Appendix/Ramblings/PowerSeriesExpressions - Revision history

Joel2: /* Displacement Function for Isothermal LAWE */

2026-09-17T17:54:37Z

Displacement Function for Isothermal LAWE

@@ Line 2,251: / Line 2,251: @@
+-
 \frac{\xi^8}{2^2 \cdot 3^2 \cdot 120}
+-
 \biggl[

2026-09-15T23:10:06Z

Displacement Function for Isothermal LAWE

@@ Line 2,333: / Line 2,333: @@
    <td align="left">
 <math>
-+ \frac{1}{3!}\biggl[ \psi^3 \biggr]
++ \frac{1}{3!}\biggl[ \frac{\xi^6}{2^3 \cdot 3^3}
-+ \frac{1}{4!}\biggl[ \psi^4 \biggr]
 +\mathcal{O}(\xi^{10})
 </math>
    </td>

2026-09-15T22:48:05Z

Displacement Function for Isothermal LAWE

@@ Line 2,301: / Line 2,301: @@
 <math>~
 + \psi + \frac{\psi^2}{2!} + \frac{\psi^3}{3!} + \frac{\psi^4}{4!} + \cdots
 </math>
    </td>

2026-09-15T22:35:52Z

Displacement Function for Isothermal LAWE

@@ Line 2,093: / Line 2,093: @@
 </td></tr></table>
 </div>
 <table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left">
@@ Line 2,226: / Line 2,224: @@
 </table>
-and that the [[Appendix/Ramblings/PowerSeriesExpressions#Exponential|exponential function gives]],
+and that,
 <table border="0" align="center" cellpadding="5">

2026-09-15T21:53:56Z

Displacement Function for Isothermal LAWE

@@ Line 2,207: / Line 2,207: @@
 + \frac{\xi^8}{2\cdot 3^4\cdot 5 \cdot 7}
 + \frac{\xi^8}{2^6\cdot 3^2\cdot 5^2}
 +\mathcal{O}(\xi^{10})
 </math>

2026-09-15T21:37:19Z

Displacement Function for Isothermal LAWE

← Older revision		Revision as of 17:37, 15 September 2026
Line 2,056:		Line 2,056:
	In summary, the desired, approximate power-series expression for the isothermal displacement function is:		In summary, the desired, approximate power-series expression for the isothermal displacement function is:
	<div align="center" id="IsothermalDisplacement">		<div align="center" id="IsothermalDisplacement">
	<table border="1" width="80%" cellpadding="8" align="center"><tr><td align="~~center~~">		<table border="1" width="80%" cellpadding="8" align="center"><tr><td align="left">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Line 2,073:		Line 2,073:
	</tr>		</tr>
	</table>		</table>
			Then, in the particular case where <math>\gamma = 1</math> and <math>\sigma_c^2=0</math> — which implies that <math>\alpha = (3-4/\gamma)=-1</math> and <math>\mathfrak{F} = (\sigma_c^2/\gamma - 2\alpha) = +2</math> — we have,
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>x_Y(\xi)</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			1 - \frac{\xi^2}{30} + \frac{\xi^4}{1260} + \cdots
			</math>
			</td>

			</tr>
			</table>

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

2026-09-15T21:05:03Z

Exponential

← Older revision		Revision as of 17:05, 15 September 2026
Line 223:		Line 223:
	<td align="left">		<td align="left">
	<math>~		<math>~
	1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots		1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \frac{x^6}{6!} \cdots
	</math>		</math>
	</td>		</td>

2026-09-15T17:50:20Z

Displacement Function for Isothermal LAWE

@@ Line 2,174: / Line 2,174: @@
 -\frac{\xi^6}{2^4\cdot 3^2\cdot 5} + \frac{\xi^8}{2^6\cdot 3^2\cdot 5^2}
 + \frac{\xi^8}{2^2\cdot 3^4\cdot 5 \cdot 7}
 +\mathcal{O}(\xi^{10})
 </math>

2026-09-15T17:45:41Z

Displacement Function for Isothermal LAWE

@@ Line 2,147: / Line 2,147: @@
 + \biggl[ \frac{\xi^6}{1890} \biggr]
 \times \biggl[ \frac{\xi^2}{6} \biggr]
 +\mathcal{O}(\xi^{10})
 </math>

2026-09-15T17:18:28Z

Displacement Function for Isothermal LAWE

← Older revision		Revision as of 13:18, 15 September 2026
Line 2,128:		Line 2,128:
	<td align="left">		<td align="left">
	<math>		<math>
	\biggl[ \frac{\xi^2}{6} - \frac{\xi^4}{120} + \frac{\xi^6}{1890} - \frac{61 \xi^8}{1,632,960} ~~+ \cdots~~ \biggr]		\biggl[ \frac{\xi^2}{6} - \frac{\xi^4}{120} + \frac{\xi^6}{1890} - \frac{61 \xi^8}{1,632,960} \biggr]
	\times \biggl[ \frac{\xi^2}{6} - \frac{\xi^4}{120} + \frac{\xi^6}{1890} - \frac{61 \xi^8}{1,632,960} + \~~cdots~~ \~~biggr]~~		\times \biggl[ \frac{\xi^2}{6} - \frac{\xi^4}{120} + \frac{\xi^6}{1890} - \frac{61 \xi^8}{1,632,960} \biggr]
			+ \mathcal{O}(\xi^{10})
	</math>		</math>
	</td>		</td>
Line 2,140:		Line 2,141:
	<math>		<math>
	\biggl[ \frac{\xi^2}{6} \biggr]		\biggl[ \frac{\xi^2}{6} \biggr]
	\times \biggl[ \frac{\xi^2}{6} - \frac{\xi^4}{120} + \frac{\xi^6}{1890} ~~- \frac{61 \xi^8}{1,632,960} + \cdots~~ \biggr]		\times \biggl[ \frac{\xi^2}{6} - \frac{\xi^4}{120} + \frac{\xi^6}{1890} \biggr]
	+		+
	\biggl[ - \frac{\xi^4}{120} \biggr]		\biggl[ - \frac{\xi^4}{120} \biggr]
	\times \biggl[ \frac{\xi^2}{6} - \frac{\xi^4}{120} + \frac{\xi^6}{1890} - \frac{61 \xi^8}{~~1,632,960~~} ~~+ \cdots~~ \biggr]		\times \biggl[ \frac{\xi^2}{6} - \frac{\xi^4}{120} \biggr]
			+ \biggl[ \frac{\xi^6}{1890} \biggr]
			\times \biggl[ \frac{\xi^2}{6} \biggr]
			+\mathcal{O}(\xi^{10})
	</math>		</math>
	</td>		</td>