Chapter 12: Q8P (page 590)

To show that $\lim_{x \to 0} x^{- 3 / 2} J_{3 / 2} (x) = 3^{- 1} \sqrt{2 / π}$ .

Short Answer

Expert verified

Hence, $\lim_{x \to 0} x^{- 3 / 2} J_{3 / 2} (x) = 3^{- 1} \sqrt{\frac{J}{ττ}}$

Step by step solution

Concept of the Property of Jp(x)

Infinite series:

$J_{p} (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{⌜ (n + 1) ⌜ (n + 1 + p)} {(\frac{x}{2})}^{2 n + p}$

Calculation of the function limx→0x-3/2J3/2(x) :

Considering the infinite series, $J_{p} (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{⌜ (n + 1) ⌜ (n + 1 + p)} {(\frac{x}{2})}^{2 n + p}$ .

Let $p = \frac{3}{2}$ in the above recursion relation.

Therefore, simplify as follows:

localid="1659257652571" $J_{\frac{3}{2}} (x) = {(\frac{x}{2})}^{\frac{3}{2}} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{⌜ (n + 1) ⌜ (n + \frac{5}{2})} {(\frac{x}{2})}^{2 n + p} = {(\frac{x}{2})}^{\frac{3}{2}} \{\frac{1}{⌜ (\frac{5}{2})} - \frac{{(\frac{x}{2})}^{2}}{\frac{5}{2} ⌜ (\frac{5}{2})} + \frac{{(\frac{x}{2})}^{4}}{2 \times \frac{7}{2} \times \frac{5}{2} ⌜ (\frac{5}{2})} . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\}$

localid="1659258104658" $\lim_{x \to 0} x^{\frac{3}{2}} J_{\frac{3}{2}} (x) = \lim_{x \to 0} {(\frac{1}{2})}^{\frac{3}{2}} = {(\frac{x}{2})}^{\frac{3}{2}} \{\frac{1}{⌜ (\frac{5}{2})} - \frac{{(\frac{x}{2})}^{2}}{\frac{5}{2} ⌜ (\frac{5}{2})} + \frac{{(\frac{x}{2})}^{4}}{2 \times \frac{7}{2} \times \frac{5}{2} ⌜ (\frac{5}{2})} . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\} = \frac{{(\frac{1}{2})}^{\frac{3}{2}}}{\frac{3}{2} \times \frac{1}{2} \times \sqrt{ττ}} = \frac{(4)}{3 {(8)}^{\frac{1}{2}} \times \sqrt{ττ}} \lim_{x \to 0} x^{\frac{3}{2}} J_{\frac{3}{2}} (x) = \frac{(4)}{3 \times 2 {(2)}^{\frac{1}{2}} \times \sqrt{ττ}} = 3^{- 1} \sqrt{\frac{2}{ττ}}$

Hence, $\lim_{x \to 0} x^{\frac{3}{2}} J_{\frac{3}{2}} (x) = 3^{- 1} \sqrt{\frac{2}{ττ}}$ is proved.

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To show that $\lim_{x \to 0} x^{- 3 / 2} J_{3 / 2} (x) = 3^{- 1} \sqrt{2 / π}$ .

Short Answer

Step by step solution

Concept of the Property of Jp(x)

Calculation of the function limx→0x-3/2J3/2(x) :

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Step by step solution

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Calculation of the function limx→0x-3/2J3/2(x) :

One App. One Place for Learning.

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To show that $\lim_{x \to 0} x^{- 3 / 2} J_{3 / 2} (x) = 3^{- 1} \sqrt{2 / π}$ .