- #1
qnach
- 155
- 4
- Homework Statement
- This is not a homewrok
- Relevant Equations
- J[1, x] = (x^2/10)*(J[1, x] + J[3, x])
Has any one any idea to solve this equation J[1, x] = (x^2/10)*(J[1, x] + J[3, x]), in which J are spherical Bessel function normally write as j_1 (x) and j_3(x)
Methods 1 serial expansion:
j_1(x) = \frac{\sin(x)}{(x)^2} - \frac{\cos(x)}{x} \approx \dfrac{x}{3} - \dfrac{(x)^3}{30} + \dfrac{(x)^5}{840} + ...
j_3 (x) \approx \dfrac{(x^3)}{105}
I have
\left(\frac{x}{3}-\frac{x^3}{30} + \frac{x^5}{840} \right) \approx \frac{x^2}{10} \left[ \frac{x}{3} - \frac{x^3}{30} + \frac{x^3}{105} \right]
This will lead to
x = \sqrt{ \frac{1 - \sqrt{1 - Y}}{Z} } \approx 0.79
Y and Z are some complicate expression.
Methods using numerical method.
The result obtained is 2.27 or so....
They are different. Has anyone any idea about what is wrong....
Methods 1 serial expansion:
j_1(x) = \frac{\sin(x)}{(x)^2} - \frac{\cos(x)}{x} \approx \dfrac{x}{3} - \dfrac{(x)^3}{30} + \dfrac{(x)^5}{840} + ...
j_3 (x) \approx \dfrac{(x^3)}{105}
I have
\left(\frac{x}{3}-\frac{x^3}{30} + \frac{x^5}{840} \right) \approx \frac{x^2}{10} \left[ \frac{x}{3} - \frac{x^3}{30} + \frac{x^3}{105} \right]
This will lead to
x = \sqrt{ \frac{1 - \sqrt{1 - Y}}{Z} } \approx 0.79
Y and Z are some complicate expression.
Methods using numerical method.
The result obtained is 2.27 or so....
They are different. Has anyone any idea about what is wrong....