Mathematical Methods for Physics and Engineering - Matematica.NET

18.5 BESSEL FUNCTIONSin subsection 18.1.2. The generating function for Bessel functions of integer orderis given by[ ( xG(x, h) =exp h − 1 )] ∞∑= J n (x)h n . (18.98)2 hn=−∞By expanding the exponential as a power series, it is straightfoward to verify thatthe functions J n (x) defined by (18.98) are indeed Bessel functions of the first kind,as given by (18.79).The generating function (18.98) is useful for finding, for Bessel functions ofinteger order, properties that can often be extended to the non-integer case. Inparticular, the Bessel function recurrence relations may be derived.◮Use the generating function to prove, for integer ν, the recurrence relation (18.97), i.e.J ν−1 (x)+J ν+1 (x) = 2ν x J ν(x).Differentiating G(x, h) with respect to h we obtain(1+ 1 )G(x, h) =h 2∂G(x, h)∂h= x 2which can be written using (18.98) again as(1+ 1 )∑ ∞h 2x2n=−∞J n (x)h n =∞∑nJ n (x)h n−1 ,n=−∞∞∑nJ n (x)h n−1 .n=−∞Equating coefficients of h n we obtainx2 [J n(x)+J n+2 (x)] = (n +1)J n+1 (x),which, on replacing n by ν − 1, gives the required recurrence relation. ◭Integral representationsThe generating function (18.98) is also useful for deriving integral representationsof Bessel functions of integer order.◮Show that for integer n the Bessel function J n (x) is given byJ n (x) = 1 π∫ π0cos(nθ − x sin θ) dθ. (18.99)By expanding out the cosine term in the integrand in (18.99) we obtain the integral∫ πI = 1 [cos(x sin θ)cosnθ + sin(x sin θ)sinnθ] dθ. (18.100)π 0Now, we may express cos(x sin θ) and sin(x sin θ) in terms of Bessel functions by settingh =expiθ in (18.98) to give[ x]∞∑exp2 (exp iθ − exp(−iθ)) =exp(ix sin θ) = J m (x)expimθ.613m=−∞