In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using the basic operations of addition, subtraction, multiplication, and division. This is in contrast to an algebraic function.[1] [2]
Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions. Equations over these expressions are called transcendental equations
Formally, an analytic function of one real or complex variable is transcendental if it is algebraically independent of that variable.[3] This means the function does not satisfy a polynomial equation. For example the function
| f(x)= | ax+b |
| cx+d |
is not transcendental but algebraic, because it satisfies the polynomial equation
(ax+b)-(cx+d)f(x)=0.
Similarly, the function
f(x)
f(x)5+f(x)=x
is not transcendental, but algebraic, even though it cannot be written as a finite expression involving the basic arithmetic operations.
This can be extended to functions of several variables.
The transcendental functions sine and cosine were tabulated from physical measurements in antiquity, as evidenced in Greece (Hipparchus) and India (jya and koti-jya). In describing Ptolemy's table of chords, an equivalent to a table of sines, Olaf Pedersen wrote:
A revolutionary understanding of these circular functions occurred in the 17th century and was explicated by Leonhard Euler in 1748 in his Introduction to the Analysis of the Infinite. These ancient transcendental functions became known as continuous functions through quadrature of the rectangular hyperbola by Grégoire de Saint-Vincent in 1647, two millennia after Archimedes had produced The Quadrature of the Parabola.
The area under the hyperbola was shown to have the scaling property of constant area for a constant ratio of bounds. The hyperbolic logarithm function so described was of limited service until 1748 when Leonhard Euler related it to functions where a constant is raised to a variable exponent, such as the exponential function where the constant base is e. By introducing these transcendental functions and noting the bijection property that implies an inverse function, some facility was provided for algebraic manipulations of the natural logarithm even if it is not an algebraic function.
The exponential function is written Euler identified it with the infinite series where denotes the factorial of .
The even and odd terms of this series provide sums denoting and, so that
ex=\coshx+\sinhx.
Let be a positive constant. The following functions are transcendental:
For the second function
f2(x)
c
e
ex
c
e
f5(x)
f5(x)=logex=lnx
The most familiar transcendental functions are the logarithm, the exponential (with any non-trivial base), the trigonometric, and the hyperbolic functions, and the inverses of all of these. Less familiar are the special functions of analysis, such as the gamma, elliptic, and zeta functions, all of which are transcendental. The generalized hypergeometric and Bessel functions are transcendental in general, but algebraic for some special parameter values.
A function that is not transcendental is algebraic. Simple examples of algebraic functions are the rational functions and the square root function, but in general, algebraic functions cannot be defined as finite formulas of the elementary functions, as shown by the example above with
f(x)5+f(x)=x
The indefinite integral of many algebraic functions is transcendental. For example, the logarithm function arose from the reciprocal function in an effort to find the area of a hyperbolic sector.
Differential algebra examines how integration frequently creates functions that are algebraically independent of some class, such as when one takes polynomials with trigonometric functions as variables.
Most familiar transcendental functions, including the special functions of mathematical physics, are solutions of algebraic differential equations. Those that are not, such as the gamma and the zeta functions, are called transcendentally transcendental or hypertranscendental functions.[4]
If is an algebraic function and
\alpha
In many instances the exceptional set is fairly small. For example,
l{E}(\exp)=\{0\},
In general, finding the exceptional set of a function is a difficult problem, but if it can be calculated then it can often lead to results in transcendental number theory. Here are some other known exceptional sets:
\alpha ≠ 0,1
\beta
\alpha\beta
| ex | |
| l{E}\left(e |
\right)=\emptyset.
f(x)=\exp(1+\pix).
While calculating the exceptional set for a given function is not easy, it is known that given any subset of the algebraic numbers, say, there is a transcendental function whose exceptional set is .[9] The subset does not need to be proper, meaning that can be the set of algebraic numbers. This directly implies that there exist transcendental functions that produce transcendental numbers only when given transcendental numbers. Alex Wilkie also proved that there exist transcendental functions for which first-order-logic proofs about their transcendence do not exist by providing an exemplary analytic function.[10]
In dimensional analysis, transcendental functions are notable because they make sense only when their argument is dimensionless (possibly after algebraic reduction). Because of this, transcendental functions can be an easy-to-spot source of dimensional errors. For example, is a nonsensical expression, unlike or . One could attempt to apply a logarithmic identity to get, which highlights the problem: applying a non-algebraic operation to a dimension creates meaningless results.