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BesselI

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:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix BesselI function using MatrixFunction:

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 $TemplateBox[{n, z}, BesselI]=(-1)^n TemplateBox[{n, {-, z}}, BesselI]$ :

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:

TraditionalForm formatting:

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:

HankelTransform:

InverseMellinTransform:

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:

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