Bessel Function of the Second Kind
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"Bessel Functions J and Y." §9.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
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24-3-2019
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Bessel Function of the Second Kind
A Bessel function of the second kind 
(e.g, Gradshteyn and Ryzhik 2000, p. 703, eqn. 6.649.1), sometimes also denoted 
(e.g, Gradshteyn and Ryzhik 2000, p. 657, eqn. 6.518), is a solution to the Bessel differential equationwhich is singular at the origin. Bessel functions of the second kind are also called Neumann functions or Weber functions. The above plot shows 
for 
, 1, 2, ..., 5. The Bessel function of the second kind is implemented in the Wolfram Language as BesselY[nu, z].
Let 
be the first solution and 
be the other one (since the Bessel differential equation is second-order, there are two linearly independent solutions). Then
Take 
(1) minus 
(2),
 |
(3)
|
 |
(4)
|
so 
, where 
is a constant. Divide by 
,
 |
(5)
|
 |
(6)
|
Rearranging and using 
gives
where 
is the so-called Bessel function of the second kind.
can be defined by
 |
(9)
|
(Abramowitz and Stegun 1972, p. 358), where 
is a Bessel function of the first kind and, for 
an integer 
by the series
 |
(10)
|
where 
is the digamma function (Abramowitz and Stegun 1972, p. 360).
The function has the integral representations
(Abramowitz and Stegun 1972, p. 360).
Asymptotic series are
where 
is a gamma function.
For the special case 
, 
is given by the series
 |
(15)
|
(Abramowitz and Stegun 1972, p. 360), where 
is the Euler-Mascheroni constant and 
is a harmonic number.
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). "Bessel Functions 
and 
." §9.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 358-364, 1972.
Arfken, G. "Neumann Functions, Bessel Functions of the Second Kind, 
." §11.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 596-604, 1985.
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 625-627, 1953.
Spanier, J. and Oldham, K. B. "The Neumann Function 
." Ch. 54 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 533-542, 1987.
Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.
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