From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: What are transcendental functions... Date: 2 Feb 2000 11:15:52 -0500 Newsgroups: sci.math Summary: [missing] In article , Just Another Victim of the Ambient Morality wrote: :I know what a transcendental number is, but I can't quite remember what :the definition of a transcendental function is. I can give you examples :of transcendental functions (sin, e^x, log(x), etc..) but I can't :characterize them apart from algebraic functions (x^2+1, for instance). :A definition for algebraic functions might be good too. : A function f (of variables x_1, ..., x_n) is algebraic if there is a polynomial P in (n+1) variables, non-constant in the last variable, such that P(x_1, ... , x_n, f(x_1, ... , x_n)) = 0 identically. So, the square root function is algebraic: P(x_1, f) = x_1 - f^2 . And so on. It is a plausible fact with a tedious proof that algebraic functions composed with algebraic function result in algebraic functions, and that includes sums, products, etc. Functions which are not algebraic are called transcendental. Some authors prefer to restrict this classification to "analytic" functions, which are locally representable by power series. The terminology is inherited from the shaky early times of calculus when people could not imagine functions other than analytic (except for singularities). This illusion was shattered initially by Fourier (his trigonometric series violated the belief in analytic continuation as the only way of extending a function), and later substantially by Weierstrass when he introduced continuous functions which are nowhere differentiable, and later yet by discoveries in functional analysis, when these supposed "monster functions" were shown to form an overwhelming majority, and analytic functions turned out to be extremely rare (but no less interesting or useful). How do I prove that a function is algebraic? Produce a polynomial mentioned in the definition, or show that it was built in an algebraic way from previously recognized algebraic functions. How do I prove that a function is transcendental? That's tougher: one has to invent a property that the given function has but algebraic functions don't. For example, sin(x) is non-constant periodic, while the only periodic algebraic functions are constants (this has to be proved), so sin(x), while being analytic, is transcendental. Hope some of it helps, ZVK(Slavek). ============================================================================== From: dlrenfro@gateway.net (Dave L. Renfro) Subject: Re: What are transcendental functions Date: 3 Feb 2000 00:14:45 -0500 Newsgroups: sci.math G.E. Ivey [sci.math 2 Feb 00 13:33:07 -0500 (EST)] wrote > Actually, you can't define transcendental functions WITHOUT > defining algebraic functions! > An "algebraic" function is any function that can be > expressed as a finite number of "algebraic" operations- adding, > subtracting, multiplying, dividing, and taking roots. Any > function that is not algebraic is transcendental. This isn't quite correct. The usual definition goes this way: y = f(x) is an ALGEBRAIC FUNCTION if y satisfies an equation of the form y^n + [R_{n-1}]*y^{n-1} + ... + [R_1]*y + R_0 = 0 for some rational functions R_0, R_1, ..., R_{n-1}. [The real-number coefficients of the rational functions can be *any* real numbers. In particular, these coefficients can be transcendental numbers.] Thus, y = sqrt(x) is algebraic since it satisfies y^2 - x = 0. Any function that can be expressed using a finite combination of additions, subtractions, multiplication, divisions, and extractions of positive integer-valued roots to the polynomial functions (equivalently, to constant functions and the function f(x) = x) is called an EXPLICIT ALGEBRAIC FUNCTION. Ivey's description gives only explicit algebraic functions. There exist algebraic functions that are not explicit algebraic. Hardy gives without proof the function defined by y^5 - y - x = 0 (see below, p. 55). It is not difficult to prove that no function with a positive minimal period can be algebraic. See Hardy (p. 57) and Pierpont (pp. 129-130). [Subtract the monic minimal degree representation equation for f(x) from the monic minimal degree representation for f(x+P), where P > 0 is a period of f.] Thus, none of the trig. functions are algebraic. Also, exp(x) is not algebraic (see Speck, p. 201). [Differentiate N times the minimal degree representation for exp(x) that employs polynomial coefficients, where N is one more than the algebraic degree of the "constant term" polynomial (i.e. the polynomial that *isn't* multiplied by a positive integer power of exp(x) in this equation), and divide both sides of the resulting equation by exp(x) to get a contradiction of minimal degree'ness.] It is now not difficult to show that exp(a*x) is not algebraic for any nonzero real number a, and thus no exponential function is linear. Finally, one can show that the inverse (when it exists) of an algebraic function is an algebraic function, and therefore it follows that the inverse trig. functions and various logarithmic functions are not algebraic. See Pierpont (p. 137) and Speck (theorem 8 on p. 202). G. H. Hardy, A COURSE OF PURE MATHEMATICS, 9'th edition, Cambridge Univ. Press, 1947. [see pp. 52-57] James Pierpont, THE THEORY OF FUNCTIONS OF REAL VARIABLES, volume 1, Ginn and Company, 1905. [see pp. 123-137] G. P. Speck, "Elementary transcendental functions", Mathematics Magazine 42 (1969), 200-202. There is also a class of functions known as the ELEMENTARY FUNCTIONS that comes up when one makes statements such as "no antiderivative of exp(-x^2) is an elementary function". This class includes all the explicitly algebraic functions, but not all the algebraic functions. The class of elementary functions also includes non-algebraic functions (trig., exponential, etc.). I believe the class of elementary functions is a proper subset of the collection of functions described in the next sentence. A larger class of functions are those functions that are solutions to a linear differential equation with rational function coefficients. I don't know what these are called (algebraically transcendental??), but this class of functions includes all the trig. functions, exponentials, etc., as well as things like Bessel functions, Legendre functions, etc. Functions not belonging to this class are sometimes called TRANSCENDENTALLY TRANSCENDENTAL functions. See the following papers for more information: R. D. Carmichael, "On transcendentally transcendental functions", Trans. Amer. Math. Soc. 14 (1913), 311-319. E. Gourin and J. F. Ritt, "An assemblage-theoretic proof of the existence of transcendentally transcendental functions", Bull. Amer. Math. Soc. 33 (1927), 182-184. J. F. Ritt, "Transcendental transcendency of certain functions of Poincar�", Bull. Amer. Math. Soc. 31 (1925), 300. J. F. Ritt, "Transcendental transcendency of certain functions of Poincare", Math. Ann. 95 (1926), 671-682. An internet search for "transcendentally transcendental" turned up Richard Daquila's Ph.D. abstract at , but his usage of the term doesn't appear to be the same as the usage in the papers I cited. Dave L. Renfro ============================================================================== From: "G. A. Edgar" Subject: Re: What are transcendental functions Date: Thu, 03 Feb 2000 08:49:17 -0500 Newsgroups: sci.math [deletia --djr] Some comments/questions/remarks on definitions... Dave L. Renfro wrote: > > This isn't quite correct. The usual definition goes this way: > y = f(x) is an ALGEBRAIC FUNCTION if y satisfies an equation > of the form > > y^n + [R_{n-1}]*y^{n-1} + ... + [R_1]*y + R_0 = 0 > > for some rational functions R_0, R_1, ..., R_{n-1}. [The > real-number coefficients of the rational functions can be > *any* real numbers. In particular, these coefficients can be > transcendental numbers.] Hi, Dave. Do you want the polynomial to be irreducible? Would you allow for example f(x) = 1 for x rational and f(x) = 0 for x irrational? [It satisfies y^2-y=0 after all.] Perhaps the definition allows only analytic functions or something? > [...] > > There is also a class of functions known as the ELEMENTARY > FUNCTIONS that comes up when one makes statements such > as "no antiderivative of exp(-x^2) is an elementary function". > This class includes all the explicitly algebraic functions, > but not all the algebraic functions. The class of elementary > functions also includes non-algebraic functions (trig., > exponential, etc.). I believe the class of elementary > functions is a proper subset of the collection of functions > described in the next sentence. Definitions I have seen include "algebraic" not merely "explicitly algebraic" ... for example J. F. Ritt, Integration in Finite Terms (1948). > > A larger class of functions are those functions that are > solutions to a linear differential equation with rational > function coefficients. I don't know what these are called > (algebraically transcendental??), but this class of functions > includes all the trig. functions, exponentials, etc., as > well as things like Bessel functions, Legendre functions, etc. > Functions not belonging to this class are sometimes called > TRANSCENDENTALLY TRANSCENDENTAL functions. Here, I would allow all algebraic differential equations, not only the linear ones... That is, 0 = a nontrivial polynomial in x, y, y', y'', ... If my function is the solution of no such DE, then call it transcendentally transcendental. An example is the gamma function. This definition is in, for example, J. F. Ritt, "Transcendental transcendency of certain functions of Poincare" Math. Ann. 95 (1926) 671-682. > See the following > papers for more information: > > R. D. Carmichael, "On transcendentally transcendental > functions", Trans. Amer. Math. Soc. 14 (1913), 311-319. > > E. Gourin and J. F. Ritt, "An assemblage-theoretic proof of > the existence of transcendentally transcendental functions", > Bull. Amer. Math. Soc. 33 (1927), 182-184. > > J. F. Ritt, "Transcendental transcendency of certain functions > of Poincar�", Bull. Amer. Math. Soc. 31 (1925), 300. > > J. F. Ritt, "Transcendental transcendency of certain functions > of Poincare", Math. Ann. 95 (1926), 671-682. > > An internet search for "transcendentally transcendental" > turned up Richard Daquila's Ph.D. abstract at > > , > > but his usage of the term doesn't appear to be the > same as the usage in the papers I cited. > -- Gerald A. Edgar edgar@math.ohio-state.edu