Q14P Use the Section 15 recursion rel... [FREE SOLUTION]

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Chapter 12: Q14P (page 598)

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 $h_{n}$
$(\frac{d}{dx}) j_{n} (x) = j_{n - 1} (x) - (n + 1) j_{n} (x) / x$

Short Answer

Expert verified

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

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:

$j_{n} (x) + x^{n} {(- \frac{1}{x} \frac{d}{dx})}^{n} (\frac{\sin (x)}{x}) y_{n} (x) = - x^{n} {(- \frac{1}{x} \frac{d}{dx})}^{n} (\frac{\cos (x)}{x}) Formula used : j_{n} (x) = {鈭�}_{r - 0}^{鈭�} \frac{{(- 1)}^{r}}{螕 (r + 1) 螕 (r + 1 + n)} {(\frac{x}{2})}^{2 r + n}$

Calculate the equation by the Spherical Bessel function:

Using the Spherical Bessel function to obtain:

$\frac{d}{dx} (j_{n} (x)) = \frac{d}{dx} (\sqrt{\frac{蟺}{2 x}} J_{n + \frac{1}{2}} (x)) = \sqrt{\frac{蟺}{x}} [- \frac{1}{2} x^{- \frac{3}{2}} J_{n + \frac{1}{2}} (x) + x^{- \frac{3}{2}} J_{n + \frac{1}{2}} (x)] = \sqrt{\frac{蟺}{2 x}} [- \frac{1}{2 x} J_{n + \frac{1}{2}} (x) + \frac{2 n + 1}{2 x} j_{n + \frac{1}{2}} (x) - J_{n + \frac{3}{2}} (x)] (J_{p} (x) = \frac{2 p}{x} J_{p} (x) - J_{p + 1} (x)) Simplify further as follows : \frac{d}{dx} (j_{n} (x)) = \sqrt{\frac{蟺}{2 x}} [\frac{2 n}{2 x} J_{n + \frac{1}{2} + 1} (x)] = [\frac{n}{x} j_{n} (x) - j_{n + 1} (x)] Therefore, \frac{d}{dx} (j_{n} (x)) = [\frac{n}{x} j_{n} (x) - j_{n + 1} (x)] .$

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91影视

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 $h_{n}$
$(\frac{d}{dx}) j_{n} (x) = j_{n - 1} (x) - (n + 1) j_{n} (x) / x$

Short Answer

Step by step solution

Concept of Spherical Bessel functions:

Calculate the equation by the Spherical Bessel function:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Linear Momentum

Electrostatics

Particle Model of Matter

Translational Dynamics

Modern Physics

Fluids

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91影视

Use the Section 15 recursion relations and (17.4) to obtain the following recursion relations for spherical Bessel functions. We have written them for jn, but they are valid for ynand for the hn(ddx)jn(x)=jn-1(x)-(n+1)jn(x)/x

Short Answer

Step by step solution

Concept of Spherical Bessel functions:

Calculate the equation by the Spherical Bessel function:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Linear Momentum

Electrostatics

Particle Model of Matter

Translational Dynamics

Modern Physics

Fluids

Study anywhere. Anytime. Across all devices.

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 $h_{n}$
$(\frac{d}{dx}) j_{n} (x) = j_{n - 1} (x) - (n + 1) j_{n} (x) / x$