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.

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  (42)

Numerical Evaluation  (5)

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:

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 TemplateBox[{0, x}, BesselI] satisfying equation TemplateBox[{0, x}, BesselI]=2:

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  (5)

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

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

Complex domain is the whole plane except :

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

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

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

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

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for integer and half-integer orders:

Formula for the ^(th) derivative:

Integration  (4)

Indefinite integral of BesselI:

Integrate expressions involving BesselI:

Definite integral of an odd integrand TemplateBox[{1, x}, BesselI] 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 TemplateBox[{0, x}, BesselI] around :

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

General term in the series expansion of BesselI:

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

Plot the first three approximations for TemplateBox[{{1, /, 2}, x}, BesselI] 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 z (TemplateBox[{{n, -, 1}, z}, BesselI] - TemplateBox[{{n, +, 1}, z}, BesselI])=2 nTemplateBox[{n, z}, BesselI]:

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):

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  (1)

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:

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:

Introduced in 1988
 (1.0)
 |
Updated in 1999
 (4.0)
2000
 (4.1)
2002
 (4.2)