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Modified Bessel functions of the first kind

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besselI(v, z)


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.


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)


v, z

arithmetical expressions

Return Values

Arithmetical expression.

Overloaded By



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:


See Also

MuPAD Functions

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