This sophism was constructed
by Swiss mathematician John Bernoulli
(1667 - 1748), who was one of the eight
outstanding mathematicians in the Bernoulli
family.
Find a mistake in the
following chain of arguments, pretending
to prove that
An
explanation:
The conclusion that
Ln(-z) = Ln(z)
is false, because
Ln(z)
= ln(|z|)
+ i[arg(z)
+2k
],
k
= 0, ±1, ±2, ... ,
Ln(-z)
= ln(|z|)
+ i[arg(z)
+(2k+1)],
k
= 0, ±1, ±2, ... ,
and none of the numbers representing
the value of Ln(z)
is the same as any of the numbers
representing Ln(-z).
The error occurs in
going from line 2 to line 3 because
Ln(-z) + Ln(-z) 
2Ln(-z),
Ln(z) + Ln(z)
2Ln(z).
The following example elucidates
the situation:
Let A be the set
of two numbers 3
and 4.
A set B=A+A is the
set of numbers 6;
7; 8 because
3+3=6, 3+4=7
and 4+4 =8.
A set C=2·A
is the set of numbers 9;
12; 16 because
3·3=9; 3·4=12
and 4·4=16.
So, a set A+A2·A
Ln(-z) + Ln(-z)
2Ln(-z),
Ln(z) + Ln(z)
2Ln(z).
by
Tetyana Butler
Reference
A.I. Markushevich, "Theory
of functions of a complex variable"
, 1–2 , Chelsea (1977) (Translated
from Russian)
.
