# BesselI BesselI[n,z]

gives the modified Bessel function of the first kind .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• satisfies the differential equation .
• BesselI[n,z] has a branch cut discontinuity in the complex z plane running from to .
• FullSimplify and FunctionExpand include transformation rules for BesselI.
• For certain special arguments, BesselI automatically evaluates to exact values.
• BesselI can be evaluated to arbitrary numerical precision.
• BesselI automatically threads over lists.
• BesselI can be used with Interval and CenteredInterval objects. »

# Examples

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## Basic Examples(5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

## Scope(50)

### Numerical Evaluation(6)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments and parameters:

Evaluate BesselI efficiently at high precision:

BesselI threads elementwise over lists and matrices:

BesselI can be used with Interval and CenteredInterval objects:

### Specific Values(4)

Value of BesselI for integer ( ) and half-integer ( ) orders at :

For half-integer orders, BesselI evaluates to elementary functions:

Limiting values at infinity:

Find the positive value of satisfying equation :

Visualize the result:

### Visualization(4)

Plot the BesselI function for integer ( , ) and half-integer ( ) orders:

Plot the real and imaginary parts of the BesselI function for half integer orders:

Plot the real part of :

Plot the imaginary part of :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(12) is defined for all real and complex values: is defined and real for all real values greater than 0:

Complex domain is the whole plane except : achieves all real values greater than 1: achieves all real positive values:

For integer , is an even or odd function in depending on whether is even or odd:

This can be expressed as : is an analytic function of for integer :

It is not analytic for noninteger orders:

BesselI is non-decreasing for odd values of n: is not injective for even values of :

It is injective for other values of : is surjective for odd values of :

It is not surjective for other values of : is non-negative for even values of n: is singular for , possibly including , when is noninteger:

The same is true of its discontinuities:

BesselI is convex for even values of n:

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for integer and half-integer orders:

Formula for the  derivative:

### Integration(4)

Indefinite integral of BesselI:

Integrate expressions involving BesselI:

Definite integral of an odd integrand over an interval centered at the origin is 0:

Definite integral of an even integrand over an interval centered at the origin:

This is twice the integral over half the interval:

### Series Expansions(6)

Taylor expansion for around :

Plot the first three approximations for around :

General term in the series expansion of BesselI:

Series expansion for around :

Plot the first three approximations for around :

Asymptotic approximation of BesselI:

Taylor expansion at a generic point:

BesselI can be applied to a power series:

### Integral Transforms(3)

Compute the Laplace transform using LaplaceTransform:

### Function Identities and Simplifications(3)

Use FullSimplify to simplify expressions with BesselI:

Recurrence relations :

Verify the identity :

### Function Representations(5)

Representation through BesselJ:

Series representation of BesselI:

Integral representation:

BesselI can be represented in terms of MeijerG:

BesselI can be represented as a DifferenceRoot:

## Applications(2)

Inductance of a solenoid of radius r and length a with fixed numbers of turns per unit length:

Inductance per unit length of the infinite solenoid:

3D relativistic, non-Markovian transition PDF that has the Gaussian non-relativistic limit:

Its normalization is computed after a change of variables contains BesselI:

## Properties & Relations(4)

Use FullSimplify to simplify expressions with BesselI:

Find limits of expressions involving BesselI:

Series representation of BesselI:

The exponential generating function for BesselI:

## Possible Issues(1)

With numeric arguments, half-integer Bessel functions are not automatically evaluated:

For symbolic arguments they are:

This can lead to major inaccuracies in machine-precision evaluation:

## Neat Examples(1)

Continued fraction with arithmetic progression terms: