In my recent question about the Fransén-Robinson constant, an answer was given using the Gamma reflection formula. However, as an AP Calculus student, I didn't quite understand how the reflection formula worked. After two days of research, I have only found explanations for the Gamma reflection formula in terms of Weierstrass products, which I don't begin to understand.
Is there a proof for the Gamma reflection formula by which I can understand, or at least begin to understand, how this formula works?
Note: This is a description from N.N. Lebedev, Special Functions and Their Applications, Dover, New York, 1972, it is not my work but it can be used as starting point.
Lebedev uses in his section 1.2 (Some Relations Satisfied by the Gamma Function) a double-integral approach. From the well-known integral formula
$$\Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}\, d t \qquad (\Re z > 0)$$
temporarily assume $ 0 < \Re z < 1,\,$ use the formula for $1-z\,$ and get
$$\Gamma(z)\Gamma(1-z) = \int_{0}^{\infty}\int_{0}^{\infty}s^{-z}t^{z-1}e^{-(s+t)}\,ds\, dt \qquad (0<\Re z <1)$$
With the new variables $u = s + t, v = t/s$ this gives $$\Gamma(z)\Gamma(1-z) = \int_{0}^{\infty}\int_{0}^{\infty}\frac{v^{z-1}}{1+v}e^{-u}\,du\, dv = \int_{0}^{\infty}\frac{v^{z-1}}{1+v} dv = \frac{\pi}{\sin \pi z}$$
For the last step he refers to Titchmarsh. Then he uses continuation to extend the formula to all $z\in \mathbb{C}$ without the integers.