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: | ||
$$J_{-n-\frac{1}{2}}(r) = (2 \pi r)^{-\frac{1}{2}} \left[ i^n e^{ir} y_n \left( -\dfrac{1}{ir} \right)+ \dfrac{e^{-ir}}{i^n} y_n\left( \dfrac{1}{ir} \right) \right],$$ | $$J_{-n-\frac{1}{2}}(r) = (2 \pi r)^{-\frac{1}{2}} \left[ i^n e^{ir} y_n \left( -\dfrac{1}{ir} \right)+ \dfrac{e^{-ir}}{i^n} y_n\left( \dfrac{1}{ir} \right) \right],$$ | ||
| − | where $J_{-n-\frac{1}{2}}$ denotes a [[Bessel J | + | where $J_{-n-\frac{1}{2}}$ denotes a [[Bessel J|Bessel function of the first kind]] and $y_n$ denotes a [[Bessel polynomial]]. |
==Proof== | ==Proof== | ||
==References== | ==References== | ||
Revision as of 20:12, 9 June 2016
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.
