MHB Differentiating Bessel functions

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The discussion focuses on differentiating the expression $x^{1/2}\left[c_1J_{1/4}(x^2/2)+c_2J_{-1/4}(x^2/2)\right]$. A general formula for the derivative of Bessel functions is provided: $J_{\nu}^{'}(x)= \frac{1}{2}\{J_{\nu-1}(x)-J_{\nu+1}(x)\}$. Participants are likely exploring the application of this formula to the given expression. The differentiation process is essential for understanding the behavior of the Bessel functions involved. The conversation emphasizes the mathematical techniques required for such differentiation tasks.
alexmahone
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Differentiate $x^{1/2}\left[c_1J_{1/4}(x^2/2)+c_2J_{-1/4}(x^2/2)\right]$.
 
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Alexmahone said:
Differentiate $x^{1/2}\left[c_1J_{1/4}(x^2/2)+c_2J_{-1/4}(x^2/2)\right]$.

A general formula is...

$\displaystyle J_{\nu}^{'}(x)= \frac{1}{2}\ \{J_{\nu-1}(x)-J_{\nu+1}(x)\}$ (1)

Kind regards

$\chi$ $\sigma$
 

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