/** Symbolic Laplace transforms.

    See:
    https://byjus.com/maths/laplace-transform/

    https://tutorial.math.lamar.edu/Classes/DE/Laplace_Table.aspx

    https://en.wikipedia.org/wiki/Laplace_transform
*/
transformations Laplace
{
   // While this whole library allows you to use different variables than "t"
   // and "s", those are very commonly used.  This rule is a magical helper
   // that turns any single rule LaplaceTransform[_a] where _a contains "t" and
   // is free of "s" into the expanded more versatile LaplaceTransform[x, t, s]
   // that all of these transformation rules expect.
   LaplaceTransform[_a] :: expressionContains[_a, t] and freeOf[_a, s] <-> LaplaceTransform[_a, t, s]
   
   // Transform of an expression independent of t
   LaplaceTransform[_a, _t, _s] :: freeOf[_a,_t] and freeOf[_a,_s]  <-> _a / _s
   InverseLaplaceTransform[_a/_s, _s, _t] :: freeOf[_a,_t] and freeOf[_a,_s]<->_a

   // Linearity properties
   LaplaceTransform[_a _b, _t, _s] :: freeOf[_a, _t] and freeOf[_a, _s] <-> _a LaplaceTransform[_b, _t, _s]
   LaplaceTransform[_a + _b, _t, _s] <-> LaplaceTransform[_a, _t, _s] + LaplaceTransform[_b, _t, _s]

   // Derivative of function
   LaplaceTransform[D[_f[_t], _t], _t, _s] <-> _s * LaplaceTransform[_f[_t], _t, _s] - _f[0]
   // TODO: How to apply the following to an arbitrary expression, especially
   // the _f[0] part?
   //  LaplaceTransform[D[_a, _t], _t, _s] :: expressionContains[_a, _t] <-> _s * LaplaceTransform[_a, _t, _s] - _f[0]
   
   // Integral of function
   // Note that the usual convention in Laplace transform literature is that the
   // integration is performed from t=0!
   LaplaceTransform[Integrate[_f[_t], _t], _t, _s] <-> (LaplaceTransform[_f[_t], _t, _s] - _f[0]) / s
   // TODO:  Integral of ordinary expression and not arbitrary function?
   // TODO: How to apply it to an arbitrary expression, especially
   // the _f[0] part?
   // See implementation of Shift property below

   // Shift property e^(a t) f[t]  <->  F[s-a]
   LaplaceTransform[e^((_a:1) _t) _b, _t, _s] :: freeOf[_a, _t] and expressionContains[_b, _t] <-> LaplaceTransform[_b, _t, _s-_a]
//   LaplaceTransform[e^((_a:1) _t) _f[_t], _t, _s] :: freeOf[_a, _t] <-> LaplaceTransform[_f[_t], _t, _s-_a]
   
   // Trigonometric functions
   // sin[a t + b]
   LaplaceTransform[sin[(_a:1) _t + (_b:0)], _t, _s] :: freeOf[_a, _t] and freeOf[_b, _t] <-> (_s sin[_b] + _a cos[_b]) / (_s^2 + _a^2)
   
   // cos[a t + b]
   LaplaceTransform[cos[(_a:1) _t + (_b:0)], _t, _s] :: freeOf[_a, _t] and freeOf[_b, _t] <-> (_s cos[_b] - _a sin[_b]) / (_s^2 + _a^2)

   // Hyperbolic functions
   // sinh[a t]
   LaplaceTransform[sinh[(_a:1) _t], _t, _s] :: freeOf[_a, _t] <-> _a / (_s^2 - _a^2)

   // cosh[a t]
   LaplaceTransform[cosh[(_a:1) _t], _t, _s] :: freeOf[_a, _t] <-> _s / (_s^2 - _a^2)
   
   // e^(a t)
   LaplaceTransform[e^((_a:1) _t), _t, _s] :: freeOf[_a, _t] <-> 1/(_s - _a)

   // e^(a t) sin[b t]
   LaplaceTransform[e^((_a:1) _t) sin[(_b:1) _t], _t , _s] :: freeOf[_a, _t] <-> _b / ((_s - _a)^2 + _b^2)
   
   // e^(a t) cos[b t]
   LaplaceTransform[e^((_a:1) _t) cos[(_b:1) _t], _t , _s] :: freeOf[_a, _t] <-> (_s - _a) / ((_s - _a)^2 + _b^2)

   // t^n expr    where n is a positive integer
   LaplaceTransform[_t^(_n:1) (_a:1), _t, _s] :: freeOf[_n, _t] and (_n >= 1) and isInteger[_n] <-> (-1)^_n D[LaplaceTransform[_a, _t, _s], _s, _n]
}

//println[transformExpression[noEval[LaplaceTransform[D[g[t], t], t, s]]]]
//println[transformExpression[noEval[LaplaceTransform[Integrate[g[t], t], t, s]]]]