Integrating Over Real Numbers: Understanding the Notation and Limitations

[eljose]

let be the integral:

\int_0^{\infty}F(x)dx with x\rightarrow\infty F(x)\rightarrow0 then we make the change of variable x=-e^{t} then the new integral would become \int_{-\infty}^{\infty+i\pi}F(-e^t)e^{t}dt my question is if we can ignore the integral from (\infty,\infty+i\pi) so we have only the integral \int_{-\infty}^{\infty}F(-e^t)e^{t}dt as for big value the F(x) tends to 0

 

[AKG]

You cannot make this substitution. Note that x > 0, and e^t is also always greater than 0, so how could x = -e^t ever be true?

 

[eljose]

why not? the same would happen with x=-1/(t+1) x>0 but t<0

 

[AKG]

How can a negative quantity equal a positive quantity?

 

[quetzalcoatl9]

i think that you lost a negative sign when making your substitution.

the example you give in post #3 is correct for x>0 t<-1

so it sounds like the unresolved question is your bounds on the integral. I'm not sure how to handle this, but keep two things in mind:

1) remember that you are dealing with a formal limit

\lim_{a \rightarrow -\infty} \int_{a}^{i\pi} f(x) dx + \lim_{b \rightarrow \infty} \int_{i\pi}^{b} f(x) dx

2) you are integrating over \mathbb{C} so you need to get a book on complex integration and see what to do. i suspect that solving the above and integrating as usual will suffice.

 

[eljose]

Another question let,s suppose we have the integral:

\int_{-\infty}^{\infty}F(x)dx and make the change of variable x=t+ai with i
=sqrt(-1) then what would be the new limits?..thanks...

 

[AKG]

I haven't taken any courses on complex analysis, so maybe I'm unfamiliar with the notation. Over what region are you integrating when you take:

\int _{-\infty} ^{\infty} F(x)dx

To me, that suggests that you're integrating over all the real numbers, i.e. x ranges over the reals. If this is the case, then you again run into the problem of equating x with t + ai because unless t = b - ai for some real b, then x will be never have a real part, and t+ai sometimes will, so the two cannot be equated.