Bessel functions. ν + 1 ; 1 = 0 for k = 0, 1, 2,..., n 1. Γ( n + k + 1) = ( 1) n J n (z). Γ(n + k + 1) k!

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1 Bessel functions The Bessel function J ν (z of the first kind of order ν is defined by J ν (z ( (z/ν ν Γ(ν + F ν + ; z 4 ( k k ( Γ(ν + k + k! For ν this is a solution of the Bessel differential equation z y (z + zy (z + ν y(z, ν ( For ν / {,,, } we have that J ν (z is a second solution of the differential equation ( the two solutions J ν (z J ν (z are clearly linearly independent For ν n {,,, } we have since n J n (z This implies that J n (z n ( k k n Γ( n + k + k! kn for k,,,, n Γ( n + k + kn ( n ( k k, Γ( n + k + k! ( k k n Γ( n + k + k! ( n+k (n+k Γ(k + (n + k! n ( k Γ(n + k + k! k ( n J n (z This implies that J n (z J n (z are linearly dependent for n {,,, } A second linearly independent solution can be found as follows Since ( n cos nπ, we see that J ν (z cos νπ J ν (z is a solution of the differential equation ( which vanishes when ν n {,,, } Now we define Y ν (z : J ν(z cos νπ J ν (z, (3 where the case that ν n {,,, } should be regarded as a limit case By l Hopital s rule we have Y n (z lim Y ν (z [ ] [ ] Jν (z ( n J ν (z ν n π ν π ν νn νn This implies that Y n (z ( n Y n (z for n {,,, } The function Y ν (z is called the Bessel function of the second kind of order ν Using the definition ( we find that [ ] Jν (z ν νn n J n (z ln ( k ψ(n + k + (n + k! k! k,

2 where For ν / {,,, } we have Now we use J ν (z ν ψ(z d dz ln Γ(z Γ (z Γ(z ν J ν (z ln + ( k ψ( ν + k + Γ( ν + k + k! k ( n lim (z + nγ(z z n n! Γ(z ( n (z + n n! for z n This implies that Hence lim ν n This implies that ψ(z Γ (z Γ(z ( n (z+n n! ( n (z+n n! z + n for z n ψ(z lim z n Γ(z ( n+ n! for n,,, ( k ψ( ν + k + Γ( ν + k + k! n ( n n ( n [ ] J ν (z ν νn (n k! k! (n k! k! k k + J n (z ln kn k + ( n ( k ψ( n + k + Γ( n + k + k! n k ( k ψ(k + k Γ(k + (n + k! ( + ( n z n n (n k! k k! k ( + ( n z n ( k ψ(k + (n + k! k! Finally we use the fact that J n (z ( n J n (z to conclude that Y n (z π J n(z ln ( n z n (n k! k π k! n π ( k (n + k! k! [ψ(n + k + + ψ(k + ] k for n {,,, } equations Compare with the theory of Frobenius for linear second differential

3 In the theory of second order linear differential equations of the form y + p(zy + q(zy, two solutions y y are linearly independent if only if y (z y (z W (y, y (z : y (z y (z y (zy (z y (zy (z This determinant is called the Wronskian of the solutions y y It can (easily be shown that this determinant of Wronski satisfies the differential equation W (z + p(zw (z This result is called Abel s theorem or the theorem of Abel-Liouville In the case of the Bessel differential equation we have p(z /z, which implies that for some constant c Now we have Theorem W (z + z W (z W (y, y (z c z W (J ν, J ν (z πz W (J ν, Y ν (z πz (4 For ν this implies that J ν (z J ν (z are linearly independent if ν / {,,, } that J ν (z Y ν (z are linearly independent for all ν Proof Note that W (J ν, Y ν (z J ν (zy ν(z J ν(zy ν (z J ν (z J ν(z cos νπ J ν(z J ν(z J ν(z cos νπ J ν (z Now we use the definition ( to obtain J ν (z J ν(zj ν(z J ν(zj ν (z ( k Γ(ν + k + k! zν+k ν+k J ν(z W (J ν, J ν (z ( k (ν + k Γ(ν + k + k! zν+k ν+k J ν (z m ( m Γ( ν + m + m! z ν+m ν+m J ν(z m ( m ( ν + m Γ( ν + m + m! z ν+m ν+m 3

4 z W (J ν, J ν (z z [ J ν (zj ν(z J ν(zj ν (z ] This implies that m m ( k+m ( ν + m Γ(ν + k + Γ( ν + m + k! m! zk+m k+m m ( k+m (ν + k Γ(ν + k + Γ( ν + m + k! m! zk+m k+m ( k+m (ν + k m Γ(ν + k + Γ( ν + m + k! m! zk+m k+m lim z W (J ν ν, J ν (z z Γ(ν + Γ( ν + ν νγ(νγ( ν π Using the definition ( we obtain d dz [zν J ν (z] d dz Similarly we have ( k Γ(ν + k + k! zν+k ν+k ( k Γ(ν + k k! zν+k ν+k zν J ν (z ( k (ν + k Γ(ν + k + k! zν+k ν+k d dz [zν J ν (z] z ν J ν (z zj ν(z + νj ν (z zj ν (z (5 d [ z ν J ν (z ] d dz dz ( k Γ(ν + k + k! z k ν+k ( k+ Γ(ν + k + k! z k+ ν+k+ z ν J ν+ (z ( k k Γ(ν + k + k! zk ν+k d [ z ν J ν (z ] z ν J ν+ (z zj dz ν(z νj ν (z zj ν+ (z (6 Elimination of J ν(z from (5 (6 gives J ν (z + J ν+ (z ν z J ν(z elimination of J ν (z from (5 (6 gives J ν (z J ν+ (z J ν(z 4

5 Special cases For ν / we have from the definition ( by using Legendre s duplication formula for the gamma function x ( k ( x k x ( k J / (x Γ(k + 3/ k! π (k +! xk sin x, x > πx for ν / we have ( k ( x k J / (x x Γ(k + / k! πx Note that the definition (3 implies that ( k (k! xk πx cos x, x > Y / (x J / (x πx cos x Y /(x J / (x πx sin x, x > First we will prove Theorem J ν (z Proof We start with Integral representations ( z ν Γ(ν + / e π izt ( t ν / dt, Re ν > / (7 e izt ( t ν / dt (iz n n n! t n ( t ν / dt Note that the latter integral vanishes when n is odd For n k we obtain using t u t k ( t ν / dt u k ( u ν / B(k + /, ν + / du u Now we use Legendre s duplication formula to find that π Γ(k π Γ(k + / k Γ(k Γ(k + k Γ(k + This proves the theorem e izt ( t ν / dt (iz k (k! Γ(ν + / π u k / ( u ν / du Γ(k + /Γ(ν + / Γ(ν + k + π (k! k k! Γ(k + /Γ(ν + / Γ(ν + k + ( k k Γ(ν + k + k! 5

6 We also have Poisson s integral representations Theorem 3 J ν (z ( z ν π Γ(ν + / e π iz cos θ (sin θ ν dθ ( z ν π Γ(ν + / cos(z cos θ (sin θ π ν dθ, Proof Use the substitution t cos θ to obtain e izt ( t ν / dt Further we have π e iz cos θ ( cos θ ν / ( sin θ dθ e iz cos θ cos(z cos θ + i sin(z cos θ sin(z cos θ(sin θ ν dθ Re ν > / e iz cos θ (sin θ ν dθ This shows that Poisson s integral representations follow from the integral representation (7 Remarks: The Fourier transform is defined by with inversion formula F (z : π f(t π e izt f(t dt e izt F (z dz This implies that the Fourier transform of the function is f(t F (z { ( t ν /, t, t > Γ(ν + / ν Jν (z Instead of the substitution t cos θ in (7 one can also use the substitution t sin θ, which leads to slightly different forms of Poisson s integral representations In fact we have ( z ν π/ J ν (z Γ(ν + / e π iz sin θ (cos θ ν dθ π/ ( z ν π/ Γ(ν + / cos(z sin θ (cos θ π ν dθ Γ(ν + / π π/ ( z ν π/ cos(z sin θ (cos θ ν dθ, Re ν > / 6

7 Integrals of Bessel functions The Hankel transform of a function f is defined by F (s tf(tj ν (st dt for functions f for which the integral converges The inversion formula is given by f(t sf (sj ν (st ds This pair of integrals is called a Hankel pair of order ν An example of such an integral is t µ e ρ t J ν (st dt Γ ( µ+ν ν+ Γ(ν + s ν ( (µ + ν/ ρ µ+ν F ; s ν + 4ρ, Re(µ + ν > It can be shown that the integral converges for Re(µ + ν > Now we use the definition ( to obtain t µ e ρ t ( k J ν (st dt Γ(ν + k + k! sν+k ν+k t µ+ν+k e ρ t dt Using the substitution ρ t u we find that t µ+ν+k e ρ t dt ρ µ ν k+ u (µ+ν+k / e u du ρ u ρ µ ν k u (µ+ν+k / e u du ( µ + ν + k ρ µ ν k Γ t µ e ρ t J ν (st dt ( k Γ ( µ+ν ρ µ ν + k s ν+k Γ(ν + k + k! ν+k ρ k Γ ( µ+ν ρ µ+ν Γ(ν + sν ν ( k ((µ + ν/ k (ν + k k! Γ ( µ+ν ν+ Γ(ν + s ν ρ µ+ν F ( (µ + ν/ ν + ; s 4ρ s k k ρ k The special case µ ν + is of special interest: in that case we have (µ + ν/ ν + This implies that the F reduces to a F which is an exponential function The result is t ν+ e ρ t J ν (st dt s ν (ρ ν+ e s /4ρ, Re ν > 7

8 Hankel functions The functions H ν ( (z H ν ( (z are defined by H ( ν (z : J ν (z + iy ν (z H ν ( (z : J ν (z iy ν (z These functions are called Hankel functions or Bessel functions of the third kind Note that these definitions imply that J ν (z H( ν (z + H ν ( (z Further we have H ( / (x J /(x + iy / (x Y ν (z H( ν (z H ν ( (z i (cos x + i sin x πx πx eix, x > H ( / (x J /(x iy / (x (cos x i sin x πx πx e ix, x > Similarly we have H ( / (x J /(x + iy / (x (sin x i cos x i πx πx eix, x > H ( / (x J /(x iy / (x (sin x + i cos x i πx πx e ix, x > Modified Bessel functions The modified Bessel function I ν (z of the first kind of order ν is defined by ( I ν (z : (z/ν Γ(ν + F ν + ; z ν 4 k Γ(ν + k + k! For ν this is a solution of the modified Bessel differential equation z y (z + zy (z + ν y(z, ν For ν / {,,, } we have that I ν (z is a second solution of this differential equation the two solutions I ν (z I ν (z are linearly independent For ν n {,,, } we have I n (z I n (z The modified Bessel function K ν (z of the second kind of order ν is defined by π K ν (z : [I ν(z I ν (z] for ν / {,,, } Now we have for x > I / (x πx sinh x, I /(x K n (z lim ν n K ν (z for n {,,, } π πx cosh x K /(x K / (x x e x 8

9 A generating function The Bessel function J n (z of the first kind of integer order n Z can also be defined by means of the generating function ( exp z ( t t J n (zt n (8 n In fact, the series on the right-h side is a so-called Laurent series at t for the function at the left-h side Using the Taylor series for the exponential function we obtain ( exp z ( t t ( zt ( exp exp z ( zt j ( k k t j! k! t a n t n j n For n {,,, } we have n+k ( k k n a n (n + k! k! ( k k Jn (z (n + k! k! This proves (8 a n If t e iθ, then we have j j ( j+n n+j ( n J n (z J n (z j! (n + j! ( t t eiθ e iθ i sin θ ( exp x ( t t cos(x sin θ + i sin(x sin θ e ix sin θ n n e ix sin θ cos(x sin θ + i sin(x sin θ J n (xe inθ n J n (x [cos(nθ + i sin(nθ] Since J n (x ( n J n (x this implies that cos(x sin θ J n (x cos(nθ J (x + J k (x cos(kθ sin(x sin θ n k J n (x sin(nθ k J k+ (x sin(k + θ For θ π/ this implies that cos x J (x + ( k J k (x sin x ( k J k+ (x For θ we also have J (x + J k (x k 9

10 The generating function (8 can be used to prove that The proof is as follows: n J n (x + y k J k (xj n k (y ( J n (x + yt n exp (x + y ( t t ( exp x ( t t ( exp y ( t t ( J k (xt k J m (yt m k m n k J k (xj n k (y t n We can also find an integral representation for the Bessel function J n (x of the first kind of integral order n starting from the generating function e ix sin θ n J n (xe inθ We use the orthogonality property of the exponential function, id est {, k Z, k e ikθ dθ π π, k π This implies that e ix sin θ e inθ dθ J n (x π The special case n reads π J (x π k e i(x sin θ nθ dθ π J k (x e ikθ e inθ dθ π J n (x π cos (x sin θ dθ π cos (x sin θ nθ dθ cos(xt t dt

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