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Q8.3-31E

Expert-verifiedFound in: Page 444

Book edition
9th

Author(s)
R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages
616 pages

ISBN
9780321977069

**(a) Use the first formula in (30) and Problem 32 to find the spherical Bessel functions \({j_1}(x)\) and \({j_2}(x)\).**

**(b) Use a graphing utility to plot the graphs of \({j_1}(x)\) and \({j_2}(x)\) in the same coordinate plane.**

(a) The values are \({j_1}(x) = \frac{1}{{{x^2}}}sinx - \frac{1}{x}cosx\) and \({j_2}(x) = \left( {\frac{3}{{{x^3}}} - \frac{1}{x}} \right)sinx - \frac{3}{{{x^2}}}cosx\).

(b) The graph has been plotted.

**Bessel functions of half-integral order areused to dene two more important functions:**

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

**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.**

Using the above formula and the result of Problem 32. Let,

\(\begin{aligned}{j_1}(x) &= \sqrt {\frac{\pi }{{2x}}} {J_{3/2}}\\ &= \sqrt {\frac{\pi }{{2x}}} \left( {\sqrt {\frac{2}{{\pi {x^3}}}} sinx - \sqrt {\frac{2}{{\pi x}}} cosx} \right)\\ &= \sqrt {\frac{\pi }{{2x}} \cdot \frac{2}{{\pi {x^3}}}} sinx - \sqrt {\frac{\pi }{{2x}} \cdot \frac{2}{{\pi x}}} cosx\\ &= \frac{1}{{{x^2}}}sinx - \frac{1}{x}cosx\end{aligned}\)

Let,

\(\begin{aligned}{j_2}(x) &= \sqrt {\frac{\pi }{{2x}}} {J_{5/2}}\\ &= \sqrt {\frac{\pi }{{2x}}} \left( {\left( {\sqrt {\frac{{18}}{{\pi {x^5}}}} - \sqrt {\frac{2}{{\pi x}}} } \right)sinx - \sqrt {\frac{{18}}{{\pi {x^3}}}} cosx} \right)\\ &= \left( {\sqrt {\frac{\pi }{{2x}} \cdot \frac{{18}}{{\pi {x^5}}}} - \sqrt {\frac{\pi }{{2x}} \cdot \frac{2}{{\pi x}}} } \right)sinx - \sqrt {\frac{\pi }{{2x}} \cdot \frac{{18}}{{\pi {x^3}}}} cosx\\ &= \left( {\sqrt {\frac{9}{{{x^6}}}} - \sqrt {\frac{1}{{{x^2}}}} } \right)sinx - \sqrt {\frac{9}{{{x^4}}}} cosx\\ &= \left( {\frac{3}{{{x^3}}} - \frac{1}{x}} \right)sinx - \frac{3}{{{x^2}}}cosx\end{aligned}\)

Let the graph of \({j_1}(x)\) and \({j_2}(x)\) be,

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