Question

Use a CAS to graph \({J_{3/2}}(x),{J_{ - 3/2}}(x),{J_{5/2}}(x),\) and \({J_{ - 5/2}}(x)\).

Short answer

The graph has been plotted.

Step-by-step solution

Define Spherical Bessel’s equation.

Bessel functions of half-integral order are used to dene two more important functions:

\(\begin{array}{l}{j_n}(x) = \sqrt {\frac{\pi }{{2x}}} {J_{n + 1/2}}(x)\\{y_n}(x) = \sqrt {\frac{\pi }{{2x}}} {Y_{n + 1/2}}(x)\end{array}\)

The function \({j_n}(x)\) is called the spherical Bessel function of the first kind and \({y_n}(x)\) is the spherical Bessel function of the second kind.

Find the graph of \({j_{3/2}}(x)\).

Use GNU Octave to plot the functions.

Find the value of \({j_{ - 3/2}}(x)\).

Let,

Find the value of \({j_{5/2}}(x)\).

Let,

Find the value of \({j_{ - 5/2}}(x)\).

Let,