User:Tohline/Appendix/Ramblings/PowerSeriesExpressions - Revision history

Tohline: /* Binomial */

2020-07-11T20:29:04Z

Binomial

← Older revision		Revision as of 20:29, 11 July 2020
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~1 - 3x + 6x^2 - 10x^3 + 15x^4 - 21x^5 + \cdots \, .</math>		<math>~1 - 3x + 6x^2 - 10x^3 + 15x^4 - 21x^5 + \cdots \, ;</math>
	</td>		</td>
	</tr>		</tr>
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	<td align="left">		<td align="left">
	<math>~1 - 4x + 10 x^2 - 20x^3		<math>~1 - 4x + 10 x^2 - 20x^3
	+ 35x^4 - 56x^5 + \cdots </math>		+ 35x^4 - 56x^5 + \cdots \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Binomial */

2020-07-11T20:28:11Z

Binomial

← Older revision		Revision as of 20:28, 11 July 2020
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	<td align="left">		<td align="left">
	<math>~1 - 3x + 6x^2 - 10x^3 + 15x^4 - 21x^5 + \cdots \, .</math>		<math>~1 - 3x + 6x^2 - 10x^3 + 15x^4 - 21x^5 + \cdots \, .</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~(1+x)^{-4}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~1 - 4x + \biggl[ \frac{4\cdot 5 }{ 2} \biggr]x^2 - \biggl[ \frac{ 4\cdot 5 \cdot 6}{ 2\cdot 3} \biggr]x^3
			+ \biggl[ \frac{4\cdot 5 \cdot 6 \cdot 7 }{2\cdot 3 \cdot 4 } \biggr]x^4 - \biggl[ \frac{4\cdot 5 \cdot 6 \cdot 7 \cdot 8}{2\cdot 3 \cdot 4 \cdot 5 } \biggr]x^5 + \cdots </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~1 - 4x + 10 x^2 - 20x^3
			+ 35x^4 - 56x^5 + \cdots </math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Binomial */

2018-10-04T15:46:52Z

Binomial

← Older revision		Revision as of 15:46, 4 October 2018
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~1 - 3x + 6x^2 - 10x^3		<math>~1 - 3x + 6x^2 - 10x^3 + 15x^4 - 21x^5 + \cdots \, .</math>
	+ 15x^4 - 21x^5 + \cdots \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Binomial */

2018-10-04T15:16:51Z

Binomial

← Older revision		Revision as of 15:16, 4 October 2018
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	</tr>		</tr>
	</table>		</table>


			Note, for example, that,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~(1+x)^{-1}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~1 - x +x^2 - x^3 + x^4 - x^5 + \cdots \, ;</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~(1+x)^{-2}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~1 - 2x + 3x^2 - 4x^3 + 5x^4 - 6x^5 + \cdots \, ;</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~(1+x)^{-3}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~1 - 3x + \biggl[ \frac{3\cdot 4 }{ 2} \biggr]x^2 - \biggl[ \frac{ 3\cdot 4 \cdot 5}{ 2\cdot 3} \biggr]x^3
			+ \biggl[ \frac{3\cdot 4 \cdot 5 \cdot 6 }{2\cdot 3 \cdot 4 } \biggr]x^4 - \biggl[ \frac{3\cdot 4 \cdot 5 \cdot 6 \cdot 7}{2\cdot 3 \cdot 4 \cdot 5 } \biggr]x^5 + \cdots </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~1 - 3x + 6x^2 - 10x^3
			+ 15x^4 - 21x^5 + \cdots \, .</math>
			</td>
			</tr>
			</table>


	See also:		See also:

Tohline: /* Binomial */

2018-03-18T00:28:21Z

Binomial

← Older revision		Revision as of 00:28, 18 March 2018
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	----		----

	'''Example:'''  Setting <math>~a = 1</math> gives,		'''Our Example:'''  Setting <math>~a = 1</math> gives,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	<td align="left">		<td align="left">
	<math>~		<math>~
	1+\binom{n}{1}b+\binom{n}{2}b^{2}~~+\dots~~+\binom{n}{~~n-1~~}b^{n-1}+b^{n}		1+\binom{n}{1}b+\binom{n}{2}b^{2}+\binom{n}{3}b^{3}+\binom{n}{4}b^{4}+\dots
	</math>		</math>
	</td>		</td>
Line 105:		Line 105:
	<td align="left">		<td align="left">
	<math>~		<math>~
	1+\~~binom~~{n}{1}b+\~~binom~~{n}{2}b^{2}+\dots+\~~binom~~{n}{n-1}b^{n-1}+b^{n}		1+\frac{n!}{(n-1)!}~b + \frac{n!}{(n-2)! 2!}~b^{2} + \frac{n!}{(n-3)! 3!}~b^{3} + \frac{n!}{(n-4)! 4!}~b^{4} + \dots
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			1+ nb + \biggl[ \frac{n(n-1)}{2!}\biggr] b^{2} + \biggl[ \frac{n(n-1)(n-2)}{3!} \biggr] b^{3} + \biggl[ \frac{n(n-1)(n-2)(n-3)}{4!} \biggr] b^{4} + \dots
	</math>		</math>
	</td>		</td>

Tohline: /* Binomial */

2018-03-17T23:13:03Z

Binomial

← Older revision		Revision as of 23:13, 17 March 2018
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	<td align="left">		<td align="left">
	<math>~		<math>~
	a^{n}+\~~genfrac{(}{)}{0.0pt}{}~~{n}{1}a^{n-1}b+\~~genfrac{(}{)}{0.0pt}{}~~{n}{2}a^{n-2}b^{2}+\dots+\~~genfrac{(}{)}{0.0pt}{}~~{n}{n-1}ab^{n-1}+b^{n},		a^{n}+\binom{n}{1}a^{n-1}b+\binom{n}{2}a^{n-2}b^{2}+\dots+\binom{n}{n-1}ab^{n-1}+b^{n},
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	where,		where, for nonnegative integer values of <math>~k</math> and <math>~n</math> and <math>~k \le n</math>, the notation,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\~~genfrac{(}{)}{0.0pt}{}~~{n}{k}</math>		<math>~\binom{n}{k}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{n!}{(n-k)!k!}=\~~genfrac~~{(}{)}{~~0.0pt~~}{}{n}{n%		\frac{n!}{(n-k)!k!}=\binom{n}{n-k}.
	-k}.		</math>
			</td>
			</tr>
			</table>

			----

			'''Example:'''  Setting <math>~a = 1</math> gives,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~(1+b)^{n}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			1+\binom{n}{1}b+\binom{n}{2}b^{2}+\dots+\binom{n}{n-1}b^{n-1}+b^{n}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			1+\binom{n}{1}b+\binom{n}{2}b^{2}+\dots+\binom{n}{n-1}b^{n-1}+b^{n}
	</math>		</math>
	</td>		</td>

Tohline: /* Binomial */

2018-03-17T22:46:33Z

Binomial

← Older revision		Revision as of 22:46, 17 March 2018
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	</table>		</table>
	</div>		</div>

			<table border="1" align="center" cellpadding="8" width="70%">
			<tr>
			<th align="center" bgcolor="yellow">
			LaTeX mathematical expressions cut-and-pasted directly from
			<br />
			NIST's ''Digital Library of Mathematical Functions''
			</th>
			</tr>
			<tr>
			<td align="left">
			As a primary point of reference, note that according to [http://dlmf.nist.gov/1.2 §1.2 of NIST's ''Digital Library of Mathematical Functions''], the binomial theorem states that,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~(a+b)^{n}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			a^{n}+\genfrac{(}{)}{0.0pt}{}{n}{1}a^{n-1}b+\genfrac{(}{)}{0.0pt}{}{n}{2}a^{n-2}b^{2}+\dots+\genfrac{(}{)}{0.0pt}{}{n}{n-1}ab^{n-1}+b^{n},
			</math>
			</td>
			</tr>
			</table>
			where,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\genfrac{(}{)}{0.0pt}{}{n}{k}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{n!}{(n-k)!k!}=\genfrac{(}{)}{0.0pt}{}{n}{n%
			-k}.
			</math>
			</td>
			</tr>
			</table>

			</td>
			</tr>
			</table>

	See also:		See also:

Tohline: /* Isothermal Lane-Emden Function */

2017-08-09T22:07:08Z

Isothermal Lane-Emden Function

← Older revision		Revision as of 22:07, 9 August 2017
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	Result:		Result:
	<div align="center" id="IsothermalLaneEmden">		<div align="center" id="IsothermalLaneEmden">
	<table border="1" width="80%" cellpadding="8" align="center"><tr><td align="center">		<table border="1" width="80%" cellpadding="8" align="center">
			<tr><th align="center">For Spherically Symmetric Configurations</th></tr>
			<tr><td align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Line 870:		Line 872:
	See also:		See also:
	* Equation (377) from §22 in Chapter IV of [[User:Tohline/Appendix/References#C67\|C67]]).		* Equation (377) from §22 in Chapter IV of [[User:Tohline/Appendix/References#C67\|C67]]).


			NOTE:  For cylindrically symmetric, rather than spherically symmetric, configurations, an analytic expression for the function, <math>~w(r)</math>, is presented as equation (56) in a paper by [http://adsabs.harvard.edu/abs/1964ApJ...140.1056O J. P. Ostriker (1964, ApJ, 140, 1056)] titled, ''The Equilibrium of Polytropic and Isothermal Cylinders''.

	===Displacement Function for Polytropic LAWE===		===Displacement Function for Polytropic LAWE===

Tohline: /* Polytropic Lane-Emden Function */

2017-08-09T22:02:04Z

Polytropic Lane-Emden Function

← Older revision		Revision as of 22:02, 9 August 2017
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	In summary, the desired, approximate power-series expression for the polytropic Lane-Emden function is:		In summary, the desired, approximate power-series expression for the polytropic Lane-Emden function is:
	<div align="center" id="PolytropicLaneEmden">		<div align="center" id="PolytropicLaneEmden">
	<table border="1" width="80%" cellpadding="8" align="center"><tr><td align="center">		<table border="1" width="80%" cellpadding="8" align="center">
			<tr><th align="center">For Spherically Symmetric Configurations</th></tr>
			<tr><td align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</td></tr></table>		</td></tr></table>
	</div>		</div>

			NOTE:  For cylindrically symmetric, rather than spherically symmetric, configurations, the analogous power-series expression appears as equation (15) in the article by [http://adsabs.harvard.edu/abs/1964ApJ...140.1056O J. P. Ostriker (1964, ApJ, 140, 1056)] titled, ''The Equilibrium of Polytropic and Isothermal Cylinders''.

	===Isothermal Lane-Emden Function===		===Isothermal Lane-Emden Function===

Tohline: /* Taylor Series (Hunter77) */

2017-07-18T19:42:36Z

Taylor Series (Hunter77)

Show changes