Difference between revisions of "User:Tohline/Math/EQ Gamma01"

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\frac{\pi}{\sin(\pi z)} \, .
\frac{\pi}{\sin(\pi z)} \, .
</math>
</math>
  </td>
  <td align="center" rowspan="3">&nbsp; &nbsp; &nbsp; </td>
  <td align="center" rowspan="3">
for example, if<br />
<math>~z \rightarrow  (m-n + \tfrac{1}{2})</math>
  </td>
  <td align="center" rowspan="3">&nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math>&nbsp; &nbsp; </td>
  <td align="center" rowspan="3">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Gamma(m-n+\tfrac{1}{2})~\Gamma(n-m+\tfrac{1}{2})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi \biggl\{\sin[\pi(m-n+\tfrac{1}{2})]  \biggr\}^{-1}</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi (-1)^{(m-n)} \, .</math>
  </td>
</tr>
</table>
   </td>
   </td>
</tr>
</tr>

Revision as of 21:22, 14 June 2018

LSU Key.png

<math>~ \Gamma(z) ~\Gamma(1-z) </math>

<math>~=</math>

<math>~ \frac{\pi}{\sin(\pi z)} \, . </math>

     

for example, if
<math>~z \rightarrow (m-n + \tfrac{1}{2})</math>

      <math>~\Rightarrow</math>   

<math>~\Gamma(m-n+\tfrac{1}{2})~\Gamma(n-m+\tfrac{1}{2})</math>

<math>~=</math>

<math>~\pi \biggl\{\sin[\pi(m-n+\tfrac{1}{2})] \biggr\}^{-1}</math>

 

<math>~=</math>

<math>~\pi (-1)^{(m-n)} \, .</math>

DLMF §5.5(ii)

Valid for:

   <math>~z \ne0, \pm 1, \pm 2, </math> …

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