Mathematica(@mathemetica):Four classic results from real analysis appear here side by side. The Gaussian ∫₋∞⁺∞ e⁻ˣ² dx = √π The arctangent ∫₋∞⁺∞ 1/(x²+1) dx = π Dirichlet’s ∫₋∞⁺∞ (sin x)/x dx = π And ∫₋∞⁺∞ cos(x)/(1+x²) dx = π/e Digital audio codecs rely on the sinc integral to enforce the Nyquist rate that prevents aliasing in compressed music files.

Mathematica

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Mathematica@mathemetica

2026.07.26 07:18

Four classic results from real analysis appear here side by side. The Gaussian ∫₋∞⁺∞ e⁻ˣ² dx = √π The arctangent ∫₋∞⁺∞ 1/(x²+1) dx = π Dirichlet’s ∫₋∞⁺∞ (sin x)/x dx = π And ∫₋∞⁺∞ cos(x)/(1+x²) dx = π/e Digital audio codecs rely on the sinc integral to enforce the Nyquist rate that prevents aliasing in compressed music files.