(Everything a physicist needs to know about) Bessel functions Jn(x) of integer order (1;2) (and also Hankel functions Hn ) Nikolai G. Lehtinen December 18, 2020 Abstract Some properties of integer-order Bessel functions Jn(x) are derived from their definition using the generating function. The results may be of use in such areas as plasma physics. Now with a Section on Hankel (1;2) functions Hn (x)! We assume that the reader knows some complex analysis (e.g., can integrate in the complex plane using residues). 1 Basic properties 1.1 Generating function We derive everything else from here, which will serve us the definition of the integer-order Bessel functions (of the first kind): +1 x (t− 1 ) X n g(x; t) = e 2 t = Jn(x)t (1) n=−∞ We immediately make our first observations that since g(x; t) is real for real x; t then Jn(x) must be real for real x. From now on, +1 X X X or just will mean n n=−∞ Z dθ Z 2π dθ will mean 2π 0 2π 1 1.2 Some useful formulas − x (t− 1 ) P n n P n 1. t ! −t in (1) =) e 2 t = (−1) Jn(x)t = Jn(−x)t =) (equate powers of t) =) n Jn(−x) = (−1) Jn(x) x 1 1 − 2 (t− t ) P n P −n 2. t ! t in (1) =) e = Jn(−x)t = Jn(x)t =) (equate powers of t and use previous expression) =) n J−n(x) = (−1) Jn(x) = Jn(−x) (2) iθ 1 1 3. t = e =) 2 t − t = i sin θ =) ix sin θ X inθ e = Jn(x)e (3) R dθ −imθ 4. 2π e × (3), switch m ! n, =) Z dθ J (x) = eix sin θ−inθ (4) n 2π 1.3 Recursion formulas @ 1 1 P 0 n 1. @x (1) =) 2 t − t g(x; t) = Jn(x)t =) P 1 n+1 n−1 P 0 n 2 Jn(x)(t − t ) = Jn(x)t =) P 1 n P 0 n 2 (Jn−1(x) − Jn+1(x))t = Jn(x)t =) 1 J 0 (x) = (J (x) − J (x)) (5) n 2 n−1 n+1 @ x 1 P n−1 2. @t (1) =) 2 1 + t2 g(x; t) = nJn(x)t , multiply by t, =) x 1 x P n−1 n+1 P n 2 t + t g(x; t) = 2 Jn(x)(t + t ) = nJn(x)t =) x P n P n 2 (Jn+1(x) + Jn−1(x))t = nJn(x)t =) x nJ (x) = (J (x) + J (x)) (6) n 2 n−1 n+1 3. Combining the two we can get n J 0 = J − J (7) n x n n+1 n J 0 = J − J (8) n n−1 x n 2 1.4 Differential equation Bessel equation arises when we solve Helmholz equation r2φ + p2φ = 0 in 2D, in cylindrical coordinates. It is more naturally understood when we go to Fourier coordinates k (see Section 3.1 below), in which operator r = ik, r2 = −k2. Here we brutally derive the Bessel equation from the recursion formulas. 0 Let us use (7) to find Jn, differentiate it again and substitute expression 0 n+1 Jn+1 = Jn − x Jn+1 derived from (8). We have J 0 = n J − J 9 n x n n+1 > 00 n 0 n+1 => Jn = x Jn − Jn + x Jn+1 = − n J + n2 J − n J − J + n+1 J x2 n x2 n x n+1 n x n+1 > 1 n n2 ;> = x (− x Jn + x Jn − xJn + Jn+1) By eliminating Jn+1 from these equations we obtain 2 00 0 2 2 x Jn + xJn + (x − n )Jn = 0 (9) 2 Sums 0 2.1 Sums of the first power of Jn(x) or Jn(x) The basic idea is to substitute t = 1 in various derivatives of g(x; t) in respect to both x and t. 1. t = 1 in (1) =) X Jn(x) = 1 (10) 2. P(6) =) X nJn(x) = x (11) P k 3. n Jn(x) may be found by differentiating g(x; t) over t and multiply- ing by t, k times. Example: P 2 n @ x 1 n Jn(x)t = t @t 2 t + t g(x; t) = ::: 3 Figure 1: Graf's addition theorem 0 2.2 Sums of the second power of Jn(x) or Jn(x) The basic idea is to multiply two g(x; t) or its derivatives in respect to both x and t. One can take the second g to be a function of u = 1=t instead. Note that for equal first arguments g(x; t)g(x; u) = 1 and we have a polynomial in t. By equating powers, we get various sums. Here are some examples. P P n−m 1. g(x; t)g(x; 1=t) = 1 = n m Jn(x)Jm(x)t =) all coefficients are zero except at t0 =) X JnJn−k = δk;0 n x+y 2. g(x; t)g(y; t) = exp 2 [t − 1=t] =) P k P P n+m k Jk(x + y)t = n m Jn(x)Jm(y)t =) X X Jk(x + y) = Jn(x)Jk−n(y) = Jn(x)Jn−k(−y) n n where we used (2). 2.3 Graf's addition theorem See Abramowitz and Stegun (1965, eq 8.1.79). From Figure 1, OB = z sin(θ + ) = OA + AB = x sin θ + y sin(φ − θ). R dθ iz sin(θ+ )−in(θ+ ) in R dθ ix sin θ+iy sin(φ−θ)−inθ From (4), Jn(z) = 2π e =) e Jn(z) = 2π e ix sin θ P ikθ iy sin(φ−θ) P im(φ−θ) Substitute e = k Jk(x)e , e = m Jm(y)e =) 4 in P P imφ R dθ −inθ+ikθ−imθ e Jn(z) = k m Jk(x)Jm(y)e 2π e R dθ ilθ Use 2π e = δl;0 for integer l =) in X imφ e Jn(z) = Jm+n(x)Jm(y)e (12) m In particular, for n = 0 p 2 2 X imφ J0(z) = J0( x + y − 2xy cos φ) = Jm(x)Jm(y)e (13) m 3 Integrals 3.1 2D Fourier transforms Define direct and inverse Fourier transform of a function F (r) in 2D position ~ space r = fx; yg into function F (k) in 2D wavevector space k = fkx; kyg as Z F~(k) = d2r e−ik·rF (r) Z d2k F (r) = eik·rF~(k) (2π)2 2 2 where d r = dx dy and d k = dkx dky and the integration is over the whole plane. Below, we also use polar coordinates r = jrj, θ = atan2(y; x) in position space and k = jkj, χ = atan2(ky; kx) in wavevector space. Let F (r) = δ(r − a)e−inθ. The Fourier image is Z Z 1 Z 2π F~(k) = d2r e−ik·rF (r) = r dr dθ δ(r − a)e−inθ−ikr cos(θ−χ) 0 0 Introducing an auxiliary angle φ = θ − χ − π=2, we get cos(θ − χ) = − sin φ and θ = φ + χ + π=2 =) Z dφ F~(k) = 2πae−inχ−inπ=2 e−inφ+ika sin φ = 2πai−ne−inχJ (ka) 2π n We can use the same formula by substituting r $ k and i $ −i and 5 taking care of coefficient (2π)2. It summary, F (r) () F~(k) δ(r − a) einθ () 2πi−nJ (ka)einχ (14) a n 2πδ(k − p) inJ (pr)einθ () einχ (15) n p 3.2 Weber's First Integral See Abramowitz and Stegun (1965, eq 11.4.28 with µ = 2; ν = 0). R 1 −q2r2 The integral I1 = 0 r dr J0(pr)e is found by representing this inte- gral in a 2D plane (2πr dr = d2r) and going to Fourier space r ! k. R 2 −q2r2 1 I1 = d r F1(r)F2(r), where F1(r) = e , F2(r) = 2π J0(pr), where R 2 R d2k ~∗ ~ r = jrj. We use the fact that d r F1(r)F2(r) = (2π)2 F1 (k)F2(k), where 2 − k ~ R 2 −ik·r ~ π 4q2 F1;2(k) = d r e F1;2(r) are the Fourier images. We have F1(k) = q2 e ~ δ(k−p) and using the Fourier transform result from above, we obtain F2(k) = p . Thus, Z 1 2 2 2 1 − p −q x 4q2 x dx J0(px)e = 2 e (16) 0 2q 3.3 Weber's Second Integral Z 1 −q2x2 I2 = x dx Jn(ax)Jn(bx)e 0 R dφ −inφ From Graf's addition theorem (13) =) Jn(x)Jn(y) = 2π J0(z)e =) p J (ax)J (bx) = R dθ J (x a2 + b2 − 2ab cos θ)e−inθ =) n n 2π 0 p R dθ −inθ R 1 2 2 −q2x2 I2 = 2π e 0 x dx J0(x a + b − 2ab cos θ)e =) [use (16)] =) 2 2 h 2 2 i − a +b h i R dθ 1 a +b −2ab cos θ 1 4q2 R dφ ab π I2 = 2π 2q2 exp − 4q2 − inθ = 2q2 e 2π exp i(−i 2q2 ) sin φ − inφ + in 2 (we substituted θ = φ − π=2) 2 2 − a +b 1 4q2 ab n I2 = 2q2 e Jn(−i 2q2 )i We use the definition of the modified Bessel function (of the first kind) n −n In(x) = i Jn(−ix) = i Jn(ix) 6 =) Z 1 2 2 2 2 1 − a +b ab −q x 4q2 x dx Jn(ax)Jn(bx)e = 2 e In 2 (17) 0 2q 2q In particular, for a = b we have Z 1 2 2 2 2 2 1 − a a 1 a 2 −q x 2q2 x dx Jn(ax)e = 2 e In 2 = 2 Λn 2 0 2q 2q 2q 2q where we introduced another function Λn whose properties are explored in more detail below in Section 5.