A function which is one of the solutions to the modified
Bessel differential equation and is closely related to the Bessel
function of the first kind . The above plot shows for , 2, ..., 5. The modified Bessel function of the first kind
is implemented in the Wolfram Language
as BesselI[nu,
z].
The modified Bessel function of the first kind can be defined by the contour
integral
|
(1)
|
where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).
In terms of ,
|
(2)
|
For a real number , the function can be computed using
|
(3)
|
where is the gamma function.
An integral formula is
|
(4)
|
which simplifies for an integer to
|
(5)
|
(Abramowitz and Stegun 1972, p. 376).
A derivative identity for expressing higher order modified Bessel functions in terms of is
|
(6)
|
where is a Chebyshev
polynomial of the first kind.
The special case of gives as the series
|
(7)
|
See also
Bessel Function of the First Kind,
Continued Fraction
Constants,
Modified Bessel
Function of the Second Kind,
Weber's Formula
Related Wolfram sites
http://functions.wolfram.com/Bessel-TypeFunctions/BesselI/
Explore with Wolfram|Alpha
References
Abramowitz, M. and Stegun, I. A. (Eds.). "Modified Bessel Functions and ." §9.6 in Handbook
of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 374-377, 1972.Arfken, G. "Modified Bessel
Functions, and ." §11.5 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 610-616,
1985.Finch, S. R. Mathematical
Constants. Cambridge, England: Cambridge University Press, 2003.Press,
W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T.
"Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions."
§6.7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 234-245, 1992.Spanier, J. and Oldham,
K. B. "The Hyperbolic Bessel Functions and " and "The General Hyperbolic Bessel Function
."
Chs. 49-50 in An
Atlas of Functions. Washington, DC: Hemisphere, pp. 479-487 and 489-497,
1987.Referenced on Wolfram|Alpha
Modified Bessel
Function of the First Kind
Cite this as:
Weisstein, Eric W. "Modified Bessel Function of the First Kind." From MathWorld--A Wolfram Web Resource.
https://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html
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