Integral between -infinity and +infinity

[rwooduk]

integral between -infinity and +infinity of

e^-iwt / (a - iw) dw


u = a - iw

du/dw = -i

dw = du / -i

e^-iwt = e^(u+a)t / -iu du


its a homework question, so just a pointer as to which method to use would be appreciated, it says substitution, but I've tried the above and doesn't get me anywhere, maybe I am not choosing the correct substitution?

thanks for any help

 

[Simon Bridge]

Have you been instructed to use substitution?
Have you tried ratonalizing the denominator?

 

[Ray Vickson]

integral between -infinity and +infinity of

e^-iwt / (a - iw) dw


u = a - iw

du/dw = -i

dw = du / -i

e^-iwt = e^(u+a)t / -iu du


its a homework question, so just a pointer as to which method to use would be appreciated, it says substitution, but I've tried the above and doesn't get me anywhere, maybe I am not choosing the correct substitution?

thanks for any help

Your exp(at) part has the wrong sign in the exponent. You should find that your integral, J, is
$$J \equiv \int \frac{e^{-iwt}}{a-iw} \, dw = i e^{-at} \int \frac{e^{ut}}{u} \, du$$
This last form reveals that the integral is "non-elementary", because $\exp(tu)/u$ cannot be integrated found as a finite expression involving elementary functions; it involves the so-called exponential integral; see http://en.wikipedia.org/wiki/Exponential_integral .