Appendix/Mathematics/Hypergeometric: Difference between revisions
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Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Hypergeometric Differential Equation= According to §9.151 (p. 1045) of GR, <font color="darkgreen">"… a hypergeometric series is one of the solutions of the differential equation, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>\frac{dH(x)}{dx}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>\delta(x) \, ,</math> </td> </tr> </t..." |
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=Hypergeometric Differential Equation= | =Hypergeometric Differential Equation= | ||
According to §9.151 (p. 1045) of | According to §9.151 (p. 1045) of [http://www.mathtable.com/gr/ Gradshteyn & Ryzhik], <font color="darkgreen">"… a hypergeometric series is one of the solutions of the differential equation, | ||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math> | <math>0</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
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</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>\ | <math> | ||
z(1-z) \frac{d^2u}{dz^2} + [\gamma - (\alpha + \beta + 1)z] \frac{du}{dz} - \alpha \beta u | |||
\, ,</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
which is called the ''hypergeometric equation."</font> | which is called the ''hypergeometric equation.''</font> And, according to §9.10 (p. 1039) of [http://www.mathtable.com/gr/ Gradshteyn & Ryzhik], <font color="darkgreen">"A ''hypergeometric series'' is a series of the form,</font> | ||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>F(\alpha, \beta; \gamma; z)</math> | |||
</td> | |||
<td align="center"> | |||
<math>=</math> | |||
</td> | |||
<td align="left"> | |||
<math> | |||
1 + \biggl[\frac{\alpha \cdot \beta}{\gamma \cdot 1} \biggr]z | |||
+ | |||
\biggl[\frac{ \alpha(\alpha+1)\beta(\beta+1) }{ \gamma(\gamma+1)\cdot 1\cdot 2 }\biggr]z^2 | |||
+ | |||
\biggl[\frac{ \alpha(\alpha+1)(\alpha+2)\beta(\beta+1)(\beta+2) }{ \gamma(\gamma+1)(\gamma+2)\cdot 1\cdot 2 \cdot 3}\biggr]z^3 | |||
+ | |||
\dots | |||
</math> | |||
</td> | |||
</tr> | |||
</table> | |||
Among other attributes, [http://www.mathtable.com/gr/ Gradshteyn & Ryzhik] note that this, <font color="darkgreen">"… series terminates if <math>\alpha</math> or <math>\beta</math> is equal to a negative integer or to zero."</font> | |||
=See Also= | =See Also= | ||
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Revision as of 17:53, 25 October 2022
Hypergeometric Differential Equation
According to §9.151 (p. 1045) of Gradshteyn & Ryzhik, "… a hypergeometric series is one of the solutions of the differential equation,
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which is called the hypergeometric equation. And, according to §9.10 (p. 1039) of Gradshteyn & Ryzhik, "A hypergeometric series is a series of the form,
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Among other attributes, Gradshteyn & Ryzhik note that this, "… series terminates if or is equal to a negative integer or to zero."
See Also
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Appendices: | VisTrailsEquations | VisTrailsVariables | References | Ramblings | VisTrailsImages | myphys.lsu | ADS | |