A hypertranscendental function or transcendentally transcendental function is a
transcendental analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
which is not the solution of an
algebraic differential equation with coefficients in
(the
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s) and with algebraic
initial conditions.
History
The term 'transcendentally transcendental' was introduced by
E. H. Moore in 1896; the term 'hypertranscendental' was introduced by
D. D. Morduhai-Boltovskoi in 1914.
Definition
One standard definition (there are slight variants) defines solutions of
differential equations of the form
:
,
where
is a polynomial with constant coefficients, as ''algebraically transcendental'' or ''differentially algebraic''. Transcendental functions which are not ''algebraically transcendental'' are ''transcendentally transcendental''.
Hölder's theorem shows that the
gamma function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
is in this category.
[Lee A. Rubel, "A Survey of Transcendentally Transcendental Functions", ''The American Mathematical Monthly'' 96:777-788 (November 1989) ]
Hypertranscendental functions usually arise as the solutions to
functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
s, for example the
gamma function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
.
Examples
Hypertranscendental functions
* The zeta functions of
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
s, in particular, the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...
* The
gamma function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
(''cf.''
Hölder's theorem)
Transcendental but not hypertranscendental functions
* The
exponential function,
logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
, and the
trigonometric and
hyperbolic
Hyperbolic may refer to:
* of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics
** Hyperbolic geometry, a non-Euclidean geometry
** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...
functions.
* The
generalized hypergeometric function
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which ...
s, including special cases such as
Bessel function
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
s (except some special cases which are algebraic).
Non-transcendental (algebraic) functions
* All
algebraic function
In mathematics, an algebraic function is a function that can be defined
as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operati ...
s, in particular
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s.
See also
*
Hypertranscendental number
Notes
References
* Loxton, J.H., Poorten, A.J. van der,
A class of hypertranscendental functions,
Aequationes Mathematicae, Periodical volume 16
*
Mahler, K., "Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen", Math. Z. 32 (1930) 545-585.
* {{citation, mr=0028347, last=Morduhaĭ-Boltovskoĭ, first=D., title=On hypertranscendental functions and hypertranscendental numbers, language=Russian, journal=Doklady Akademii Nauk SSSR , series=New Series, volume=64, year=1949, pages= 21–24
Analytic functions
Mathematical analysis
Types of functions
Ordinary differential equations