User:Tohline/Appendix/Ramblings/PowerSeriesExpressions
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Approximate PowerSeries Expressions
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Broadly Used Mathematical Expressions (shown here without proof)
Binomial


for 
LaTeX mathematical expressions cutandpasted directly from



As a primary point of reference, note that according to §1.2 of NIST's Digital Library of Mathematical Functions, the binomial theorem states that,
where, for nonnegative integer values of and and , the notation,
Our Example: Setting gives,

Note, for example, that,












See also:
Exponential



Expressions with Astrophysical Relevance
Polytropic LaneEmden Function
We seek a powerseries expression for the polytropic, LaneEmden function, — expanded about the coordinate center, — that approximately satisfies the LaneEmden equation,
A general powerseries should be of the form,



First derivative:



Lefthandside of LaneEmden equation:



Righthandside of LaneEmden equation (adopt the normalization, , then use the binomial theorem recursively):



where,






First approximation: Assume that , in which case the LHS contains terms only up through . This means that we must ignore all terms on the RHS that are of higher order than ; that is,









Expressions for the various coefficients can now be determined by equating terms on the LHS and RHS that have like powers of . Remembering to include a negative sign on the RHS, we find:
Term  LHS  RHS  Implication 
















By including higher and higher order terms in the series expansion for , and proceeding along the same line of deductive reasoning, one finds:
 Expressions for the four coefficients, , remain unchanged.
 The coefficient is zero for all other terms that contain odd powers of ; specifically, for example, .
 The coefficients of and are, respectively,






In summary, the desired, approximate powerseries expression for the polytropic LaneEmden function is:
For Spherically Symmetric Configurations  


NOTE: For cylindrically symmetric, rather than spherically symmetric, configurations, the analogous powerseries expression appears as equation (15) in the article by J. P. Ostriker (1964, ApJ, 140, 1056) titled, The Equilibrium of Polytropic and Isothermal Cylinders.
Isothermal LaneEmden Function
Here we seek a powerseries expression for the isothermal, LaneEmden function — expanded about the coordinate center — that approximately satisfies the isothermal LaneEmden equation; making the variable substitution (sorry for the unnecessary complication!), , the governing ODE is,



A general powerseries should be of the form,



Derivatives:






Put together, then, the lefthandside of the isothermal LaneEmden equation becomes:






Drawing on the above powerseries expression for an exponential function, and adopting the convention that , the righthandside becomes,

































Expressions for the various coefficients can now be determined by equating terms on the LHS and RHS that have like powers of . Beginning with the highest order terms, we initially find,
Term  LHS  RHS  Implication 
















With this initial set of coefficient values in hand, we can rewrite (and significantly simplify) our approximate expression for the RHS, namely,






Continuing, then, with equating terms with like powers on both sides of the equation, we find,
Term  LHS  RHS  Implication 
















Result:
For Spherically Symmetric Configurations  


See also:
 Equation (377) from §22 in Chapter IV of C67).
NOTE: For cylindrically symmetric, rather than spherically symmetric, configurations, an analytic expression for the function, , is presented as equation (56) in a paper by J. P. Ostriker (1964, ApJ, 140, 1056) titled, The Equilibrium of Polytropic and Isothermal Cylinders.
Displacement Function for Polytropic LAWE
The LAWE for polytropic spheres may be written as,






where, is the polytropic LaneEmden function describing the configuration's unperturbed radial density distribution, and , , and are constants. Here we seek a powerseries expression for the displacement function, , expanded about the center of the configuration, that approximately satisfies this LAWE.
First we note that, near the center, an accurate powerseries expression for the polytropic LaneEmden function is,



Hence,



Therefore, near the center of the configuration, the LAWE may be written as,









where, for present purposes, we have kept terms in the series no higher than . Let's now adopt a powerseries expression for the displacement function of the form,






and,



Substituting these expressions into the LAWE gives,



Expressions for the various coefficients can now be determined by equating terms on the LHS and RHS that have like powers of .
Term  LHS  RHS  Implication 
















In summary, the desired, approximate powerseries expression for the polytropic displacement function is:

Displacement Function for Isothermal LAWE
The LAWE for isothermal spheres may be written as,



where, is the isothermal LaneEmden function describing the configuration's unperturbed radial density distribution, and , , and are constants. Here we seek a powerseries expression for the displacement function, , expanded about the center of the configuration, that approximately satisfies this LAWE.
First we note that, near the center, an accurate powerseries expression for the isothermal LaneEmden function is,



Hence,



Therefore, near the center of the configuration, the LAWE may be written as,



Let's now adopt a powerseries expression for the displacement function of the form,






and,



Substituting these expressions into the LAWE gives,



Keeping terms only up through leads to the following simplification:



where,
Finally, balancing terms of like powers on both sides of the equation leads us to conclude the following:
Term  LHS  RHS  Implication 
















In summary, the desired, approximate powerseries expression for the isothermal displacement function is:

Taylor Series (Hunter77)
First (Unsuccessful) Try
First:









Note that, replacing the term with the expression derived in the Second step, below, gives,


















Then, replacing the term with the expression derived in the Third step, below, gives,


















Second:
























Now, replacing the term with the expression derived in the Third step, below, gives,


















Third:




































And, finally:




































Result:
Definitely WRONG! 



When I used an Excel spreadsheet to test this out against a parabola, the integration quickly became wildly unstable, strongly suggesting that there is an error in the derivation. My first attempt to uncover this error produced a new coefficient on the , namely,
Somewhat Improved 



Although it showed improvement, this expression still blows up. So I have not bothered to revise the original (definitely WRONG!) derivation. Instead, let's start all over and approach it with a more gradual derivation.
Second Try
We will work from the following foundation expression in which is the variable that we desire to evaluate, and the "known" quantities are: , , , , and .



Let's use similar Taylorseries expansions for , , etc. in order to eliminate the term, the term, etc.









First:













This expression works very well for a parabola.
Second:


















This also allows us to improve the expression for the term, as initially derived in the "First" subsection, above. Namely,









Hence, an improved expression for is,













Third:






























Hence,


















And,















Finally, then:














































© 2014  2020 by Joel E. Tohline 