Perturbations`LaplaceCoefficients`

Laplace introduced the following expansion:

(((1 + (α^2) - 2 α cos (x)))^(-s)) = (1/2) (∑_(j = -∞)^(+∞))((Subscript[b, s])^((j))) (α) cos (j x)
\(\(\(\(b\_\(s, \(-j\)\)\) \((α)\) = \(b\_\(s, 
            j\)\) \((α)\)\)\(
\)\)	\)
s = (1/2), (3/2), (5/2),  ... 
j = intero 
α = (a/a1)          0 < α < 1

In the expansions of the disturbing function, the following symbols are used, where the value of α is ignored:

\(\(\(b[s, j, 0] = \(b\_s\^\((j)\)\) \((α)\)\)\(
\)
    \)\)
b[s, j, 
        p] = (((d^p) ((Subscript[b, s])^((j))) (α))/d)(α^p)              p ≥ 
        1

The package defines the following functions, which allow calculation of the numerical value in the expressions:

({{"\!\(\*StyleBox[\"\\\"LaplaceCoefficient[\\\"\", \"MR\"]\)\!\(\*StyleBox[\"\\\"s\\\"\", \"TI\"]\)\!\(\*StyleBox[\"\\\",\\\"\", \"MR\"]\) \!\(\*StyleBox[\"\\\"j\\\"\", \"TI\"]\)\!\(\*StyleBox[\"\\\",\\\"\", \"MR\"]\) \!\(\*StyleBox[\"\\\"p\\\"\", \"TI\"]\)\!\(\*StyleBox[\"\\\",\\\"\", \"MR\"]\) \!\(\*StyleBox[\"\\\"alpha\\\"\", \"TI\"]\)\!\(\*StyleBox[\"\\\"]\\\"\", \"MR\"]\) ", "starting from the value of
 \!\(\*StyleBox[\"\\\"alpha\\\"\", \"TI\"]\) calculates the numerical value of the Laplace
coefficient"}, {"\!\(\*StyleBox[\"\\\"NumericalValue[\\\"\", \"MR\"]\)\!\(\*StyleBox[\"\\\"expr\\\"\", \"TI\"]\)\!\(\*StyleBox[\"\\\",\\\"\", \"MR\"]\) \!\(\*StyleBox[\"\\\"alpha\\\"\", \"TI\"]\)\!\(\*StyleBox[\"\\\"]\\\"\", \"MR\"]\) ", "replaces the numerical value of \!\(\*StyleBox[\"\\\"alpha\\\"\", \"TI\"]\) in the expression
\!\(\*StyleBox[\"\\\"expr\\\"\", \"TI\"]\)"}, {"\!\(\*StyleBox[\"\\\"NumericalValue[\\\"\", \"MR\"]\)\!\(\*StyleBox[\"\\\"expr\\\"\", \"TI\"]\)\!\(\*StyleBox[\"\\\",\\\"\", \"MR\"]\) \!\(\*StyleBox[\"\\\"alpha\\\"\", \"TI\"]\)\!\(\*StyleBox[\"\\\",\\\"\", \"MR\"]\) \!\(\*StyleBox[\"\\\"rule\\\"\", \"TI\"]\)\!\(\*StyleBox[\"\\\"]\\\"\", \"MR\"]\) ", "evaluates the expression \!\(\*StyleBox[\"\\\"expr\\\"\", \"TI\"]\), with the numerical value of \!\(\*StyleBox[\"\\\"alpha\\\"\", \"TI\"]\), and with the replacements \!\(\*StyleBox[\"\\\"rule\\\"\", \"TI\"]\)"}})

Calculation of Laplace coefficients.

This loads the package.

<<Perturbations`LaplaceCoefficients`

LaplaceCoefficient

The syntax is the same used for symbols b, but in this case, the numerical value of α must be added, too.

LaplaceCoefficient[(1/2), 1, 0, 0.6689]

LaplaceCoefficient[(7/2), -3, 5,  .7001]

Should the fourth argument be a symbol, an exact expression is supplied (generally in terms of HypergeometricPFQ). Mathematica does not automatically apply simplification from HypergeometricPFQ to Hypergeometric2F1. Instead of using the instruction FullSimplify, one may write the appropriate replacement:

LaplaceCoefficient[(1/2), 0, 0, α]

LaplaceCoefficient[(3/2), 2, 0, α]

lap = LaplaceCoefficient[(3/2), 1, 1, α]

lap // FullSimplify

lap /. 
      HypergeometricPFQ[{a_, b_}, {c_}, (α^2)] :> 
        Hypergeometric2F1[a, b, c, (α^2)]

NumericalValue

NumericalValue[expr, alpha] carries out the replacement expr / . α -> alpha. For example, considering the expression:

ex = (1/8) (17  b[(1/2), 1, 0] + 
                10 α b[(1/2), 1, 1] + (α^2) 
                  b[(1/2), 1, 2]) - (27/8) α;

Here is the numerical value for a=0.480597:

NumericalValue[ex, 0.480597]

NumericalValue[expr, alpha] carries out the replacement expr /. α->alpha /. rule, where rule represents one or more replacements of the following type: 
e->number, e1->number, etc.
For example, considering the expression:

ex = -(((e^5) (σ^6))/1228) (4731447  (α^3) b[(7/2), 15, 0] + 1163365  (α^4) b[(7/2), 15, 1] + 
              110950  (α^5) b[(7/2), 15, 2] + 
              5130  (α^6) b[(7/2), 15, 3] + 
              115  (α^7) b[(7/2), 15, 4] + (α^8) 
                b[(7/2), 15, 5]);

Let us replace α=0.5331, e=0.2323, σ=0.1025:

NumericalValue[ex, 
      0.5331, {e -> 0.2323, σ -> 0.1025}]