Substitution of variables in improper integrals

[Rasalhague]

What principles apply when making a substitution of variables in an improper integral. I gather that a substitution of variables can change an impoper integral to a proper integral. Can substitution also change a proper integral into an improper integral? Suppose I know that a pair of integrals are each to be evaluated over the whole real line, and that one can be obtained from the other by a change of variable; in that case, can I ignore the issue of how, if the domain of integration had been finite, the limits would change?

 

[kdbnlin78]

Without giving it a whole lot of thought I am going to say yes. I think that in some occasions it may be pertinent to change variable of integration so as to create an improper integral (for instance, when one considers smooth functions with compact support in distribution theory) as one may then use such techniques like Causcy Principal value to assign values to integrals that would otherwise be unvalued.

A long sentence but I hope it in some way answers your question!

Regards,
kdbnlin

 

[nickalh]

For the most part, I echo kdbnlin78.
I was taught in Calculus II, if the integral does not converge, then substitutions which make it converge are invalid. Unfortunately, I believe someone elsewhere (possibly a greater authority on the subject) on this forum said some substitutions allow us to define values for certain improper integrals.

My opinion about the reverse (proper integral to improper) would be:
If the substitution creates a divergent integral out of a convergent integral, the substitution is invalid. Likewise for creating a convergent integral out of a divergent integral.

Summary & simplification of my opinion/ understanding:
The substitution may not change from convergent to divergent or divergent to convergent.