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The converse is not true: there are entire transcendental functions f such that f (α) is an algebraic number for any algebraic α. [6] For a given transcendental function the set of algebraic numbers giving algebraic results is called the exceptional set of that function. [7] [8] Formally it is defined by:
A modern, abstract point of view contrasts large function spaces, which are infinite-dimensional and within which most functions are 'anonymous', with special functions picked out by properties such as symmetry, or relationship to harmonic analysis and group representations. See also List of types of functions
John Herschel, Description of a machine for resolving by inspection certain important forms of transcendental equations, 1832. In applied mathematics, a transcendental equation is an equation over the real (or complex) numbers that is not algebraic, that is, if at least one of its sides describes a transcendental function. [1] Examples include:
A transcendental number is a (possibly complex) number that is not the root of any integer polynomial. Every real transcendental number must also be irrational, since a rational number is the root of an integer polynomial of degree one. [17] The set of transcendental numbers is uncountably infinite.
Self-concordant function; Semi-differentiability; Semilinear map; Set function; List of set identities and relations; Shear mapping; Shekel function; Signomial; Similarity invariance; Soboleva modified hyperbolic tangent; Softmax function; Softplus; Splitting lemma (functions) Squeeze theorem; Steiner's calculus problem; Strongly unimodal ...
Higher Transcendental Functions - Volume I - Based, in part, on notes left by Harry Bateman (PDF). Bateman Manuscript Project. Vol. I (1 ed.). New York / Toronto / London: McGraw-Hill Book Company, Inc. LCCN 53-5555. Contract No. N6onr-244 Task Order XIV. Project Designation Number: NR 043-045. Order No. 19545.
Mahler proved that the exponential function sends all non-zero algebraic numbers to S numbers: [28] [29] this shows that e is an S number and gives a proof of the transcendence of π. This number π is known not to be a U number. [30] Many other transcendental numbers remain unclassified.
Painlevé (1900, 1902) found that forty-four of the fifty equations are reducible in the sense that they can be solved in terms of previously known functions, leaving just six equations requiring the introduction of new special functions to solve them. There were some computational errors, and as a result he missed three of the equations ...