Do These Integrals Always Exist Under Certain Conditions?

[lokofer]

Let be the integrals:

$$\int_{-\infty}^{\infty}dx Cos(uf(x))$$ (or the same but a sine) and

$$\int_{-\infty}^{\infty}dxe^{-ag(x)}$$

Where "a" is a a>0 positive constant, u can be either positive or negative.. and g(x)>0 for every real x.. my question is will these integrals "always2 exist under these conditions?..what would happen if we take the limit a-->oo and u-->oo ? are in this case equal to 0?

 

[arildno]

For the second case, set g(x)=1 (for all x). What do you see?

 

[lokofer]

Oh..then sorry "Arildno".. then for similar case with f(x) and g(x) different from f(x)=C (C a real constant) and g(x)>0 i think the integrals should tend to 0 for big u and a, and that they exist..for example for the WKB approach in Physics if you definte the "Action" S of the system the approximate wave function can be written as $$\psi(x)= ACos(S(x)/\hbar)$$ so in the "Semi-classical " limit (h-->0 ) the Wave function is 0 , A is a normalization constant.

- for the case of real exponential if you take g(x)=h(x) so h(x) is the "inverse" of g(x) the function becomes a "Laplace transform" of h'(x) that tends to 0 for big a