Galileo's paradox
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
Math Interesting Facts
.
There is a chain of arguments,
pretending to prove that
arctg(1) = 0. 
,
i
=
=
=
=
=
=
0
),
k
= 0, ±1, ±2, ... 
and the paradox disappears.
=
= 
