Chapter 12: Q10P (page 604)

Use the table above and the definitions in Section 17 to find approximate formulas for large x for $h_{n}^{(2)} (i x)$ :

Short Answer

Expert verified

The approximate value for large value of x is $i^{n} x^{- 1} e^{x}$ .

Step by step solution

Take out the equations.

The given term is $h_{n}^{(2)} (i x)$ .

$h_{n}^{(2)} (x) = j_{n} (x) - i y_{n} (x)$

As, $h_{n}^{(2)} (x) = j_{n} (x) - i y_{n} (x)$ .

Therefore, $h_{n}^{(2)} (i x) = j_{n} (i x) - i y_{n} (i x)$ …… (1)

For larger values of x as follows:

$j_{n} (x) = \frac{1}{x} \sin (x - \frac{n π}{2}) + O (x^{- 2})$ …… (2)

$y_{n} (x) = - \frac{1}{x} \cos (x - \frac{n π}{2}) + O (x^{- 2})$ …… (3)

Use equations (1), (2), and (3) for calculation.

Using equation (1), (2), and (3) as follows:,

$h_{n}^{(2)} (i x) = j_{n} (i x) - i y_{n} (i x) = \frac{1}{i x} \sin (i x - \frac{n π}{2}) - i (- \frac{1}{i x} \cos (i x - \frac{n π}{2})) = \frac{1}{i x} \sin (i x - \frac{n π}{2}) + i \frac{1}{i x} \cos (i x - \frac{n π}{2}) = \frac{- i^{2}}{i x} \sin (i x - \frac{n π}{2}) + \frac{1}{x} \cos (i x - \frac{n π}{2})$

Simplify further as follows:

$h_{n}^{(2)} (i x) = \frac{- i}{x} \sin (i x - \frac{n π}{2}) + \frac{1}{x} \cos (i x - \frac{n π}{2}) = \frac{1}{x} [\cos (i x - \frac{n π}{2}) - i \sin (i x - \frac{n π}{2})] = x^{- 1} [\cos (i x - \frac{n π}{2}) - i \sin (i x - \frac{n π}{2})]$

As, $e^{- i θ} = \cos θ - i \sin θ$

Therefore, calculate:

$h_{n}^{(2)} (i x) = x^{- 1} [\cos (i x - \frac{n π}{2}) - i \sin (i x - \frac{n π}{2})] = x^{- 1} e^{- i (i x - \frac{n π}{2})} = x^{- 1} e^{(- i^{2} x + i \frac{n π}{2})} = x^{- 1} e^{(x + i \frac{n π}{2})} = x^{- 1} e^{x} (e^{i \frac{n π}{2}})$

Simplifying further to get:

$h_{n}^{(2)} (i x) = x^{- 1} e^{x} (i)^{n} = i^{n} x^{- 1} e^{x}$

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Use the table above and the definitions in Section 17 to find approximate formulas for large x for $h_{n}^{(2)} (i x)$ :

Short Answer

Step by step solution

Take out the equations.

Use equations (1), (2), and (3) for calculation.

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Use the table above and the definitions in Section 17 to find approximate formulas for large x for hn(2)(ix):

Short Answer

Step by step solution

Take out the equations.

Use equations (1), (2), and (3) for calculation.

One App. One Place for Learning.

Most popular questions from this chapter

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Use the table above and the definitions in Section 17 to find approximate formulas for large x for $h_{n}^{(2)} (i x)$ :