Difference between revisions of "Bessel at n+1/2 in terms of Bessel polynomial"
From specialfunctionswiki
| Line 2: | Line 2: | ||
The following formula holds: | 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 J]] and $y_n$ denotes a [[Bessel polynomial]]. | + | where $J_{n+\frac{1}{2}}$ denotes a [[Bessel J]], $i$ denotes the [[imaginary number]], $e^{ir}$ denotes the [[exponential]], and $y_n$ denotes a [[Bessel polynomial]]. |
==Proof== | ==Proof== | ||
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, $i$ denotes the imaginary number, $e^{ir}$ denotes the exponential, and $y_n$ denotes a Bessel polynomial.
