Differentiating Bessel functions

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SUMMARY

The discussion focuses on differentiating the expression $x^{1/2}\left[c_1J_{1/4}(x^2/2)+c_2J_{-1/4}(x^2/2)\right]$, where $J_{\nu}(x)$ represents Bessel functions. The general formula for the derivative of Bessel functions is provided as $J_{\nu}^{'}(x)= \frac{1}{2}\ \{J_{\nu-1}(x)-J_{\nu+1}(x)\}$. This formula is crucial for calculating the derivative of the given expression accurately.

PREREQUISITES
  • Understanding of Bessel functions, specifically $J_{\nu}(x)$
  • Knowledge of differentiation techniques in calculus
  • Familiarity with mathematical notation and expressions
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the properties and applications of Bessel functions in mathematical physics
  • Learn advanced differentiation techniques for special functions
  • Explore the derivation and proof of the Bessel function derivative formula
  • Investigate numerical methods for evaluating Bessel functions
USEFUL FOR

Mathematicians, physicists, and engineering students who require a deeper understanding of Bessel functions and their derivatives for applications in wave equations and signal processing.

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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