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Mathematical paradoxes
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Paradox of Bernoulli and Leibniz

Complex Analysis. FreeTutorial

A cense of the paradox:
arctg(1) =pi/4. There is a chain of arguments, pretending to prove that arctg(1) = 0.

arctg(x) =arctg(x) = integral dx/1+x*x

Let us factorize 1/1+x*x

1/1+x*x=factorize 1/1+x*x, i =root (-1) = i

arctg(x) = integral dx/1+x*x=1/2i ln(i-x)/(i+x)

Let x = 1
arctg(1) = 1/2i ln(i-x)/(i+x) = solution 1/2i ln(i-x)/(i+x)=

= solution 1/2i ln(i-x)/(i+x) continue= = solution= 0

But arctg(1) =pi/4

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) = explanation of the paradox= =

arctg(1) =

by Tetyana Butler

Complex functions Tutorial
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.
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