Difference between revisions of "Bessel Y"
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{{:Bessel J sub nu and Y sub nu solve Bessel's differential equation}} | {{:Bessel J sub nu and Y sub nu solve Bessel's differential equation}} | ||
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+ | <div class="toccolours mw-collapsible mw-collapsed"> | ||
+ | <strong>Theorem:</strong> The following formula holds for $n\in\mathbb{Z}$: | ||
+ | $${\small Y_n(z)=\dfrac{2}{\pi} \left[ \log \left( \dfrac{z}{2} \right)+\gamma-\dfrac{1}{2}\displaystyle\sum_{k=1}^n \dfrac{1}{k} \right]J_n(x) - \dfrac{1}{\pi} \displaystyle\sum_{k=0}^{\infty} (-1)^k \dfrac{1}{k!(n+k)!} \left(\dfrac{z}{2}\right)^{n+2k}\displaystyle\sum_{j=1}^k \left( \dfrac{1}{j} + \dfrac{1}{j+n} \right) - \dfrac{1}{\pi}\displaystyle\sum_{k=0}^{n-1} \dfrac{(n-k-1)!}{k!} \left( \dfrac{z}{2} \right)^{-n+2k},}$$ | ||
+ | where $Y_n$ denotes the [[Bessel Y sub nu|Bessel function of the second kind]], $\log$ denotes the [[logarithm]], $\gamma$ denotes the [[Euler-Mascheroni constant]], and $J_n$ denotes the [[Bessel J sub nu|Bessel function of the first kind]]. | ||
+ | <div class="mw-collapsible-content"> | ||
+ | <strong>Proof:</strong> █ | ||
+ | </div> | ||
+ | </div> | ||
<center>{{:Bessel functions footer}}</center> | <center>{{:Bessel functions footer}}</center> |
Revision as of 06:14, 10 June 2015
Bessel functions (of the second kind) $Y_{\nu}$ are defined via the formula $$Y_{\nu}(z)=\dfrac{J_{\nu}(z)\cos(\nu \pi)-J_{-\nu}(z)}{\sin(\nu \pi)}.$$
Domain coloring of analytic continuation of $Y_0(z)$.
Bessel functions from Abramowitz&Stegun
Properties
Theorem: The following formula holds for $n\in\mathbb{Z}$: $${\small Y_n(z)=\dfrac{2}{\pi} \left[ \log \left( \dfrac{z}{2} \right)+\gamma-\dfrac{1}{2}\displaystyle\sum_{k=1}^n \dfrac{1}{k} \right]J_n(x) - \dfrac{1}{\pi} \displaystyle\sum_{k=0}^{\infty} (-1)^k \dfrac{1}{k!(n+k)!} \left(\dfrac{z}{2}\right)^{n+2k}\displaystyle\sum_{j=1}^k \left( \dfrac{1}{j} + \dfrac{1}{j+n} \right) - \dfrac{1}{\pi}\displaystyle\sum_{k=0}^{n-1} \dfrac{(n-k-1)!}{k!} \left( \dfrac{z}{2} \right)^{-n+2k},}$$ where $Y_n$ denotes the Bessel function of the second kind, $\log$ denotes the logarithm, $\gamma$ denotes the Euler-Mascheroni constant, and $J_n$ denotes the Bessel function of the first kind.
Proof: █