Chapter 12: Q11P (page 598)

From (17.4), show that, $h_{n}^{(1)} (ix) = - e^{- x} / x$ .

Short Answer

Expert verified

The figures show the behavior of Bessel functions of different kinds.

Step by step solution

Concept of Spherical Bessel functions:

Spherical Bessel function is given as follows:

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

With:

$j_{n} (x) + x^{n} {(- \frac{1}{x} \frac{d}{dx})}^{n} (\frac{\sin (x)}{x}) y_{n} (x) = - x^{n} {(- \frac{1}{x} \frac{d}{dx})}^{n} (\frac{\cos (x)}{x}) Formula used : j_{n} (x) = \sum_{r = 0}^{\infty} \frac{{(- 1)}^{r}}{Γ (r + 1) Γ (r + 1 + n)} {(\frac{x}{2})}^{2 r + n}$

Calculate the equation by the Spherical Bessel function:

Using the Spherical Bessel function to obtain:

$h_{n}^{(1)} (x) = j_{n} (x) + {iy}_{n} (x) = h_{n}^{(1)} (ix) = j_{n} (ix) + {iy}_{n} (ix) = h_{0}^{(1)} (ix) Simplify further as follows : h_{n}^{(1)} (x) = j_{n} (ix) + {iy}_{0} (ix) = [\frac{\sin ix}{ix} - i \frac{\cos ix}{ix}] = [\frac{\sinh x}{x} - i \frac{\cosh x}{x}] = - \frac{e^{- x}}{x} Hence, h_{0}^{(1)} (ix) = - \frac{e^{- x}}{x} .$

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From (17.4), show that, $h_{n}^{(1)} (ix) = - e^{- x} / x$ .

Short Answer

Step by step solution

Concept of Spherical Bessel functions:

Calculate the equation by the Spherical Bessel function:

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From (17.4), show that,hn(1)(ix)=-e-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

Solid State Physics

Energy Physics

Fluids

Circular Motion and Gravitation

Measurements

Physical Quantities And Units

Study anywhere. Anytime. Across all devices.

From (17.4), show that, $h_{n}^{(1)} (ix) = - e^{- x} / x$ .