/** This program tries to help you find a Taylor series for a particular function around x0. */ use functionUtils.frink use allTransforms.frink symbolicMode[true] showApproximations[false] /** This function finds a Taylor series for a particular function around x0. x0 can be a symbol, in which case you probably want to wrap it in a noEval[] block. */ TaylorSeries[f, x0=0, terms=6, debug=false] := { body = functionBody[f] vars = functionArgumentsAsSymbols[f] var1 = vars@0 if debug println["var1 is \$var1"] sum = substituteExpression[body, var1, x0] if debug println["sum = " + inputForm[sum]] d0 = body for i = 1 to terms-1 { d0 = transformExpression[constructExpression["FunctionCall", ["D", d0, var1]]] if debug println["d0 = " + inputForm[d0]] d1 = transformExpression[substituteExpression[d0, var1, x0]] if debug println["d1 = " + inputForm[d1]] df = transformExpression[d1 (var1-x0)^i / i!] sum = sum + df if debug println["df = " + inputForm[df] + "\n"] } return transformExpression[sum] } // This is the equation for total relativistic energy. If you expand it with // a Taylor series, you will get the Einsteinian mass-energy relation // // E = m0 c^2 // // where m0 is the rest mass and c is the speed of light // // plus the Newtonian kinetic energy // // KE = 1/2 m v^2 // // plus it also contains relativistic expansion terms which you'll see if you // run this program through a Taylor series. Relativity gives an infinite // expansion of corrective terms! // // If you run this, you'll see that the Taylor series of this is: // c^2 m0 + 1/2 m0 v^2 + 3/8 c^-2 m0 v^4 + 5/16 c^-4 m0 v^6 + 35/128 c^-6 m0 v^8 // // All of this is derived through the Lorentz equation (1/sqrt[1 - v^2/c^2]) f = {|v| (m0 c^2) (1/sqrt[1 - v^2/c^2]) } // This is the equation for relativistic kinetic energy. It is the // relativistic total energy (above) minus the rest-mass energy "m c^2". // The Taylor series expansion is actually important. The original equation // is valid for all velocities but has numerical issues if evaluated naively // for low velocities in the given form below. That's why, for low velocities, // we expand it to a Taylor series which is closer to the classical 1/2 m v^2 // (which you can actually see in the Taylor series expansion) but with // interesting correction terms. // // The Taylor series expansion is: // // 1/2 m v^2 + 3/8 c^-2 m v^4 + 5/16 c^-4 m v^6 + 35/128 c^-6 m v^8 ... // // which converges quickly for low values of v, (so we choose the Taylor series // for low values of v) but converges slowly for high values of v (so we choose // the sqrt[...] formulation for high values of v, which has large errors for // small values of v) which are accurate for high // values of v which don't lose numerical precision from taking a huge // total energy minus a huge restmass-energy. // // An empirical threshold is found around v = 0.13 c, depending on how many // terms of the Taylor series are taken. f = {|v| (m c^2) (1/sqrt[1 - v^2/c^2] - 1) } // This is the equation for the Lorentz transformation. //f = {|v| 1 / sqrt[1 - v^2/c^2] } // This has an elegant Taylor series expansion. //f = {|x| 1/(1-x) } // This is the classic example //f = {|x| e^x } println["The Taylor series for " + inputForm[functionBody[f]] + " is:"] println[inputForm[TaylorSeries[f, 0, 10, false]]]