Function Repository Resource:

FullSimplifyFunction

Convert a pure function into a simpler form

Contributed by: E. Chan-López, Carlos Francisco Arias Méndez & Ulrich Neumann

ResourceFunction["FullSimplifyFunction"][expr]

applies a full simplification to all Function expressions within expr.

Details and Options

ResourceFunction["FullSimplifyFunction"] threads over lists.

ResourceFunction["FullSimplifyFunction"] operates on subexpressions with head Function, accepting either a single symbol or a list of parameters.

ResourceFunction["FullSimplifyFunction"] takes the same options as FullSimplify.

Examples

Basic Examples (4)

Use FullSimplifyFunction to simplify a simple Function:

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Use FullSimplifyFunction to simplify a simple function given as a rule:

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Use FullSimplifyFunction to simplify a more complicated function:

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$ResourceFunction["FullSimplifyFunction"][ f -> Function[{\[Theta], \[CurlyPhi]}, a*Cos[\[CurlyPhi]] Sin[\[Theta]] - a*Cos[\[Theta]] Sin[\[CurlyPhi]]]]$

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Use FullSimplifyFunction to simplify multiple function rules:

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$ResourceFunction[ "FullSimplifyFunction"][{y -> Function[x, 1/2 (E^-x + E^x)], y -> Function[x, (E^x - E^-x)/(E^x + E^-x)]}]$

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Scope (2)

Use FullSimplifyFunction to simplify expressions involving special functions:

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Use FullSimplifyFunction with more complicated function rules:

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ResourceFunction["FullSimplifyFunction"][
x -> Function[t, 1/6 E^(-Sqrt[3] t) (3 - 2 Sqrt[3] + 3 E^(2 Sqrt[3] t) + 2 Sqrt[3] E^(2 Sqrt[3] t)) C[1] - (
E^(-Sqrt[3] t) (-1 + E^(2 Sqrt[3] t)) C[2])/(2 Sqrt[3])]]

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Options (2)

Use FullSimplifyFunction with Assumptions:

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$ResourceFunction["FullSimplifyFunction"][ a -> Function[ x, -Log[(1 - E^(x k[1] + (k[1] (2 I \[Pi] C[1]))/k[1]))/k[1]]], Assumptions -> C[1] \[Element] Integers]$

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Use FullSimplifyFunction to simplify a function involving an EllipticF using assumptions:

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Applications (3)

Use FullSimplifyFunction with the results of DSolve:

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Use FullSimplifyFunction on the solutions:

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Verify the solutions:

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Solve an initial value problem for an inhomogeneous linear hyperbolic system with constant coefficients:

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$pdes = {D[u[x, t], t] == D[v[x, t], x] + 1, D[v[x, t], t] == -D[u[x, t], x] - 1}; ic = {u[x, 0] == Cos[x]^2, v[x, 0] == Sin[x]}; sol = DSolve[{pdes, ic}, {u, v}, {x, t}]$

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Use FullSimplifyFunction on the solutions:

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Verify the solutions:

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Use FullSimplifyFunction in the solutions of the Euler-Lagrange equations governing the motion of a rolling wheel on an inclined plane. In this particular problem, the translational coordinate y and the rotational coordinate θ are treated as independent variables. Define the equation of motion for the translational coordinate and its initial conditions:

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$eqny = {2/3 m g Sin[\[Alpha]] - m y''[t] == 0};$

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The solution for equation of motion using DSolve:

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Apply FullSimplifyFunction:

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Verify the solution:

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Use the resource function SymbolToSubscript to format some parts of the solution as subscripts:

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The equation of motion for the rotational coordinate and its initial conditions:

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$eqn\[Theta] = {1/3 m R g Sin[\[Alpha]] - 1/2 m R^2 \[Theta]''[t] == 0};$

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$icv\[Theta] = {\[Theta][0] == \[Theta]0, \[Theta]'[0] == \[Omega]0};$

The solution for this equation of motion using DSolve:

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$First@DSolve[Join[eqn\[Theta], icv\[Theta]], \[Theta], t]$

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Verify the solution:

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$Part[ReplaceAll[%][eqn\[Theta]], 1]$

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Use SymbolToSubscript to format some parts of the solution as subscripts:

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$ResourceFunction["SymbolToSubscript"][ ResourceFunction["FullSimplifyFunction"][ First@DSolve[Join[eqn\[Theta], icv\[Theta]], \[Theta], t]]]$

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Properties and Relations (2)

Use FullSimplifyFunction with the resource function SolutionRulesToFunctions:

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$First@DSolve[ Join[{1/3 g m R Sin[\[Alpha]] - 1/2 m R^2 (\[Theta]^\[Prime]\[Prime])[t] == 0}, {\[Theta][0] == \[Theta]0, Derivative[1][\[Theta]][0] == \[Omega]0}], \[Theta][t], t]$

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Apply FullSimplifyFunction on the solution:

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Publisher

Ramón Eduardo Chan López

Requirements

Wolfram Language 13.0 (December 2021) or above

Version History

1.0.0 – 02 August 2023

Source Metadata

Citation:
- Baumann, G. (2006). Mathematica for theoretical physics: classical mechanics and nonlinear dynamics (Vol. 1). Springer Science & Business Media. DOI: 10.1007/b106777

Related Resources

Author Notes

This function is a generalization of the responses found in a post (titled “Simplifying the exponential expression within the Function”) on the Mathematica StackExchange platform. It incorporates ideas from that discussion and was developed in response to the author's query.

A relevant reference on handling variable scoping using Block can be found in the Stack Overflow Community post, “How to Block Symbols without evaluating them?”, with a significant contribution from Leonid Shifrin.

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License

Wolfram Function Repository

FullSimplifyFunction

Details and Options