An Elementary Approach to Euler’s Reflection Formula and Its Role in the Infinite Product of the Sine Function and the Basel Problem

We present a proof, using elementary methods, of the Euler reflection formula for the Gamma function, based on an integral computed by Laplace and on the Euler–Gauss infinite product representation of Gamma. This way, we reverse the classical path, and, using the reflection formula as a starting point, we obtain the representation of the sine as an infinite product and that of the cotangent in partial fractions, which, as is known, allows the explicit calculation of the zeta function with an even argument: all this without resorting to complex analysis or the Herglotz trick. We can present a teaching proposal that illustrates the complete proof of this fundamental formula using undergraduate-level mathematical analysis tools, such as the derivation of parametric integrals, the second Mean Value Theorem for Integrals (Bonnet formula), and the convergence criterion for Dirichlet oscillatory integrals.