Series 215 problem 08 (Bessel).mws

1. Bessel functions.

For this activity you will need the Bessel function of the first kind of order 1, known to Maple as BesselJ(1,x), which is defined by

J[1](x) = Sum((-1)^n*x^(2*n+1)/(2^(2*n+1)*n!*(n+1)!...

Hence we have

> J[1]:=x->BesselJ(1,x);

J[1] := proc (x) options operator, arrow; BesselJ(1...

Submission:

(a) Find the domain of J[1] , i.e. what is the radius of convergence?

(b) Graph the first several partial sums (use s[1], s[5], s[10] and s[20] of the Bessel function along with J[1] itself on the

same coordinate axes. Use the viewing window [ -10 .. 10 ] by [ -4 .. 4 ] .

Submission worksheet:

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2. More Bessel functions.

The Bessel function of the first kind of order 0 is given by

J[0](x) = Sum((-1)^n*x^(2*n)/(2^(2*n)*n!^2),n = 0 ....

Maple knows this function as BesselJ(0,x) . That is we can define this function by

> J[0]:=x->BellelJ(0,x);

J[0] := proc (x) options operator, arrow; BellelJ(0...

Submission:

(a) Show that J[0] satisfies the differential equation:

x^2*`@@`(D,2)(J[0])(x)+x*D(J[0])(x)+x^2*J[0](x) = 0...

(b) Evaluate Int(J[0](x),x = 0 .. 1) correct to three dedcimal places.

Submission worksheet:

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