Bessel's equation of order

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Special functions and orthogonal polynomials

Conditions of Validity, Ch Series solutions, singular points and where the special functions come from

Problem

Bessel's equation of order ν\nu, written as y′′+1xy′+(1−ν2x2)y=0y''+\dfrac1x y'+\left(1-\dfrac{\nu^2}{x^2}\right)y=0, has its only singular point, a regular singular point, at x=0x=0; every other finite point, real or complex, is ordinary. ((a)) A power-series solution is wanted about the ordinary point x0=5x_0=5. Predict the radius within which every such series is guaranteed to converge, from the location of the nearest singularity of the coefficients. ((b)) For a non-negative integer nn, the Frobenius series built on the exponent −n-n has, at order kk, coefficient (−1)kk! Γ(k−n+1)(x2)2k−n\dfrac{(-1)^k}{k!\,\Gamma(k-n+1)}\left(\dfrac{x}{2}\right)^{2k-n}; the terms with k<nk<n vanish because 1/Γ(k−n+1)=01/\Gamma(k-n+1)=0 there. Relabel k=m+nk=m+n in the surviving terms and show that the result is (−1)nJn(x)(-1)^nJ_n(x), so that J−n=(−1)nJnJ_{-n}=(-1)^nJ_n is not an independent second solution. ((c)) Take n=2n=2. The series for J2J_2 is J2(x)=∑m=0∞(−1)mm! (m+2)!(x2)2m+2.J_2(x)=\sum_{m=0}^\infty\frac{(-1)^m}{m!\,(m+2)!}\left(\frac{x}{2}\right)^{2m+2}. Sum its first four terms (m=0,1,2,3m=0,1,2,3) at x=3.000x=3.000 to obtain a numerical estimate of J2(3.000)J_2(3.000), compare it with the tabulated value J2(3.000)=0.4861J_2(3.000)=0.4861, and use the identity of part (b) to state the value of J−2(3.000)J_{-2}(3.000).

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