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The Transcendence of | pi and e

1904
*
The American mathematical monthly
*

*THE*

*TRANSCENDENCE*

*OF*X AND

*e*, By DR. OSWALD VEBLEN,

*The*University

*of*Chicago. ?1.

*The*proof that Xr is a transcendental number is ordinarily arranged as follows. ...

*The*proof* that (C) is impossible is so difficult for most students that it *

*The*prlncipal references in English on

*the*subject

*of*

*the*

*transcendence*

*of*7r and

*e*seem to be

*the*translation by W. W. ...

##
###
Extension of Hurwitz's Proof for the Transcendence of e to the Transcendence of π

1900
*
Annals of Mathematics
*

EXTENSION

doi:10.2307/2007181
fatcat:5cekwfdl4rdy3kbgx52t43wuqq
*OF*HURWITZ'S PROOF FOR*THE**TRANSCENDENCE**OF**e*TO*THE**TRANSCENDENCE**OF*7r.* By ROBERT*E*. MORITZ. ...*The*application*of**the*law*of**the*mean to*the*function*e*-xF(x) leads to*the*relation F(x) -exF(O) = -xe(l-*e*)xf(Ox) 0 < 6 < 1. ...##
###
On the Irrationality and Transcendence of Rational Powers of e

2021
*
Asian Research Journal of Mathematics
*

A number that can't be expressed as

doi:10.9734/arjom/2021/v17i230277
fatcat:bqh7wx446rcs5k3w4ssnltiznq
*the*ratio*of*two integers is called an irrational number. Euler and Lambert were*the*first mathematicians to prove*the*irrationality and*transcendence**of**e*. ... In this article we review various proofs*of*irrationality and*transcendence**of*rational powers*of**e*founded by mathematicians over*the*time. ... cte tv ( ∫ ∞ t f k*e*−vx dx) (4.2) Q = n ∑ t=1 cte tv ( ∫ t 0 f k*e*−vx dx) (4.3) We now derive two lemmas to prove*the**transcendence**of**e*. ...##
###
Numerical Results on the Transcendence of Constants Involving π, e, and Euler's Constant

1988
*
Mathematics of Computation
*

*The*recursive form

*of*

*the*Ferguson-Forcade algorithm has been implemented with multiprecision arithmetic on

*the*Cray-2 supercomputer at NASA Ames Research C

*e*nter. ... These computations established that

*the*following constants cannot satisfy any algebraic equation

*of*degree eight or less with integer coe cients whose Euclidean norm is 10 9 or less:

*e*=

*e*+ log

*e*

*e*= ... Acknowledgement

*The*author wishes to acknowledge

*the*patient assistance

*of*Prof. Helaman Ferguson in suggesting this work and in explicating

*the*details

*of*his algorithm. ...

##
###
Numerical results on the transcendence of constants involving $\pi,e$, and Euler's constant

1988
*
Mathematics of Computation
*

*The*resulting computer program has been used to probe

*the*question

*of*whether or not certain constants involving it,

*e*, and 7 satisfy any simple polynomial equations. ... These computations established that

*the*following constants cannot satisfy any algebraic equation

*of*degree eight or less with integer coefficients whose Euclidean norm is 109 or less:

*e*/tr,

*e*+ it, loge ...

*The*author wishes to acknowledge

*the*patient assistance

*of*Professor Helaman Ferguson in suggesting this work and in explicating

*the*details

*of*his algorithm. ...

##
###
The Gnoseological Transcendence in Nicolai Hartmann's Metaphysics of Cognition

1966
*
The Thomist
*

*e*deal here then with two propositions which are equallysignificant as expression

*of*

*the*phenomenon, "but which are at

*the*same time contradictory to each other. ...

*The*

*transcendent*concept

*of*cognition necessarily comprises

*the*

*transcendent*concept

*of*truth. ... gnoseological

*transcendence*stretches -although in varying degrees -from

*the*lowest to

*the*highest levels

*of*human intellection. ...

##
###
A Galois-theoretic proof of the differential transcendence of the incomplete Gamma function

2013
*
Journal of Algebra
*

Our criteria imply, in particular,

doi:10.1016/j.jalgebra.2013.04.037
fatcat:yb7reatosvdivnwpbed4opkn4i
*the*∂/∂ t-*transcendence**of**the*incomplete Gamma function γ(t,x), generalizing a result*of*Johnson, Reinhart, and Rubel [9]. ... We give simple necessary and sufficient conditions for*the*∂/∂ t-*transcendence**of**the*solutions to a parameterized second order linear differential equation*of**the*form ∂^2 Y/∂ x^2 - p ∂ Y/∂ x = 0, where ... It is a pleasure to acknowledge their essential contributions to*the*present form*of*this work. I am deeply grateful to*the*anonymous referees for their many helpful suggestions and encouragement. ...##
###
On Lerch's transcendent and the Gaussian random walk

2007
*
The Annals of Applied Probability
*

Symp. on Congestion Theory (1965) 137--169]

doi:10.1214/105051606000000781
fatcat:i3s6a2mxffbv7oba6wsgokreyi
*of**the*mean for β↓0. We build upon*the*work*of*Chang and Peres [Ann. ... We present explicit expressions for*the*mean and variance*of**the*maximum M={S_n:n≥0}. ... We are grateful to Bert Zwart for discussions that motivated this paper and for pointing us to*the*references [10, 18] . ...##
###
Transcending the replacement paradigm of solid-state lighting

2008
*
Optics Express
*

*The*field

*of*photonics starts with

*the*efficient generation

*of*light. ... We will show that such future, "smart" light sources, can harness

*the*huge potential

*of*LEDs by offering multi-dimensional controllability that will enhance

*the*functionality and performance

*of*light sources ... Ideally,

*the*forward energy (

*e*V f ) would be equal to

*the*photon energy, so that η Vf = 100%. ...

##
###
Implications of Quantum Logic to the Notion of Transcendence

2015
*
Pensamiento. Revista de Investigación e Información Filosófica
*

This article attempts to study

doaj:72fa1c15601b4b99affb485fa82f5869
fatcat:meffoth3snfnjinep7vm56ddfm
*the*notion*of**Transcendence*from*the*perspective*of*Quantum Mechanics. ... A movement toward*the*deeper reality*of*God is what is behind*the*notion*of**transcendence*. ... In fuzzy logic A ∩ A -≠ Ø and A ∪ A -≠*E*. J. MANYAHI -F. XAVIER, QUANTUM LOGIC AND NOTION*OF**TRANSCENDENCE*951 J. MANYAHI -F. ...##
###
Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions
[article]

2019
*
arXiv
*
pre-print

We apply our criteria to prove a new result to

arXiv:1809.05416v2
fatcat:u3rid2z4cbewfbczf4tu7gws7i
*the*effect that most elliptic hypergeometric functions are differentially transcendental. ... We develop general criteria that ensure that any non-zero solution*of*a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases*of*interest, such ... For this reason, we prove two related but distinct results on differential*transcendence*: (A) differential*transcendence**of*solutions*of**the*elliptic hypergeometric equation (5.4) , where we think*of*...##
###
Meromorphic Painlevé III transcendents and the Joukowski correspondence

2019
*
Journal of Integrable Systems
*

From

doi:10.1093/integr/xyz001
fatcat:lp3h3soqjrct3prgdce5bkwchm
*the*point*of*view*of*generalized monodromy data,*the*condition is equivalent to triviality*of**the*Stokes matrices and half-integral exponents*of*formal monodromy. ... We obtain canonically defined representations in terms*of*a Birkhoff factorization whose entries are related to*the*data at*the*origin and*the*Painlevé constants. ... Second, we express*the*system equation (29) in terms*of**the*J-matrix, so that we can see how*the*Painlevé*transcendent*is given in terms*of*its components. 9 In order to show this, we multiply equation ...##
###
Transcendence of factorial series with periodic coefficients

1949
*
Bulletin of the American Mathematical Society
*

BIBLIOGRAPHY Thus, since

doi:10.1090/s0002-9904-1949-09316-3
fatcat:mbaj455zlrektlwep6nszjqd6a
*the*sum*of**the*rth powers*of*all*the*mth roots*of*unity is zero if r is not divisible by m, we obtain: m g m ]£*e*» h '* = m + o+ --• +0+ m -... + O + Ö M ---+ 0+.-. . ... Let m be*the*"length"*of**the*period, that is,*the*number*of*coefficients a n belonging to*the*period, and let co be a primitive mth root*of*unity. ...##
###
A Pastoral da Saúde e sua importância no mundo da saúde: da presença solidária ao transcender a dor e o sofrimento

The Pastoral of Health and its relevance in the world of health: from solidary presence to transcending pain and suffering

2010
*
O Mundo da Saúde
*

The Pastoral of Health and its relevance in the world of health: from solidary presence to transcending pain and suffering

a Pastoral da saúde

doi:10.15343/0104-7809.20104547552
fatcat:mdhcuqo2zzb2jglu7q4guyppp4
*e*sua imPortância no mundo da saúde: da Presença solidária ao*transcender*a dor*e*o sofrimento ...*transcendê*--lo. ... por meio da solidariedade*e*da busca humana pelo sentido da sua existência, sentido capaz de levar até a*transcender*o sofrimento. ...##
###
Human Openness to Divine Transcendence in the Anthropology of V. E. Frankl

La apertura humana a la trascendencia divina en la antropología de V. E. Frankl

2016
*
Scripta Theologica
*

La apertura humana a la trascendencia divina en la antropología de V. E. Frankl

*E*. ...

*E*. ...

*E*. ...

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