Let n > 0 be a fixed nonnegative integer and recall that 0! := 1. The Bessel

Question:

Let n > 0 be a fixed nonnegative integer and recall that 0! := 1. The Bessel function of order n is the function defined by

a) Show that Bn(x) converges pointwise on R and uniformly on any closed interval [a,b.
b) Prove that y = Bn(x) satisfies the differential equation
x2y" + xy' + (x2 - n2)y = 0
for x ˆˆ R.
c) Prove that
(xnBn(x))' = xnBn-l(x)
for n ˆˆ N and x ˆˆ R.