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The Transcendence of | pi and e

Oswald Veblen
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 &#960

Robert E. Moritz
1900 Annals of Mathematics
EXTENSION 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

Sourangshu Ghosh
2021 Asian Research Journal of Mathematics
A number that can't be expressed as 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

David H. Bailey
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

David H. Bailey
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

Caroline E. Schuetzinger
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

Carlos E. Arreche
2013 Journal of Algebra
Our criteria imply, in particular, the ∂/∂ t-transcendence of the incomplete Gamma function γ(t,x), generalizing a result of Johnson, Reinhart, and Rubel .  ...  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

A. J. E. M. Janssen, J. S. H. van Leeuwaarden
2007 The Annals of Applied Probability
Symp. on Congestion Theory (1965) 137--169] 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

Jong Kyu Kim, E. Fred Schubert
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

Jerome Manyahi, Francis Xavier
2015 Pensamiento. Revista de Investigación e Información Filosófica
This article attempts to study 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]

Carlos E. Arreche, Thomas Dreyfus, Julien Roques
2019 arXiv   pre-print
We apply our criteria to prove a new result to 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

Andrea E V Ferrari, Lionel Mason
2019 Journal of Integrable Systems
From 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

Verne E. Dietrich, Arthur Rosenthal
1949 Bulletin of the American Mathematical Society
BIBLIOGRAPHY Thus, since 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.  ...