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Modified Bessel Function of the First Kind
A function
which is one of the solutions to themodified Bessel differential equation and is closely related to theBessel function of the first kind
. The above plot shows
for
, 2, ..., 5. The modified Bessel function of the first kind is implemented in theWolfram Language asBesselI[nu,z].
The modified Bessel function of the first kind
can be defined by thecontour integral
 | (1) |
where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).
In terms of
,
 | (2) |
For areal number
, the function can be computed using
 | (3) |
where
is thegamma function. An integral formula is
 | (4) |
which simplifies for
aninteger
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 aChebyshev 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 FormulaRelated Wolfram sites
http://functions.wolfram.com/Bessel-TypeFunctions/BesselI/Explore with Wolfram|Alpha
More things to try:
References
Abramowitz, M. and Stegun, I. A. (Eds.). "Modified Bessel Functions
and
." §9.6 inHandbook 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 inMathematical 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 inNumerical 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 inAn Atlas of Functions. Washington, DC: Hemisphere, pp. 479-487 and 489-497, 1987.Referenced on Wolfram|Alpha
Modified Bessel Function of the First KindCite this as:
Weisstein, Eric W. "Modified Bessel Function of the First Kind." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html