[PDF] J Bessel function of the rst kind of order

J Bessel function of the rst kind of order

The Bessel function J n(x), n2N, called the Bessel function of the rst kind of order n, is de ned by the absolutely convergent in nite series J n(x) = xn X m 0 ( 21)mxm 22m+nm(n+ m) for all x2R: (1) It satis es the Bessel di erential equation x2 J00 n (x) + xJ0 n (x) + (x2 n2)J n(x) = 0: (2) The Bessel functions most relevant to this course are J

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Representation of signals as series of orthogonal functions

Series expansions involving Bessel functions : Neumann series Fourier-Bessel series (can be generalized to real indexes Jν) A reference book about Bessel functions (800 pages) α np is the nth positive root of J p Convenient for expansions on [-∞,+∞] Convenient for expansions on [0,1] with boundary condition atx=1

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Eliminating Z(r) and P(r) yields a differential equation for R(r): r2 d2R dr2 +r dR dr − 1+(kr)2 R = 0 (18) This corresponds to modified Bessel Equation of order 1, whose solutions may be written in terms of the modified Bessel functions of the first and second kind, respectively, I 1(kr) and K 1(kr) We note that K

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Course notes

where no convergence condition is imposed, and de ne asymptoticity by the following De nition 1 2 A function f is asymptotic to the formal series f~ as tt 0 (once more, the approach of t 0 may have to be restricted to a generally complex curve) if f(t) XM k=0 f k(t) = o(f M(t)) (8M2N or 8M6 M 0 2N)(1 3)

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Notes on Bessel functions

The Bessel functionJn(x),n2N, called theBessel function of the rst kind of ordern, is dened by the absolutely convergent innite series J n(x) =xnX m0(1)mx2m22m+nm!(n+m)!for allx2R:(1)

It satises the Bessel dierential equation

x

2J00n(x) +xJ0n(x) + (x2n2)Jn(x) = 0:(2)

The Bessel functions most relevant to this course areJ0(x) and the closed related function J

1(x). The functionJ0(x) is an even function, whileJ1(x) is odd; similarly for otherJn(x)'s,

depending on the parity ofn. We have J

00(x) =J1(x); J01(x) =J0(x)1xJ1(x):(3)

Using the dierential equations (3) and (2), it is not dicult to show that Z xJ

20(x)dx=x22[J20(x) +J21(x)] + Const:(4)

for all2R, and (22)Z xJ

0(x)J0(x)dx=x[J00(x)J0(x) J00(x)J0(x)] + Const:(5)

for all;2R. From (5) and (4) one deduces Z 1 0 xJ

0(x)J0(x)dx= 0 (6)

ifJ0() =J0() = 0,; >0, and6=. Moreover Z 1 0 xJ20(x)dx=12[J20() +J21()] (7) ifJ0() = 0.

Exercise 1.Verify the equations (4), (5).

Exercise 2.Use the equation (2) to show that ifis a repeated root ofJ0(x) (i.e.J0() = J

00() = 0), thenJ(n)() = 0 for alln0. Conclude thatJ0(x) has no multiple root. (Hint:

Ifis a multiple root, the Bessel dierential equation implies that the second derivative of J

0(x) vanishes at. Dierentiate the Bessel dierential equation, use it to conclude that the

third derivative of the Bessel dierential equation vanishes at. Similarly for higher order derivatives.) 1 For large values ofx,Jn(x) behaves like a damped harmonic oscillator: J n(x)r2xcos(xn24);(8) in the sense that lim x!1q2xcos(xn24)Jn(x)= 1: Exercise 3.Use Maple to demonstrate the following statements. (a) For large values ofx2R, theenvelopeofJ0(x) is 2x 12cos x4 (Use several frames with dierent ranges of (large) values ofx.) (b) For large values ofx2R, the dierence of consecutive zeroes ofJ0(x) is close to. (c) The zeroes ofJ1(x)interlacewith the zeroes ofJ0(x). 2quotesdbs_dbs2.pdfusesText_2