Question

(Complex Variable) Determine for which values of a and b you can assure that the following integrals converge:

Answer 1

a) \[\int\limits_{-1}^{1} \frac{\cos(it)}{t^a}\]
I have b) and c) but I want help with one and I'll see if I can solve the other ones.

Answer 2

I think it converges for all a with Re(a)>-1

Answer 3

Sorry, Re(a)<1, I didn't think it as a fraction.

Answer 4

It's an interesting problem, I have some doubts about it, but may be it should be,
\[Re(a)\leq1\]?

Answer 5

Does it converge when Re(a)=1?

Answer 6

I cannot assure you, but I would say yes (I think there should be a hypergeometric function around there)

Answer 7

I also was doing the following integrals:
\[\int\limits_{1}^{\infty} \frac{t^a}{i+t^b} dt\]
And
\[\int\limits_{0}^{1} \frac{t^a}{i+t^b}\]
The second one I think it's integrable if the denominator never tends to zero (I think it could be integrable even if it went to zero but it would require more analysis).
So b real part can never be 1 if sin(ln(t)Im(b))=-1
In the other one I think that Re(a)<Re(b)+1 otherwise the integral I'm using to delimit it will not converge.