I was wondering if the following integral is able to be evaluated using the Gamma Function.
∫∞0t−12exp[−a(t+t−1)]dt
I already have a tedious solution that doesn't exceed the scope of the first few semesters of calculus, but I want to tackle this with the Gamma Function. I just don't know how or if it's even possible.
If anyone can give a hint, I'd really like to finish it on my own.
EDIT:
You are allowed to use the fact that
∫∞−∞exp(−x2)dx=√π
Answer
Let t=u2, and the integral becomes
2∫∞0due−a(u2+1u2)=2e2a∫∞0due−a(u+1u)2
Let v=u+1/u, then
u=12(v±√v2−4)
du=12(1±v√v2−4)dv
Now, it should be understood that as u traverses from 0 to ∞, v traverses from ∞ down to a min of 2 (corresponding to u∈[0,1]), then from 2 back to ∞ (corresponding to u∈[1,∞)). Therefore the integral is
e2a∫2∞dv(1−v√v2−4)e−av2+e2a∫∞2du(1+v√v2−4)e−av2
which is
2e2a∫∞2dvv√v2−4e−av2=e2a∫∞4dy√y−4e−ay=e−2a∫∞0dqq−1/2e−aq
I guess the gamma function comes from this integral, but I find it easier to refer to gaussian integrals.
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