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:

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:

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 :

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:

TraditionalForm formatting:

First derivative:

Higher derivatives:

Plot higher derivatives for integer and half-integer orders:

Formula for the derivative:

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:

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:

Compute the Laplace transform using LaplaceTransform:

HankelTransform:

InverseMellinTransform:

Use FullSimplify to simplify expressions with BesselI:

Recurrence relations :

Verify the identity :

Representation through BesselJ:

Series representation of BesselI:

Integral representation:

BesselI can be represented in terms of MeijerG:

BesselI can be represented as a DifferenceRoot: