Elementary function
The elementary functions called in mathematics keep cropping up, basic functions, from which can be many other features make using the basic arithmetic, concatenation, differentiation or integration. In this case, there is no generally accepted definition of when a function is called elementary and when.
The elementary functions often arise as solutions of simple differential or functional equation, and therefore - even more than the specific functions - for many natural sciences such as physics or chemistry fundamentally because they occur repeatedly in a variety of contexts.
From elementary integrable functions is used when the antiderivative of an elementary function itself is elementary. Major non- elemental integrable functions are the error function and the integral sine. Also, this way of speaking is not exact.
The manufacturer of the computer algebra system Mathematica, Wolfram Research Inc. which is one of the basic functions of the following:
- The power functions
- The Square root or the square root as the inverse of the exponential functions.
- The exponential function to the base ( the Euler's number )
- On a general basis with
- The natural logarithm as an inverse function of the exponential function.
- The trigonometric functions sine
- Cosine
- Tangent
- Cotangent
- Secant
- Cosecant
- The Inverse Trigonometric Functions as inverse functions of the trigonometric functions. arcsine
- Arccosine
- Arctangent
- Inverse cotangent
- Arkussekans
- Arkuskosekans
- The hyperbolic functions hyperbolic sine
- Hyperbolic cosine
- Hyperbolic tangent
- Hyperbolic cotangent
- Hyperbolic secant
- Hyperbolic cosecant
- The area functions as inverses of the hyperbolic functions. Area hyperbolic sine
- Areakosinus hyperbolic
- Area hyperbolic tangent
- Areakotangens hyperbolic
- Areasekans hyperbolic
- Areakosekans hyperbolic
- The Min-/Max-Funktion
- The Lambert W function, also called the Produktlogarithmus (but see below definition attempts)
Attempts to define
Since it has become despite all the uncertainty, to speak of " elementary functions ", in its wake also actual mathematical questions have emerged, repeated attempts have been made to provide accurate definitions of the term.
For example, is referred to in some sources such functions as "elementary " that are in finite number of steps alone using
- The four basic arithmetic operations,
- Exponentiation or Radizierens,
- Exponentialbildung or the logarithm and
- The concatenation
Can be formed ( ie the quotient of two polynomials ) of a rational function.
This definition allows to consider a mapping rule immediately whether it is elementary, and so make all the functions listed above - except for the Lambert W function - alone with the help of these operations express about
In addition, the so-called Risch algorithm was developed on the basis of this definition, which makes it possible to decide whether a given elementary function also has a basic primitive. Axiom is still the only computer algebra system with a complete implementation of the Risch algorithm.
Evidence that certain functions are not elementary, however, is possible only on the basis of a precise definition. For example, the primitive functions of or according to a set of Joseph Liouville not "elementary " in the above sense.