BesselI
✖
BesselI
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
open allclose allBasic Examples (5)Summary of the most common use cases
https://wolfram.com/xid/0cq0os16-bmxhvn
Plot over a subset of the reals:
https://wolfram.com/xid/0cq0os16-qt9p
Plot over a subset of the complexes:
https://wolfram.com/xid/0cq0os16-kiedlx
Series expansion at the origin:
https://wolfram.com/xid/0cq0os16-n0jahq
Series expansion at Infinity:
https://wolfram.com/xid/0cq0os16-laddhh
Scope (50)Survey of the scope of standard use cases
Numerical Evaluation (6)
https://wolfram.com/xid/0cq0os16-l274ju
https://wolfram.com/xid/0cq0os16-d147gk
The precision of the output tracks the precision of the input:
https://wolfram.com/xid/0cq0os16-fakb0c
Evaluate for complex arguments and parameters:
https://wolfram.com/xid/0cq0os16-evb6r
Evaluate BesselI efficiently at high precision:
https://wolfram.com/xid/0cq0os16-di5gcr
https://wolfram.com/xid/0cq0os16-bq2c6r
Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:
https://wolfram.com/xid/0cq0os16-oncdfq
https://wolfram.com/xid/0cq0os16-lmyeh7
Or compute average-case statistical intervals using Around:
https://wolfram.com/xid/0cq0os16-cw18bq
Compute the elementwise values of an array:
https://wolfram.com/xid/0cq0os16-thgd2
Or compute the matrix BesselI function using MatrixFunction:
https://wolfram.com/xid/0cq0os16-o5jpo
Specific Values (4)
Value of BesselI for integer () and half-integer () orders at :
https://wolfram.com/xid/0cq0os16-nww7l
For half-integer orders, BesselI evaluates to elementary functions:
https://wolfram.com/xid/0cq0os16-dshvgy
https://wolfram.com/xid/0cq0os16-bdij6w
https://wolfram.com/xid/0cq0os16-drqkdo
Find the positive value of satisfying equation :
https://wolfram.com/xid/0cq0os16-kd878f
https://wolfram.com/xid/0cq0os16-qsfg2q
Visualization (4)
Plot the BesselI function for integer (, ) and half-integer () orders:
https://wolfram.com/xid/0cq0os16-ecj8m7
Plot the real and imaginary parts of the BesselI function for half integer orders:
https://wolfram.com/xid/0cq0os16-v4w3jm
https://wolfram.com/xid/0cq0os16-i11ev
https://wolfram.com/xid/0cq0os16-epyliw
https://wolfram.com/xid/0cq0os16-bbra1u
https://wolfram.com/xid/0cq0os16-bddcjk
Function Properties (12)
is defined for all real and complex values:
https://wolfram.com/xid/0cq0os16-cl7ele
https://wolfram.com/xid/0cq0os16-de3irc
is defined and real for all real values greater than 0:
https://wolfram.com/xid/0cq0os16-rrrwu
Complex domain is the whole plane except :
https://wolfram.com/xid/0cq0os16-bthhjz
achieves all real values greater than 1:
https://wolfram.com/xid/0cq0os16-evf2yr
achieves all real positive values:
https://wolfram.com/xid/0cq0os16-fphbrc
For integer , is an even or odd function in depending on whether is even or odd:
https://wolfram.com/xid/0cq0os16-0plvl3
https://wolfram.com/xid/0cq0os16-o0x1x6
https://wolfram.com/xid/0cq0os16-jg288r
is an analytic function of for integer :
https://wolfram.com/xid/0cq0os16-gva6yl
It is not analytic for noninteger orders:
https://wolfram.com/xid/0cq0os16-g6a0aw
BesselI is non-decreasing for odd values of n:
https://wolfram.com/xid/0cq0os16-rmb7f
is not injective for even values of :
https://wolfram.com/xid/0cq0os16-fxi9f9
It is injective for other values of :
https://wolfram.com/xid/0cq0os16-1txqlb
https://wolfram.com/xid/0cq0os16-zf7zy
is surjective for odd values of :
https://wolfram.com/xid/0cq0os16-hfzo01
It is not surjective for other values of :
https://wolfram.com/xid/0cq0os16-04cjac
https://wolfram.com/xid/0cq0os16-q6hnks
is non-negative for even values of n:
https://wolfram.com/xid/0cq0os16-fxktl8
is singular for , possibly including , when is noninteger:
https://wolfram.com/xid/0cq0os16-fyfbxx
The same is true of its discontinuities:
https://wolfram.com/xid/0cq0os16-ww8ef2
BesselI is convex for even values of n:
https://wolfram.com/xid/0cq0os16-duxck
TraditionalForm formatting:
https://wolfram.com/xid/0cq0os16-h1a36h
Differentiation (3)
https://wolfram.com/xid/0cq0os16-mmas49
https://wolfram.com/xid/0cq0os16-nfbe0l
Plot higher derivatives for integer and half-integer orders:
https://wolfram.com/xid/0cq0os16-fxwmfc
https://wolfram.com/xid/0cq0os16-yr7817
https://wolfram.com/xid/0cq0os16-odmgl1
Integration (4)
Indefinite integral of BesselI:
https://wolfram.com/xid/0cq0os16-bponid
Integrate expressions involving BesselI:
https://wolfram.com/xid/0cq0os16-bc8x91
Definite integral of an odd integrand over an interval centered at the origin is 0:
https://wolfram.com/xid/0cq0os16-b9jw7l
Definite integral of an even integrand over an interval centered at the origin:
https://wolfram.com/xid/0cq0os16-ea53fk
This is twice the integral over half the interval:
https://wolfram.com/xid/0cq0os16-ewk9yc
Series Expansions (6)
https://wolfram.com/xid/0cq0os16-ewr1h8
Plot the first three approximations for around :
https://wolfram.com/xid/0cq0os16-binhar
General term in the series expansion of BesselI:
https://wolfram.com/xid/0cq0os16-eap6ac
https://wolfram.com/xid/0cq0os16-6qmn
Plot the first three approximations for around :
https://wolfram.com/xid/0cq0os16-c36g5d
Asymptotic approximation of BesselI:
https://wolfram.com/xid/0cq0os16-fz3d0l
Taylor expansion at a generic point:
https://wolfram.com/xid/0cq0os16-jwxla7
BesselI can be applied to a power series:
https://wolfram.com/xid/0cq0os16-x4c8x
Integral Transforms (3)
Compute the Laplace transform using LaplaceTransform:
https://wolfram.com/xid/0cq0os16-3dq6om
https://wolfram.com/xid/0cq0os16-iu6kuy
https://wolfram.com/xid/0cq0os16-gig890
Function Identities and Simplifications (3)
Use FullSimplify to simplify expressions with BesselI:
https://wolfram.com/xid/0cq0os16-bd1dz5
https://wolfram.com/xid/0cq0os16-cpyrfv
https://wolfram.com/xid/0cq0os16-d8ooz4
Function Representations (5)
Representation through BesselJ:
https://wolfram.com/xid/0cq0os16-jij1d
Series representation of BesselI:
https://wolfram.com/xid/0cq0os16-83m66
https://wolfram.com/xid/0cq0os16-uo5jqf
BesselI can be represented in terms of MeijerG:
https://wolfram.com/xid/0cq0os16-blxks
https://wolfram.com/xid/0cq0os16-kdqz0r
BesselI can be represented as a DifferenceRoot:
https://wolfram.com/xid/0cq0os16-bh2w29
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:
https://wolfram.com/xid/0cq0os16-hhlp8e
Inductance per unit length of the infinite solenoid:
https://wolfram.com/xid/0cq0os16-q56t8x
3D relativistic, non-Markovian transition PDF that has the Gaussian non-relativistic limit:
https://wolfram.com/xid/0cq0os16-kwtoe9
Its normalization is computed after a change of variables contains BesselI:
https://wolfram.com/xid/0cq0os16-eqtsvs
Properties & Relations (4)Properties of the function, and connections to other functions
Use FullSimplify to simplify expressions with BesselI:
https://wolfram.com/xid/0cq0os16-n75jr3
Find limits of expressions involving BesselI:
https://wolfram.com/xid/0cq0os16-fvpcx
Series representation of BesselI:
https://wolfram.com/xid/0cq0os16-ow9gxk
The exponential generating function for BesselI:
https://wolfram.com/xid/0cq0os16-gaiyeu
Possible Issues (1)Common pitfalls and unexpected behavior
With numeric arguments, half-integer Bessel functions are not automatically evaluated:
https://wolfram.com/xid/0cq0os16-saxmw
For symbolic arguments they are:
https://wolfram.com/xid/0cq0os16-dlr12e
This can lead to major inaccuracies in machine-precision evaluation:
https://wolfram.com/xid/0cq0os16-i33017
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
]}
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
]}