I want to calculate

$$ int_ {0} ^ {+ infty} enac { sin x} { sqrt {x}} dx $$

using the gamma function.

I know that by variable change, $ y = sqrt {x} $, we obtain

$$ int_ {0} ^ {+ infty} frac { sin x} { sqrt {x}} dx = 2 int_ {0} ^ {+ infty} sin y ^ 2 dy = frac { sqrt {2 pi}} {2} $$

by the Fresnel Integral.

I try it by considering this:

$$ int_ {0} ^ {+ infty} x ^ {- frac {1} {2}} e ^ {ix} dx $$

It converges for both real and imaginary parts using the Dirichlet test, and $ 0 It's not a problem here. Let the square root take the main branch where $ sqrt {1} = $ 1. Let $ y = -ix $then

$$ int_ {0} ^ {+ infty} x ^ {- frac {1} {2}} e ^ {ix} dx = sqrt {i} int_ {0} ^ {- i infty } y ^ {- frac {1} {2}} e ^ {- y} dy = ( frac { sqrt {2}} {2} + frac { sqrt {2}} {2} i ) Gamma ( frac {1} {2}) = frac { sqrt {2}} {2} + frac { sqrt {2 pi}} {2} i $$

And that coincides with the final answer!

My problem is, suppose $ L $ is a radius from $ 0 and has an angle $ phi $ with the $ x $-axis, and leave $ phi in (0.2 pi) $.

I want to argue that (maybe that's incorrect though)

$$ Gamma (z) = int_ {L} t ^ {z-1} e ^ {- t} dt $$

I know first that it converges when $ Re (z)> 0 $. Choose an outline as the area, leave $ L_1 = {z = x + iy: y = 0, r <x <R } $, $ L_2 = {z = Re ^ {i theta}: 0 < theta < phi $, $ L_3 = {z = xe ^ {i phi}: r <x <R } $ and $ L_4 = {z = re ^ {i theta}: 0 < theta < phi $, or $ r <R $ and the outline is counterclockwise. According to Cauchy's theorem, the contour integral must be $ 0.

Easy to see that $ z = x + iy $)

$$

lim_ {r to0 +, R to + infty} int_ {L_1} t ^ {z-1} e ^ {- t} dt = Gamma (z)

$$

$$

| int_ {L_4} t ^ {z-1} e ^ {- t} dt | = | int _ { phi} ^ {0} e ^ {- re} (re ^ {i theta}) ^ {z-1} ire ^ {i theta} d theta | leq int_ {0} ^ { phi} e ^ {- r cos theta} | r ^ {z} e ^ {i theta (z-1)} | d theta = int_ {0} ^ { phi} e ^ {- r cos theta} r ^ {x} e ^ {- theta y} d theta to 0, r to 0 +

$$

But when we consider $ L_2 $:

$$

| int_ {L_2} t ^ {z-1} e ^ {- t} dt | leq int_ {0} ^ { phi} e ^ {- R cos theta} R ^ {x} e ^ {- theta y} d theta

$$

when for example, $ frac {3 pi} {2}> phi> frac { pi} {2} $, we have $ cos theta <0 $ and I failed to prove that the integral above goes to zero when $ R to + infty $.

Is my use of the Gamma function to calculate the original integral a coincidence to get the correct result or is there a way to prove my argument?

Thank you so much!