There are many cases with
complex functions which appear to be
paradoxes. The familiar rules of algebra
and trigonometry of real numbers may
break down when applied to complex numbers.
We may not, for example, use the rule
,
when a
and b
are negative, for otherwise we would
have
.
There is another example of An
improper use of SQRT(-1)
Complex
sine (cosine) functions could be equal
to 2 or 5. Is it a paradox? Look
at Sin(z)
= 2
arctg(1)
=
.
If to consider arctg(x)=
and
forget that complex logarithm is a multi-valued
function, you will get "arctg(1)
= 0". Look
at Paradox
of Bernoulli and Leibniz
Swiss mathematician John
Bernoulli constructed a beautiful sophism
with the chain of arguments, pretending
to prove that Ln(-z)
= Ln(z) for any
.
Look at Bernoulli's
sophism
Top
An
improper use of SQRT(-1)
Another example
as an improper use of
=
i leads to an error.
i
= i
(1)
=
-1 = 1
(2)
The reason for the fallacy
(2) is that rule for computing the quotient
of radicals does not apply to .
