limit(1/besselI(2, x^2 + 1)*sqrt(x), x = infinity)`

`series(besselI(3, x)/x, x = infinity, 3)`

## Parameters

 `v`, `z` arithmetical expressions

## Return Values

Arithmetical expression.

`z`

## Algorithms

The modified Bessel functions Iv(z) and Kv(z) satisfy the modified Bessel equation:

.

When the index v is an integer, the modified Bessel functions of the first kind are governed by reflection formulas:

.