Basic Notions: The functional equation $f(x+a)f(x-a)=f(x)^2$ and Cauchy¹s residue theorem (Rinat Kashaev, UNIGE)

31.05.2018 16:15

Cauchy's residue theorem provides a powerful method for calculating integrals. If an integral can be calculated somehow, most likely, it can also be calculated by the residue method. Nonetheless, it is not always clear how the residue method can be used in some particular cases. A notable example is the famous gaussian integral $\int_{-\infty}^{+\infty}e^{-x^2}dy$, known to be equal to $\sqrt{\pi}$.
Remarkably, the gaussian exponential solves the functional equation in the title with some complex $a\ne0 $ and that fact, due to a simple lemma found in the context of Quantum Topology, allows to calculate the gaussian integral by the residue method.

PS. The Basic Notions Colloquium will be followed by an aperitif

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