Q15P Question: Use the Section 15 rec... [FREE SOLUTION]

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Chapter 12: Q15P (page 598)

Question: Use the Section 15 recursion relations and (17.4) to obtain the following recursion relations for spherical Bessel functions. We have written them for $j_{n}$ , but they are valid for $y_{n}$ and for the $h_{n} (\frac{d}{dx}) " [" x^{n + 1} j_{n} {(x)}^{"}] = x^{n + 1} j_{n - 1} (x)$

Short Answer

Expert verified

The resultant answer is $\frac{d}{dx} (x^{n + 1} j_{n} (x)) = (x^{n + 1} j_{n - 1} (x))$ .

Step by step solution

Concept of Spherical Bessel functions:

The spherical Bessel function is given as follows:

$h_{n}^{(1)} (x) = j_{n} (x) + {iy}_{n} (x)$

With:

localid="1659265307597" $j_{n} (x) = x^{n} {(- \frac{1}{x} \frac{d}{d x})}^{n} (\frac{s i n (x)}{x}) y_{n} (x) = - x^{n} {(- \frac{1}{x} \frac{d}{dx})}^{n} (\frac{\cos (x)}{x})$

Formula used:

$j_{n} (x) = {鈭�}_{r = 0}^{鈭�} \frac{{(- 1)}^{r}}{螕 (r + 1) 螕 (r + 1 + n)} {(\frac{x}{2})}^{2 r + n}$

Use the Spherical Bessel function to calculate the function:

Using the spherical Bessel function to calculate, obtain:

$\frac{d}{dx} [x^{n + 1} j_{n} (x)] = x^{n + 1} j_{n - 1} (x) \frac{d}{dx} (x^{n + 1} j_{n} (x)) = \frac{d}{dx} (x^{n + 1} \sqrt{\frac{蟺}{2 x}} J_{n + \frac{1}{2}} (x)) \frac{d}{dx} [x^{n + 1} j_{n} (x)] = \frac{d}{dx} (x^{n + \frac{1}{2}} \sqrt{\frac{蟺}{2}} J_{n + \frac{1}{2}} (x)) = \sqrt{\frac{蟺}{2}} (x^{n + \frac{1}{2}} J_{n + \frac{1}{2} - 1} (x))$

As known that,

$(\frac{d}{d x} (x^{p} J_{p} (x)) = x^{p} J_{p - 1} (x))$ .

Therefore,

$\frac{d}{dx} [x^{n + 1} j_{n} (x)] = \sqrt{\frac{蟺}{2 x}} (x^{n + 1} J_{n + \frac{1}{2} - 1} (x)) = [x^{n + 1} j_{n - 1} (x)]$

Hence, it is proved that $\frac{d}{dx} (x^{n + 1} j_{n} (x)) = (x^{n + 1} j_{n - 1} (x))$ .

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91影视

Question: Use the Section 15 recursion relations and (17.4) to obtain the following recursion relations for spherical Bessel functions. We have written them for $j_{n}$ , but they are valid for $y_{n}$ and for the $h_{n} (\frac{d}{dx}) " [" x^{n + 1} j_{n} {(x)}^{"}] = x^{n + 1} j_{n - 1} (x)$

Short Answer

Step by step solution

Concept of Spherical Bessel functions:

Use the Spherical Bessel function to calculate the function:

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

Question: Use the Section 15 recursion relations and (17.4) to obtain the following recursion relations for spherical Bessel functions. We have written them forjn, but they are valid forynand for thehn(ddx)"["xn+1jnx"]=xn+1jn-1(x)

Short Answer

Step by step solution

Concept of Spherical Bessel functions:

Use the Spherical Bessel function to calculate the function:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Engineering Physics

Astrophysics

Solid State Physics

Magnetism and Electromagnetic Induction

Force

Electricity and Magnetism

Study anywhere. Anytime. Across all devices.

Question: Use the Section 15 recursion relations and (17.4) to obtain the following recursion relations for spherical Bessel functions. We have written them for $j_{n}$ , but they are valid for $y_{n}$ and for the $h_{n} (\frac{d}{dx}) " [" x^{n + 1} j_{n} {(x)}^{"}] = x^{n + 1} j_{n - 1} (x)$