“Bessel Function of the First Kind”的概念、定义、习题解题方法及翻译-数学辞典

单词

Bessel Function of the First Kind

释义

Bessel Function of the First Kind

The Bessel functions of the first kind are defined as the solutions to the Bessel Differential Equation

(1)

which are nonsingular at the origin. They are sometimes also called Cylinder Functions orCylindrical Harmonics. The above plot shows

for

, 2, ..., 5.

To solve the differential equation, apply Frobenius Method using a series solution of the form

(2)

Plugging into (1) yields



	(3)


	(4)

The Indicial Equation, obtained by setting

, is

(5)

Since

is defined as the first Nonzero term,

, so

. Now, if

(6)

(7)

(8)

(9)

First, look at the special case

, then (9) becomes

(10)

(11)

Now let

, where

, 2, ....



			(12)

which, using the identity

, gives

(13)

Similarly, letting

(14)

which, using the identity

, gives

(15)

Plugging back into (2) with

gives





			(16)

The Bessel Functions of order

are therefore defined as

			(17)
			(18)

so the general solution for

(19)

Now, consider a general

. Equation (9) requires

(20)

(21)

for

, 3, ..., so

			(22)
			(23)

for

, 3, .... Let

, where

, 2, ..., then


			(24)

where

is the function of

and

obtained by iterating the recursion relationship down to

. Now let

, where

, 2, ..., so


			(25)

Plugging back into (9),





			(26)

Now define

(27)

where the factorials can be generalized to Gamma Functions for nonintegral

. Theabove equation then becomes

(28)

Returning to equation (5) and examining the case

(29)

However, the sign of

is arbitrary, so the solutions must be the same for

and

. We are therefore free toreplace

with

, so

(30)

and we obtain the same solutions as before, but with

replaced by

(31)

We can relate

and

(when

is an Integer) by writing

(32)

Now let

. Then


			(33)

But

for

, so the Denominator is infinite and the terms on the right are zero. Wetherefore have

(34)

Note that the Bessel Differential Equation is second-order, so there must be two linearly independent solutions.We have found both only for . For a general nonintegral order, the independent solutions are and. When is an Integer, the general (real) solution is of the form

(35)

where

is a Bessel function of the first kind,

(a.k.a.

) is the Bessel Function of the Second Kind(a.k.a. Neumann Function or Weber Function), and

and

are constants. Complexsolutions are given by the Hankel Functions (a.k.a. Bessel Functions of the ThirdKind).

The Bessel functions are Orthogonal in with respect to the weight factor . Exceptwhen is a Negative Integer,

(36)

where

is the Gamma Function and

is a Whittaker Function.

In terms of a Confluent Hypergeometric Function of the First Kind, the Bessel function is written

(37)

A derivative identity for expressing higher order Bessel functions in terms of

(38)

where

is a Chebyshev Polynomial of the First Kind. Asymptotic forms for the Bessel functions are

(39)

for

and

(40)

for

. A derivative identity is

(41)

An integral identity is

(42)

Some sum identities are

(43)

(44)

and the Jacobi-Anger Expansion

(45)

which can also be written

(46)

The Bessel function addition theorem states

(47)

Roots of the Function are given in the following table.

zero
1	2.4048	3.8317	5.1336	6.3802	7.5883	8.7715
2	5.5201	7.0156	8.4172	9.7610	11.0647	12.3386
3	8.6537	10.1735	11.6198	13.0152	14.3725	15.7002
4	11.7915	13.3237	14.7960	16.2235	17.6160	18.9801
5	14.9309	16.4706	17.9598	19.4094	20.8269	22.2178

Let be the th Root of the Bessel function , then

(48)

(Le Lionnais 1983).

The Roots of its Derivatives are given in the following table.

zero
1	3.8317	1.8412	3.0542	4.2012	5.3175	6.4156
2	7.0156	5.3314	6.7061	8.0152	9.2824	10.5199
3	10.1735	8.5363	9.9695	11.3459	12.6819	13.9872
4	13.3237	11.7060	13.1704	14.5858	15.9641	17.3128
5	16.4706	14.8636	16.3475	17.7887	19.1960	20.5755

Various integrals can be expressed in terms of Bessel functions

			(49)
			(50)

which is Bessel's First Integral,

			(51)
			(52)

for

, 2, ...,

(53)

for

, 2, ...,

(54)

for

. Integrals involving

include

(55)

(56)

(57)

See also Bessel Function of the Second Kind, Debye's Asymptotic Representation, Dixon-Ferrar Formula,Hansen-Bessel Formula, Kapteyn Series, Kneser-Sommerfeld Formula, Mehler's Bessel FunctionFormula, Nicholson's Formula, Poisson's Bessel Function Formula, Schläfli's Formula, Schlömilch's Series, Sommerfeld's Formula, Sonine-Schafheitlin Formula, Watson's Formula, Watson-Nicholson Formula, Weber's DiscontinuousIntegrals, Weber's Formula, Weber-Sonine Formula, Weyrich's Formula

References

Abramowitz, M. and Stegun, C. 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. ``Bessel Functions of the First Kind, '' and ``Orthogonality.'' §11.1 and 11.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 573-591 and 591-596, 1985.

Lehmer, D. H. ``Arithmetical Periodicities of Bessel Functions.'' Ann. Math. 33, 143-150, 1932.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 25, 1983.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 619-622, 1953.

Spanier, J. and Oldham, K. B. ``The Bessel Coefficients and '' and ``The Bessel Function .'' Chs. 52-53 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 509-520 and 521-532, 1987.

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