Difference between revisions of "Bessel polynomial in terms of Bessel functions"
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| − | <strong>Theorem:</strong> The following formula holds: | + | <strong>[[Bessel polynomial in terms of Bessel functions|Theorem]]:</strong> The following formula holds: |
$$y_n\left( \dfrac{1}{ir} \right) = \left(\dfrac{\pi r}{2} \right)^{\frac{1}{2}} e^{ir} \left[ \dfrac{J_{n +\frac{1}{2}}(r)}{i^{n+1}}+i^nJ_{-n-\frac{1}{2}}(r) \right],$$ | $$y_n\left( \dfrac{1}{ir} \right) = \left(\dfrac{\pi r}{2} \right)^{\frac{1}{2}} e^{ir} \left[ \dfrac{J_{n +\frac{1}{2}}(r)}{i^{n+1}}+i^nJ_{-n-\frac{1}{2}}(r) \right],$$ | ||
where $y_n$ denotes a [[Bessel polynomial]] and $J_{\nu}$ denotes a [[Bessel function]]. | where $y_n$ denotes a [[Bessel polynomial]] and $J_{\nu}$ denotes a [[Bessel function]]. | ||
Revision as of 10:20, 23 March 2015
Theorem: The following formula holds: $$y_n\left( \dfrac{1}{ir} \right) = \left(\dfrac{\pi r}{2} \right)^{\frac{1}{2}} e^{ir} \left[ \dfrac{J_{n +\frac{1}{2}}(r)}{i^{n+1}}+i^nJ_{-n-\frac{1}{2}}(r) \right],$$ where $y_n$ denotes a Bessel polynomial and $J_{\nu}$ denotes a Bessel function.
Proof: █
