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To study the approximations in the table, a computer plot on the same axes the given function together with its small x approximation and its asymptotic approximation. Use an interval large enough to show the asymptotic approximation agreeing with the function for large x. If the small x approximation is not clear, plot it alone with the function over a small interval J1(x).

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Step by step solution

01

The concept of Wolfram Mathematica built-in functions

The Wolfram Mathematica built -in functions are used because they are very convenient. The Mathematica function Bessel J[p, x]returns the values ofJP(x) .

02

Use the Wolfram Mathematica built-in function to plot the graphs.

The approximate for small x is, JP(x)=1Γ(p+1)x2p+Oxp+2.

The approximate formula for large x is, JP(x)=2Ï€xcosx-2p+14Ï€+Ox-3/2 .

The plot of the functions J1(x) together with its small 'x'approximation and its asymptotic approximation can be drawn as:

The plot of the functions J1(x)together with its small 'x'approximation over a small interval can be drawn as:

The plot of the functions J1(x) together with its asymptotic approximation over a small interval can be drawn as:

It can be seen that the asymptotic approximate is agreeing with the function of large x.

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