written and published by Renzo Diomedi
The method here used to prove the transcendence of exp(e) is very similar to the one already used
to demonstrate the transcendence of exp(algebraic number) (1).
is a holomorphic function, differentiable over the complex numbers (2) in every point of its domain,
and its integrand is a polynomial of a certain degree , in which the exponent of the denominator
slides all domain values [0,t]
Its input variables are conjugates indexed by a table made of r=rows and s=columns
compatible with the modular form (3)
We exploit all the properties of complex conjugation including those about the odd-degree
polynomials, in accordance with the complex conjugate root theorem.
We assume , i.e. a conjugate in polar form, with k integer.
Each of the infinite values that can assume, is closely related to
or its multiple or its fraction.
For this reason we must assume that the value of the multiple or the value of the fraction of
can also be non-algebraic.
Then returns algebraic values, (4)
Integrating by parts and assuming , we get:
with = degree of
and = j-th derivative of f .
Let a symmetric polynomial
with degree ,
with , , ,
and is a Prime sufficiently large.
are distinct algebraic complex conjugate linearly independent.
Appropriate coefficients and make
This polynomial is never negative.
Then we use next polynomial with integers non-zero , to verify the possibility of
existence of an algebraic result
= = (derivative of = null).
Considering that by derivations, and assuming ,
we can extract from , by derivations, the polynomial ,
then , the minimal polynomial
is divisible by and it follows that
Considering that a , n and t have not infinite values ,
so is defined in a bounded set, therefore there must be a number greater than J .
This number could be an arbitrary . So we have :
let us avoid the immediate and trivial demonstration :
The use of complex numbers ensures that every non-constant polynomial has a root, since the Fundamental
Theorem of Algebra states that every non-constant polynomial with coefficients in
C , has zeros in C , false in R (typical instance:
has zero in complex numbers only).
a redundance of complex numbers in upper half-plane in which each point
of each of the two axes is intersected by a two-dimensional table composed of complex numbers, id est
an object in four spatial dimensions, which returns only positive values, not drawable on graph.
So, we get exp(algebraic number) again!