Transforms

Currently the only tranform defined is LaplaceTransform(), which has the calling convention:

LaplaceTransform(var1,var2,func)

It has been setup much like the integration algorithm. If the transformation algorithm cannot perform the transform, the expression (in theory) is returned unsimplified. Some cases may still erroneously return Undefined or Infinity.

The LaplaceTransform() algorithm

This section describes the steps taken in doing a Laplace transform.

General structure

LaplaceTransform() is immediately handed off to LapTran(). This is done because if the last LapTran() rule is met, the Laplace transform couldn’t be found and it can then return LaplaceTransform() unevaluated.

Operational properties

The first rules that are matched against utilize the various operational properties of LaplaceTransform(), such as:

• Linearity Properties

• Shift properties, i.e. multiplying the function by an exponential

• \(\mathcal{L}\lbrace yx^n\rbrace = (-1)^n \frac{d^n}{dx^n} \mathcal{L}\lbrace y\rbrace\)

• \(\mathcal{L}\lbrace \frac{y}{x}\rbrace = \int_s^\infty\mathcal{L}\lbrace y\rbrace(\sigma)d\sigma\)

The last operational property dealing with integration is not yet fully bug-tested, it sometimes returns Undefined or Infinity if the integral returns such.

Transform tables

For elementary functions, yacas uses transform tables. For instance, the fact that the Laplace transform of \(cos(t)\) is \(\frac{s}{s^2+1}\) is declared in a transform table.

For the purpose of setting up the transform table, a few declaration functions have been defined, which use some generalized pattern matchers to be more flexible in recognizing expressions that are transformable.

Transforming simple functions

For functions like \(\sin(t)\) the transform can be declared with the function LapTranDef().

The calling sequence for LapTranDef() is:

LapTranDef(in, out)

Currently in must be a variable of _t and out must be a function of s. For instance, for the function \(\cos(t)\) there is a declaration:

LapTranDef(Cos(_t), s/(s^2+1));

The fact that the first argument is a pattern means that each occurrence of the variable to be matched should be referred to as _t, as in the example above.

LapTranDef() generalizes the transform implicitly, in that it will set up the system to actually recognize expressions of the form \(\cos(at)\) and \(\cos(\frac{t}{a})\) , and return the appropriate answer. The way this is done is by three separate rules for case of t itself, a*t and t/a. This is similar to the MatchLinear() function that Integrate() uses, except LaplaceTransforms() must have b=0.

Further Directions

Currenlty \(\sin(t)\cos(t)\) cannot be transformed, because it requires a convolution integral. This will be implemented soon. The inverse Laplace transform will be implement soon also.