A hypertranscendental function or transcendentally transcendental function is a
transcendental
Transcendence, transcendent, or transcendental may refer to:
Mathematics
* Transcendental number, a number that is not the root of any polynomial with rational coefficients
* Algebraic element or transcendental element, an element of a field exten ...
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
In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used.
The intention is to i ...
with coefficients in Z (the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
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
Dmitry Dmitrievich Morduhai-Boltovskoi (russian: Дми́трий Дми́триевич Мордуха́й-Болтовско́й; Pavlovsk, August 8, 1876 – Rostov-on-Don, February 7, 1952) was a Russian mathematician, best known for his wo ...
in 1914.
Definition
One standard definition (there are slight variants) defines solutions of
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
s 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 In mathematics, Hölder's theorem states that the gamma function does not satisfy any algebraic differential equation whose coefficients are rational functions. This result was first proved by Otto Hölder in 1887; several alternative proofs have s ...
shows that the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...
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 mea ...
s, for example the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...
.
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
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...
(''cf.''
Hölder's theorem In mathematics, Hölder's theorem states that the gamma function does not satisfy any algebraic differential equation whose coefficients are rational functions. This result was first proved by Otto Hölder in 1887; several alternative proofs have s ...
)
Transcendental but not hypertranscendental functions
* The
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
,
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
, and the
trigonometric
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
and
hyperbolic
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
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, whic ...
s, including special cases such as
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
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 a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additi ...
s, in particular
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
s.
See also
*
Hypertranscendental number A complex number is said to be hypertranscendental if it is not the value at an algebraic point of a function which is the solution of an algebraic differential equation with coefficients in \mathbb /math> and with algebraic initial conditions.
...
Notes
References
* Loxton,J.H., Poorten,A.J. van der,
A class of hypertranscendental functions,
Aequationes Mathematicae
''Aequationes Mathematicae'' is a mathematical journal. It is primarily devoted to functional equations, but also publishes papers in dynamical systems, combinatorics, and geometry. As well as publishing regular journal submissions on these topic ...
, 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