Notes about Heaviside

The maple function for SMath's sign function isn't "sign", it's signum!

Note: I guess that the maple plugin author can correct this issue.

Because that, and the definition of Heaviside in StatiscalTools plugin

is why it desappear taking the maple derivatives.

The maple derivatives for signum and abs returns some unusual functions

defined as

Because I don't use StatiscalTools I handle in a very different way this things with maple.

Redefining Heaviside with a bump function, like yours, it's a good solution for derivatives, which is a punctual propertie of a function. But for integrals, which are global properties of a function, the situation is different. That math things have actual physics consequences: derivative circuits are unstable, while integral ones are stables, for instance.

Statiscal tools introduces Dirac functions too, which is not a function, it is a distribution, introduced by the russian Sóbolev in 1935 and the french Schwartz in 1940. So, there are no way to define it as any bump function. For using it with maple, it seems that it return an error at a generic point, but infinity for zero, just where it is interesting.

And Dirac is neccessary for represent concentrated puncthual loads.

Conclusion

Ask to Viacheslav -the maple tools plugin author- to correct the maple sign bug and see if there are a way to handle sign and abs with two args in SMath.