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A transcendental number is a real or complex number that is not alg...
Ivan Niven was a Canadian-American mathematician, specializing in n...
It's interesting that we know $e$ and $\pi$ are transcendental numb...
It's actually not obvious that the product (or sum) of 2 algebraic ...
Is $i$ an algebraic number?
We showed above that the sum of 2 algebraic numbers is also an alge...
Note that if we expand expression (2) we get $$ 1+ ...+e^{\alp...
$$ \frac{d(e^{-x}F(x))}{dx}=-e^{-x}F(x)+e^{-x}F'(x)=\\ =-e^{-x}F(...
If we make the substitution $\xi = \tau x $ the integration limits ...
How does he end up with $kF(0)$?
This is not difficult to see because of the $x^{p-1}$ factor and al...
Note that $f(x)$ can be written as $$ f(x)=\frac{c^sx^{p-1}(cx...
Using expression (11) $$ F(\beta_j)=f(\beta_j)+f^{(1)}(\beta_j)...
As shown in a comment above, $f(x)$ can be written as $$ f(x)=...
Isn't a $\tau$ missing? I mean, shouldn't the integral be $$\int_...
We have finally reached a contradiction since the left side of (12)...
1939]
THE
TRANSCENDENCE
OF
ir
469
for
XI-
X_X2,
with
(x,
y)
in
a
neighborhood
of
CO,
and
y-yo(x)
and
y'-yoi(x)
not
both
zero,
then
CO
furnishes
a
proper
(or
improper)
strong
relative
minimum
for
Jf=
f2f(x,
y,
y')dx,
G(x,
y,
y,
y,
y')-f(x,
y
y,
y')-
)-(y-y)fV,(x,
y,
y')
-
(y'
-')fy,
(x,
y,
y'). Examples
were
given
which
satisfy
these
sufficient
condi-
tions
but
which
do
not
yield
to
classical
methods.
3.
The finite
abstract
groups
in
which
every
subgroup
is
invariant
(Hamil-
tonian
groups)
are
well
known,
as are
also
the
groups
in
which
every
subgroup
is
abelian.
Classifying
the
finite
abstract
groups
according
to the
number
of
complete
sets
of
non-invariant
conjugate
subgroups
contained
in
them,
Pro-
fessor
Sigley
discussed
the
class
of
groups
which
contain
a
single
complete
set
of
non-invariant
conjugate
subgroups.
Necessary
and
sufficient
conditions
that
a
group
belong
to
this
class
were
obtained.
4.
It
was s
hown
by
Professor
Finkel
how
the
derivatives
of
ax
and
sin-'
x
may
be
derived
directly
by
the
A-process,
without
reverting
to
the
derivatives
of
the
logarithmic
and sine
functions.
5.
Mr.
Gillam
sketched
a
development
of
three-dimensional
analytic
euclid-
ean
geometry
(free
of
coordinates)
with
"point"
as
the
single
undefined
element
and
"distance"
as
the
only
primitive
relation.
Lines
and
planes
were
defined
as
determinantal
loci.
6.
Mr.
Wulftange
presented
a
new
method
for
the
numerical
solution
of
quadratic
congruences
which
is
believed
to
possess
certain
practical
advantages.
L.
M.
BLUMENTHAL,
Secretary
THE
TRANSCENDENCE
OF
c
IVAN
NIVEN,*
University
of
Pennsylvania
Among
the
proofs
of
the
transcendence
of
e,
which
are
in
general
variations
and
simplifications
of
the
original
proof
of
Hermite,
perhaps
the
simplest
is
that
of
A.
Hurwitz.t
His
solution
of
the
problem
contains
an
ingenious
device
which
we
now
employ
to
obtain
a
relatively
simple
proof
of
the
transcendence
of
7r.
We
assume
that
ir
is
an
algebraic
number,
and
show
that
this
leads
to
a
con-
tradiction.
Since
the
product
of
two
algebraic
numbers
is
an
algebraic
number,
the
quantity
i7r
is
a
root
of
an
algebraic
equation
with
integral
coefficients
(1)
01(x)
=
O,
whose
roots
are
aci=ir,
a2,
a3,
* *
a.n
Using
Euler's
relation
e$r+1
=0,
we
have
(2)
(eal
+
1)(ea2
+
1)
*
(ean
+
1)
=
0.
We
now
construct
an
algebraic
equation
with
integral
coefficients
whose
roots
are
the
exponents
in
the
expansion
of
(2).
First
consider
the
exponents
(3)
ai
+
a2,
ai
+
as,
a2
+
C
.
.
*
an-1
+
an
*
Harrison
Research
Fellow.
t
A.
Hurwitz,
Beweis
der
Transendenz
der
Zahl
e,
Mathematische
Annalen,
vol.
43,
1893,
pp.
220-221
(also
in
his
Mathematische
Werke,
vol.
2,
pp.
134-135).