WOLFRAM

BesselI[n,z]

gives the modified Bessel function of the first kind TemplateBox[{n, z}, BesselI].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • TemplateBox[{n, z}, BesselI] 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

open allclose all

Basic Examples  (5)Summary of the most common use cases

Evaluate numerically:

Out[1]=1

Plot over a subset of the reals:

Out[1]=1

Plot over a subset of the complexes:

Out[1]=1

Series expansion at the origin:

Out[1]=1

Series expansion at Infinity:

Out[1]=1

Scope  (50)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

Out[1]=1

Evaluate to high precision:

Out[1]=1

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

Out[2]=2

Evaluate for complex arguments and parameters:

Out[1]=1

Evaluate BesselI efficiently at high precision:

Out[1]=1
Out[2]=2

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

Out[1]=1
Out[2]=2

Or compute average-case statistical intervals using Around:

Out[3]=3

Compute the elementwise values of an array:

Out[1]=1

Or compute the matrix BesselI function using MatrixFunction:

Out[2]=2

Specific Values  (4)

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

Out[1]=1

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

Out[1]=1

Limiting values at infinity:

Out[1]=1
Out[2]=2

Find the positive value of TemplateBox[{0, x}, BesselI] satisfying equation TemplateBox[{0, x}, BesselI]=2:

Out[1]=1

Visualize the result:

Out[2]=2

Visualization  (4)

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

Out[1]=1

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

Out[1]=1

Plot the real part of :

Out[1]=1

Plot the imaginary part of :

Out[2]=2

Plot the real part of :

Out[1]=1

Plot the imaginary part of :

Out[2]=2

Function Properties  (12)

TemplateBox[{0, z}, BesselI] is defined for all real and complex values:

Out[1]=1
Out[2]=2

TemplateBox[{{1, /, 2}, z}, BesselI] is defined and real for all real values greater than 0:

Out[1]=1

Complex domain is the whole plane except :

Out[2]=2

TemplateBox[{0, x}, BesselI] achieves all real values greater than 1:

Out[1]=1

TemplateBox[{{1, /, 2}, x}, BesselI] achieves all real positive values:

Out[2]=2

For integer , TemplateBox[{n, z}, BesselI] is an even or odd function in depending on whether is even or odd:

Out[1]=1
Out[2]=2

This can be expressed as TemplateBox[{n, z}, BesselI]=(-1)^n TemplateBox[{n, {-, z}}, BesselI]:

Out[3]=3

TemplateBox[{n, x}, BesselI] is an analytic function of for integer :

Out[1]=1

It is not analytic for noninteger orders:

Out[2]=2

BesselI is non-decreasing for odd values of n:

Out[1]=1

TemplateBox[{n, z}, BesselI] is not injective for even values of :

Out[1]=1

It is injective for other values of :

Out[2]=2
Out[3]=3

TemplateBox[{n, z}, BesselI] is surjective for odd values of :

Out[1]=1

It is not surjective for other values of :

Out[2]=2
Out[3]=3

TemplateBox[{n, z}, BesselI] is non-negative for even values of n:

Out[1]=1

TemplateBox[{n, z}, BesselI] is singular for , possibly including , when is noninteger:

Out[1]=1

The same is true of its discontinuities:

Out[2]=2

BesselI is convex for even values of n:

Out[1]=1

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Out[1]=1

Higher derivatives:

Out[1]=1

Plot higher derivatives for integer and half-integer orders:

Out[2]=2
Out[3]=3

Formula for the ^(th) derivative:

Out[1]=1

Integration  (4)

Indefinite integral of BesselI:

Out[1]=1

Integrate expressions involving BesselI:

Out[1]=1

Definite integral of an odd integrand TemplateBox[{1, x}, BesselI] over an interval centered at the origin is 0:

Out[1]=1

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

Out[1]=1

This is twice the integral over half the interval:

Out[2]=2

Series Expansions  (6)

Taylor expansion for TemplateBox[{0, x}, BesselI] around :

Out[1]=1

Plot the first three approximations for TemplateBox[{0, x}, BesselI] around :

Out[2]=2

General term in the series expansion of BesselI:

Out[1]=1

Series expansion for TemplateBox[{{1, /, 2}, x}, BesselI] around :

Out[1]=1

Plot the first three approximations for TemplateBox[{{1, /, 2}, x}, BesselI] around :

Out[2]=2

Asymptotic approximation of BesselI:

Out[1]=1

Taylor expansion at a generic point:

Out[1]=1

BesselI can be applied to a power series:

Out[1]=1

Integral Transforms  (3)

Compute the Laplace transform using LaplaceTransform:

Out[1]=1

HankelTransform:

Out[1]=1

InverseMellinTransform:

Out[1]=1

Function Identities and Simplifications  (3)

Use FullSimplify to simplify expressions with BesselI:

Out[1]=1

Recurrence relations z (TemplateBox[{{n, -, 1}, z}, BesselI] - TemplateBox[{{n, +, 1}, z}, BesselI])=2 nTemplateBox[{n, z}, BesselI]:

Out[1]=1

Verify the identity TemplateBox[{{n, +, 1}, z}, BesselI] TemplateBox[{{-, n}, z}, BesselI]-TemplateBox[{n, z}, BesselI] TemplateBox[{{{-, n}, -, 1}, z}, BesselI]=(2 sin(pi n))/(pi z):

Out[1]=1

Function Representations  (5)

Representation through BesselJ:

Out[1]=1

Series representation of BesselI:

Out[1]=1

Integral representation:

Out[1]=1

BesselI can be represented in terms of MeijerG:

Out[1]=1
Out[2]=2

BesselI can be represented as a DifferenceRoot:

Out[1]=1

Applications  (2)Sample problems that can be solved with this function

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

Out[1]=1

Inductance per unit length of the infinite solenoid:

Out[2]=2

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

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

Out[2]=2

Properties & Relations  (4)Properties of the function, and connections to other functions

Use FullSimplify to simplify expressions with BesselI:

Out[1]=1

Find limits of expressions involving BesselI:

Out[1]=1

Series representation of BesselI:

Out[1]=1

The exponential generating function for BesselI:

Out[1]=1

Possible Issues  (1)Common pitfalls and unexpected behavior

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

Out[1]=1

For symbolic arguments they are:

Out[2]=2

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

Out[3]=3

Neat Examples  (1)Surprising or curious use cases

Continued fraction with arithmetic progression terms:

Out[1]=1
Wolfram Research (1988), BesselI, Wolfram Language function, https://reference.wolfram.com/language/ref/BesselI.html (updated 2022).
Wolfram Research (1988), BesselI, Wolfram Language function, https://reference.wolfram.com/language/ref/BesselI.html (updated 2022).

Text

Wolfram Research (1988), BesselI, Wolfram Language function, https://reference.wolfram.com/language/ref/BesselI.html (updated 2022).

Wolfram Research (1988), BesselI, Wolfram Language function, https://reference.wolfram.com/language/ref/BesselI.html (updated 2022).

CMS

Wolfram Language. 1988. "BesselI." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/BesselI.html.

Wolfram Language. 1988. "BesselI." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/BesselI.html.

APA

Wolfram Language. (1988). BesselI. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BesselI.html

Wolfram Language. (1988). BesselI. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BesselI.html

BibTeX

@misc{reference.wolfram_2024_besseli, author="Wolfram Research", title="{BesselI}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/BesselI.html}", note=[Accessed: 08-January-2025 ]}

@misc{reference.wolfram_2024_besseli, author="Wolfram Research", title="{BesselI}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/BesselI.html}", note=[Accessed: 08-January-2025 ]}

BibLaTeX

@online{reference.wolfram_2024_besseli, organization={Wolfram Research}, title={BesselI}, year={2022}, url={https://reference.wolfram.com/language/ref/BesselI.html}, note=[Accessed: 08-January-2025 ]}

@online{reference.wolfram_2024_besseli, organization={Wolfram Research}, title={BesselI}, year={2022}, url={https://reference.wolfram.com/language/ref/BesselI.html}, note=[Accessed: 08-January-2025 ]}