Difference between revisions of "Bessel at n+1/2 in terms of Bessel polynomial"
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$J_{n +\frac{1}{2}}(r) = (2\pi r)^{-\frac{1}{2}} \left[\dfrac{e^{ir}}{i^{n+1}} y_n \left( -\dfrac{1}{ir} \right) + i^{n+1}e^{-ir}y_n\left( \dfrac{1}{ir} \right) \right],$$ | $$J_{n +\frac{1}{2}}(r) = (2\pi r)^{-\frac{1}{2}} \left[\dfrac{e^{ir}}{i^{n+1}} y_n \left( -\dfrac{1}{ir} \right) + i^{n+1}e^{-ir}y_n\left( \dfrac{1}{ir} \right) \right],$$ | ||
| − | where $J_{n+\frac{1}{2}}$ denotes a [[Bessel | + | where $J_{n+\frac{1}{2}}$ denotes a [[Bessel J]], $\pi$ denotes [[pi]], $i$ denotes the [[imaginary number]], $e^{ir}$ denotes the [[exponential]], and $y_n$ denotes a [[Bessel polynomial]]. |
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
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| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 05:31, 16 September 2016
Theorem
The following formula holds: $$J_{n +\frac{1}{2}}(r) = (2\pi r)^{-\frac{1}{2}} \left[\dfrac{e^{ir}}{i^{n+1}} y_n \left( -\dfrac{1}{ir} \right) + i^{n+1}e^{-ir}y_n\left( \dfrac{1}{ir} \right) \right],$$ where $J_{n+\frac{1}{2}}$ denotes a Bessel J, $\pi$ denotes pi, $i$ denotes the imaginary number, $e^{ir}$ denotes the exponential, and $y_n$ denotes a Bessel polynomial.
