The Trig Hiding Inside the Factorials (And Harmonic Numbers)

The digamma function ψ(x), despite originating from harmonic number extension, exhibits surprising structural similarity to the tangent function observable in its graph.

Euler’s reflection formula Γ(x)·Γ(1-x) = π/sin(πx) connects the gamma function to sine, revealing trigonometric structure within factorial extension.

Reflection formulas combine a function f(x) with its reflected version f(1-x) to produce new functions exhibiting enhanced symmetry or periodic behavior.