Algebra of divergent integrals

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Summary:

Call to discuss an extension of real numbers that includes divergent integrals and series

Main Question or Discussion Point

Hello, guys!

I would like to know your opinion and discuss this extension of real numbers:
https://mathoverflow.net/questions/115743/an-algebra-of-integrals/342651#342651

In essence, it extends real numbers with entities that correspond to divergent integrals and series.
By adding the rules derived from Faulhaber's formula, it allows some wonderful things to be done,
such as expressing a derivative of an analytic function without using limits or infinitesimals, and also expressing trigonometric functions via inverse trigonometric in closed form.

Some physical expressions, especially those derived via regularization become simplier.
For instance, the mean energy of quantum harmonic oscillator can be written instead of
$$\varepsilon ={\frac {h\nu }{2}}+{\frac {h\nu }{e^{h\nu /kT}-1}}$$
as
$$\varepsilon =h\nu\omega_++kT e^{\frac{h\nu\omega_-}{kT}} $$
with underlying implication that the full energy (before regularization) is infinite.

Your thoughts? I mostly interested in the opinion of
1 Algebraists - on what algebraic properties such extension should have
2 Physicists - on physical applications in the areas that require regularization
 

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