Mathematics Database Programming Web Design Price List     Possibly the greatest paradox is that mathematics has paradoxes... Bernoulli's sophism Paradox of Bernoulli and Leibniz Paradox of even (odd) and natural numbers Paradox of Hilbert’s hotel Ross-Littlewood paradox Paradox of wizard and mermaid Paradox of enchantress and witch Paradox of Tristram Shandy Barber paradox Achilles and tortoise  # Paradox of Bernoulli and Leibniz Complex Analysis. FreeTutorial A cense of the paradox:arctg(1) = . There is a chain of arguments, pretending to prove that arctg(1) = 0. arctg(x) = Let us factorize  = , i =  = Let x = 1arctg(1) = = = = = = = 0 But arctg(1) = An explanation of the paradox: The complex logarithm is a multi-valued function. Ln(z) = ln|z| + i(arg z + 2k ), k = 0, ±1, ±2, ... Ln(1) = ln|1| + i(arg (1) + 2k ), k = 0, ±1, ±2, ... If we consider a branch Ln(1), where k = 1 then Ln(1) = and the paradox disappears. arctg(1) = = = arctg(1) = by Tetyana Butler Top  Complex analysis is studying the most unexpected, surprising, even paradoxical ideas in mathematics. The familiar rules of algebra and trigonometry of real numbers may break down when applied to complex numbers. Free Lessons Contact us