Difference between revisions of "Bessel at -n-1/2 in terms of Bessel polynomial"
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<strong>[[Bessel at -n-1/2 in terms of Bessel polynomial|Theorem]]:</strong> The following formula holds: | <strong>[[Bessel at -n-1/2 in terms of Bessel polynomial|Theorem]]:</strong> The following formula holds: | ||
$$J_{-n-\frac{1}{2}}(r) = (2 \pi r)^{-\frac{1}{2}} \left[ i^n e^{ir} y_n \left( -\dfrac{1}{ir} \right)+ \dfrac{e^{-ir}}{i^n} y_n\left( \dfrac{1}{ir} \right) \right],$$ | $$J_{-n-\frac{1}{2}}(r) = (2 \pi r)^{-\frac{1}{2}} \left[ i^n e^{ir} y_n \left( -\dfrac{1}{ir} \right)+ \dfrac{e^{-ir}}{i^n} y_n\left( \dfrac{1}{ir} \right) \right],$$ | ||
| − | where $J_{-n-\frac{1}{2}}$ denotes a [[Bessel function]] and $y_n$ denotes a [[Bessel polynomial]]. | + | where $J_{-n-\frac{1}{2}}$ denotes a [[Bessel J sub nu|Bessel function of the first kind]] and $y_n$ denotes a [[Bessel polynomial]]. |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
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</div> | </div> | ||
Revision as of 05:59, 10 June 2015
Theorem: The following formula holds: $$J_{-n-\frac{1}{2}}(r) = (2 \pi r)^{-\frac{1}{2}} \left[ i^n e^{ir} y_n \left( -\dfrac{1}{ir} \right)+ \dfrac{e^{-ir}}{i^n} y_n\left( \dfrac{1}{ir} \right) \right],$$ where $J_{-n-\frac{1}{2}}$ denotes a Bessel function of the first kind and $y_n$ denotes a Bessel polynomial.
Proof: █
