Approximate Power-Series Expressions

Broadly Used Mathematical Expressions (shown here without proof)

Binomial

<math>~ 1 ~\pm ~nx + \biggl[\frac{n(n-1)}{2!}\biggr]x^2 ~\pm~ \biggl[\frac{n(n-1)(n-2)}{3!}\biggr]x^3 + \biggl[\frac{n(n-1)(n-2)(n-3)}{4!}\biggr]x^4 ~~\pm ~~ \cdots </math> for <math>~(x^2 < 1)</math>

Exponential

<math>~ 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots </math>

Expressions with Astrophysical Relevance

Polytropic Lane-Emden Function

We seek a power-series expression for the polytropic, Lane-Emden function, <math>~\Theta_\mathrm{H}(\xi)</math> — expanded about the coordinate center, <math>~\xi = 0</math> — that approximately satisfies the Lane-Emden equation,

<math>~\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\Theta_H}{d\xi} \biggr) = - \Theta_H^n</math>

A general power-series should be of the form,

<math>~\Theta_H</math>

<math>~ \theta_0 + a\xi + b\xi^2 + c\xi^3 + d\xi^4 + e\xi^5 + f\xi^6 + g\xi^7 + h\xi^8 + \cdots </math>

First derivative:

<math>~\frac{d\Theta_H}{d\xi}</math>

<math>~ a + 2b\xi + 3c\xi^2 + 4d\xi^3 + 5e\xi^4 + 6f\xi^5 + 7g\xi^6 + 8h\xi^7 + \cdots </math>

Left-hand-side of Lane-Emden equation:

<math>~\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\Theta_H}{d\xi} \biggr)</math>

<math>~ \frac{2a}{\xi} + 2\cdot 3b + 2^2\cdot 3c\xi + 2^2\cdot 5d\xi^2 + 2\cdot 3\cdot 5e\xi^3 + 2\cdot 3\cdot 7f\xi^4 + 2^3\cdot 7g\xi^5 + 2^3\cdot 3^2h\xi^6 + \cdots </math>

Right-hand-side of Lane-Emden equation (adopt the normalization, <math>~\theta_0=1</math>, then use the binomial theorem recursively):

<math>~\Theta_H^n</math>

<math>~ 1 ~+ ~nF + \biggl[\frac{n(n-1)}{2!}\biggr]F^2 ~+~ \biggl[\frac{n(n-1)(n-2)}{3!}\biggr]F^3 + \biggl[\frac{n(n-1)(n-2)(n-3)}{4!}\biggr]F^4 ~~+ ~~ \cdots </math>

where,

<math>~F</math>	<math>~\equiv</math>	<math>~ a\xi + b\xi^2 + c\xi^3 + d\xi^4 + e\xi^5 + f\xi^6 + g\xi^7 + h\xi^8 + \cdots </math>
	<math>~=</math>	<math>~ a\xi\biggl[1 + \frac{b}{a}\xi + \frac{c}{a}\xi^2 + \frac{d}{a}\xi^3 + \frac{e}{a}\xi^4 + \frac{f}{a}\xi^5 + \frac{g}{a}\xi^6 + \frac{h}{a}\xi^7 + \cdots\biggr] \, . </math>

First approximation: Assume that <math>~e=f=g=h=0</math>, in which case the LHS contains terms only up through <math>~\xi^2</math>. This means that we must ignore all terms on the RHS that are of higher order than <math>~\xi^2</math>; that is,

<math>~\Theta_H^n</math>	<math>~\approx</math>	<math>~ 1 ~+ ~nF + \biggl[\frac{n(n-1)}{2!}\biggr]F^2 </math>
	<math>~\approx</math>	<math>~ 1 ~+ ~n(a\xi+b\xi^2) + \biggl[\frac{n(n-1)}{2!}\biggr]a^2\xi^2 </math>
	<math>~\approx</math>	<math>~ 1 ~+~na\xi + ~\biggl[n b + \frac{n(n-1)a^2}{2}\biggr]\xi^2\, . </math>

Expressions for the various coefficients can now be determined by equating terms on the LHS and RHS that have like powers of <math>~\xi</math>. Remembering to include a negative sign on the RHS, we find:

Term	LHS	RHS	Implication
<math>~\xi^{-1}:</math>	<math>~2a</math>	<math>~0</math>	<math>~\Rightarrow ~~~a=0</math>
<math>~\xi^{0}:</math>	<math>~2\cdot 3 b</math>	<math>~-1</math>	<math>~\Rightarrow ~~~b=- \frac{1}{6}</math>
<math>~\xi^{1}:</math>	<math>~2^2\cdot 3 c</math>	<math>~-na</math>	<math>~\Rightarrow ~~~c=0</math>
<math>~\xi^{2}:</math>	<math>~2^2\cdot 5 d</math>	<math>~-\biggl[n b + \frac{n(n-1)a^2}{2}\biggr]</math>	<math>~\Rightarrow ~~~d=+\frac{n}{120}</math>

By including higher and higher order terms in the series expansion for <math>~\Theta_H</math>, and proceeding along the same line of deductive reasoning, one finds:

Expressions for the four coefficients, <math>~a, b, c, d</math>, remain unchanged.
The coefficient is zero for all other terms that contain odd powers of <math>~\xi</math>; specifically, for example, <math>~e = g = 0</math>.
The coefficients of <math>~\xi^6</math> and <math>~\xi^8</math> are, respectively,

<math>~f</math>	<math>~=</math>	<math>~- \frac{n}{378}\biggl(\frac{n}{5}-\frac{1}{8} \biggr) \, ;</math>
<math>~h</math>	<math>~=</math>	<math>~\frac{n(122n^2 -183n + 70)}{3265920} \, .</math>

In summary, the desired, approximate power-series expression for the polytropic Lane-Emden function is:

<math>~\theta</math>

<math>~ 1 - \frac{\xi^2}{6} + \frac{n}{120} \xi^4 - \frac{n}{378} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) \xi^6 + \biggl[ \frac{n(122n^2 -183n + 70)}{3265920} \biggr] \xi^8 + \cdots </math>

Isothermal Lane-Emden Function

We seek a power-series expression for the isothermal, Lane-Emden function, <math>~w(r)</math> — expanded about the coordinate center, <math>~r = 0</math> — that approximately satisfies the isothermal Lane-Emden equation,

A general power-series should be of the form,

<math>~ w_0 + ar + br^2 + cr^3 + dr^4 + er^5 + fr^6 + gr^7 + hr^8 +\cdots </math>

Derivatives:

<math>~\frac{dw}{dr}</math>	<math>~=</math>	<math>~ a + 2br + 3cr^2 + 4dr^3 + 5er^4 + 6fr^5 + 7gr^6 + 8hr^7 +\cdots \, ; </math>
<math>~\frac{d^2w}{dr^2}</math>	<math>~=</math>	<math>~ 2b + 2\cdot 3cr + 2^2\cdot 3dr^2 + 2^2\cdot 5er^3 + 2\cdot 3 \cdot 5fr^4 + 2\cdot 3 \cdot 7gr^5 + 2^3\cdot 7hr^6 +\cdots \, . </math>

Put together, then, the left-hand-side of the isothermal Lane-Emden equation becomes:

<math>~\frac{d^2w}{dr^2} +\frac{2}{r} \frac{d w}{dr} </math>	<math>~=</math>	<math>~ 2b + 2\cdot 3cr + 2^2\cdot 3dr^2 + 2^2\cdot 5er^3 + 2\cdot 3 \cdot 5fr^4 + 2\cdot 3 \cdot 7gr^5 + 2^3\cdot 7hr^6 + \frac{2}{r}\biggl[ a + 2br + 3cr^2 + 4dr^3 + 5er^4 + 6fr^5 + 7gr^6 + 8hr^7 \biggr] + \cdots </math>
	<math>~=</math>	<math>~\frac{2a}{r} + r^0(6b) + r^1(2^2\cdot 3c) + r^2(2^2\cdot 3d + 2^3d) + r^3(2^2\cdot 5e + 2\cdot 5e) + r^4(2\cdot 3\cdot 5 f + 2^2\cdot 3f) + r^5(2\cdot 3\cdot 7 g+ 2\cdot 7g) + r^6(2^3 \cdot 7 h + 2^4 h) + \cdots </math>

Drawing on the above power-series expression for an exponential function, and adopting the convention that <math>~w_0 = 0</math>, the right-hand-side becomes,

<math>~e^{-w}</math>	<math>~=</math>	<math>~ e^{0}\cdot e^{-ar} \cdot e^{-br^2} \cdot e^{-cr^3} \cdot e^{-dr^4} \cdot e^{-er^5} \cdot e^{-fr^6} \cdot e^{-gr^7} \cdot e^{-hr^8} \cdots </math>
	<math>~=</math>	<math>~ \biggl[ 1 -ar + \frac{a^2r^2}{2!} - \frac{a^3r^3}{3!} + \frac{a^4r^4}{4!} - \frac{a^5r^5}{5!} + \frac{a^6r^6}{6!} + \cdots \biggr] </math>
		<math>~ \times \biggl[ 1 -br^2 + \frac{b^2r^4}{2!} - \frac{b^3r^6}{3!} + \cdots \biggr] \times \biggl[ 1 -cr^3 + \frac{c^2r^6}{2!} + \cdots \biggr] \times \biggl[1 - dr^4\biggr] \times \biggl[1 - er^5\biggr]\times \biggl[1 - fr^6\biggr] </math>
	<math>~\approx</math>	<math>~ \biggl[ 1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} + \frac{a^4r^4}{24} - \frac{a^5r^5}{5\cdot 24} + \frac{a^6r^6}{30\cdot 24} \biggr] \times \biggl[ 1 -cr^3 + \frac{c^2r^6}{2} -br^2 + bcr^5 + \frac{b^2r^4}{2} - \frac{b^3r^6}{6} \biggr] \times \biggl[1 - dr^4 - er^5 - fr^6\biggr] </math>
	<math>~\approx</math>	<math>~\biggl\{ \biggl[ 1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} + \frac{a^4r^4}{24} - \frac{a^5r^5}{5\cdot 24} + \frac{a^6r^6}{30\cdot 24} \biggr] - dr^4 \biggl[ 1 -ar + \frac{a^2r^2}{2} \biggr] - er^5 \biggl[ 1 -ar \biggr] - fr^6 \biggr\} </math>
		<math>~ \times \biggl[ 1 -br^2 -cr^3 + \frac{b^2r^4}{2} + bcr^5 + r^6\biggl(\frac{c^2}{2}- \frac{b^3}{6}\biggr) \biggr] </math>
	<math>~\approx</math>	<math>~\biggl[ 1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} + \frac{a^4r^4}{24} - \frac{a^5r^5}{5\cdot 24} + \frac{a^6r^6}{30\cdot 24} - dr^4 + adr^5 - \frac{a^2d r^6}{2} - er^5 + aer^6 - fr^6 \biggr] </math>
		<math>~ \times \biggl[ 1 -br^2 -cr^3 + \frac{b^2r^4}{2} + bcr^5 + r^6\biggl(\frac{c^2}{2}- \frac{b^3}{6}\biggr) \biggr] </math>

Result:

<math>~\frac{r^2}{6} - \frac{r^4}{120} + \frac{r^6}{1890} + \cdots \, .</math>

Displacement Function for Polytropic LAWE

Displacement Function for Isothermal LAWE

User:Tohline/Appendix/Ramblings/PowerSeriesExpressions

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