E565 in the Enestrom index. Translated from the Latin original, "De plurimis
quantitatibus transcendentibus quas nullo modo per formulas integrales
exprimere licet" (1775).
Euler does not prove any results in this paper. It seems to me like he is
trying to develop some general ideas about special functions. He gives some
examples of numbers he claims but does not prove cannot be represented by
definite integrals of algebraic functions. Euler has the idea that if we knew
more about the function with the power series \$x^t\_n\$ where \$t\_n\$ is the
\$n\$th triangular number, this could lead to a proof of Fermat's theorem that
every positive integer is the sum of three triangular numbers. This doesn't end
of being fruitful for Euler, but in fact later Jacobi proves a lot of results
like this with his theta functions. The last paragraph (\S 9) is not clear to
me. My best reading is that there are infinitely many "levels" of
transcendental numbers and that this is unexpected or remarkable.
%0 Generic
%1 citeulike:3036240
%A Euler, Leonhard
%D 2007
%K Vor1800 available-in-tex-format mathematics pre1800 special-functions
%T On highly transcendental quantities which cannot be expressed by integral formulas
%U http://arxiv.org/abs/0711.4986
%X E565 in the Enestrom index. Translated from the Latin original, "De plurimis
quantitatibus transcendentibus quas nullo modo per formulas integrales
exprimere licet" (1775).
Euler does not prove any results in this paper. It seems to me like he is
trying to develop some general ideas about special functions. He gives some
examples of numbers he claims but does not prove cannot be represented by
definite integrals of algebraic functions. Euler has the idea that if we knew
more about the function with the power series \$x^t\_n\$ where \$t\_n\$ is the
\$n\$th triangular number, this could lead to a proof of Fermat's theorem that
every positive integer is the sum of three triangular numbers. This doesn't end
of being fruitful for Euler, but in fact later Jacobi proves a lot of results
like this with his theta functions. The last paragraph (\S 9) is not clear to
me. My best reading is that there are infinitely many "levels" of
transcendental numbers and that this is unexpected or remarkable.
@misc{citeulike:3036240,
abstract = {E565 in the Enestrom index. Translated from the Latin original, "De plurimis
quantitatibus transcendentibus quas nullo modo per formulas integrales
exprimere licet" (1775).
Euler does not prove any results in this paper. It seems to me like he is
trying to develop some general ideas about special functions. He gives some
examples of numbers he claims but does not prove cannot be represented by
definite integrals of algebraic functions. Euler has the idea that if we knew
more about the function with the power series \$\sum x^{t\_n}\$ where \$t\_n\$ is the
\$n\$th triangular number, this could lead to a proof of Fermat's theorem that
every positive integer is the sum of three triangular numbers. This doesn't end
of being fruitful for Euler, but in fact later Jacobi proves a lot of results
like this with his theta functions. The last paragraph (\S 9) is not clear to
me. My best reading is that there are infinitely many "levels" of
transcendental numbers and that this is unexpected or remarkable.},
added-at = {2009-08-02T17:14:35.000+0200},
archiveprefix = {arXiv},
author = {Euler, Leonhard},
biburl = {https://www.bibsonomy.org/bibtex/2a798e46f5c10d15d59db392b513cfd6a/rwst},
citeulike-article-id = {3036240},
citeulike-linkout-0 = {http://arxiv.org/abs/0711.4986},
citeulike-linkout-1 = {http://arxiv.org/pdf/0711.4986},
description = {my bookmarks from citeulike},
eprint = {0711.4986},
interhash = {591578d4ff9b0c0e24b36f5e9b2afb0c},
intrahash = {a798e46f5c10d15d59db392b513cfd6a},
keywords = {Vor1800 available-in-tex-format mathematics pre1800 special-functions},
month = Nov,
posted-at = {2008-07-23 08:33:12},
priority = {2},
timestamp = {2009-08-05T17:05:59.000+0200},
title = {On highly transcendental quantities which cannot be expressed by integral formulas},
url = {http://arxiv.org/abs/0711.4986},
year = 2007
}