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

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(Created page with '<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> [[Image:LSU_Key.png|25px|link=http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templat…')
 
 
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[[Image:LSU_Key.png|25px|link=http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#Special_Function_Relationships]]
[[Image:LSU_Key.png|25px|link=http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#Special_Function_Relationships]]
</td>  
</td>  
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<math>~
<math>~
\Gamma(z) ~\Gamma(1-z)
\Gamma(z) ~\Gamma(1-z)
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<math>~
<math>~
\frac{\pi}{\sin(\pi z)} \, .
\frac{\pi}{\sin(\pi z)}  
</math>
</math>
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  <td align="center" rowspan="1"><math>~\biggl|</math></td>
  <td align="center" rowspan="3">
for example, if<br />
<math>~z \rightarrow  (m-n + \tfrac{1}{2})</math>
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<math>~\Rightarrow ~~~\Gamma(m-n+\tfrac{1}{2})~\Gamma(n-m+\tfrac{1}{2})</math>
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<math>~=</math>
  </td>
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<math>~\pi \biggl\{\sin\biggl[ \frac{\pi}{2} + \pi(m-n) \biggr]  \biggr\}^{-1}</math>
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[https://dlmf.nist.gov/5.5#ii DLMF]:&nbsp; &sect;5.5(ii)  
&nbsp;
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<math>~=</math>
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<math>~\pi (-1)^{m-n} </math>
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   </td>
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[https://dlmf.nist.gov/5.5#ii DLMF &sect;5.5(ii)]
  </td>
  <td align="center" rowspan="1"><math>~\biggl|</math></td>
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<tr>
<tr>
   <td align="right">Valid for: &nbsp; &nbsp;</td>
   <td align="right" colspan="2">Valid for: </td>
   <td align="left">
   <td align="left" colspan="3">
<math>~z \ne0, \pm 1, \pm 2, </math> &hellip;
&nbsp; &nbsp;<math>~z \ne0, \pm 1, \pm 2, </math> &hellip;
   </td>
   </td>
  <td align="center" rowspan="1"><math>~\biggl|</math></td>
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Latest revision as of 00:08, 15 June 2018

LSU Key.png

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

<math>~=</math>

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

<math>~\biggl|</math>

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

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

<math>~=</math>

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

 

<math>~=</math>

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

DLMF §5.5(ii)

<math>~\biggl|</math>
Valid for:

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

<math>~\biggl|</math>
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