Interesting Expressions of Pi
2 December 2024
#1: \((0.5!\times2)^2=\pi\)
#2: \(\frac{\text{ln}(-1)}{i}=\pi\)
This is derived from Euler's formula, \(e^{i\pi}+1=0\).
#3: \(\ \int_{-1}^{1} \frac{4}{1+x^2} \, dx =\pi \)
Archimedes' Arctangent formula (four times the area under the curve on [-1, 1] on the derivative of \(\arctan(x) \)
#4: \(\sqrt{-1} \int_{-\infty}^{\infty} e^{-x^2} \, dx = \pi \)
Gaussian Integral
#5: \( 2\arccos(0)=\pi \)
Trig identity using \( \cos(\frac {\pi}{2})=0 \)
More coming soon...
—GL