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 only the basic operations of addition, subtraction, multiplication, and division (without the need of taking limits). This is in contrast to an algebraic function. The most familiar transcendental functions are the exponential, trigonometric, and hyperbolic functions, and their inverses, such as the logarithm and inverse trigonometric functions. All special functions such as the gamma, error, bessel, and Riemann zeta functions are transcendental. Equations including transcendental functions are transcendental equations.
Definition Formally, an analytic function f {\displaystyle f} of one real or complex variable is transcendental if it is algebraically independent of that variable. This means the function does not satisfy any polynomial equation. For example, the function f {\displaystyle f} given by
f ( x ) = a x + b c x + d {\displaystyle f(x)={\frac {ax+b}{cx+d}}} for all x ≠ − d c {\displaystyle x\neq -{\frac {d}{c}}}
is not transcendental, but algebraic, because it satisfies the polynomial equation
( a x + b ) − ( c x + d ) f ( x ) = 0 {\displaystyle (ax+b)-(cx+d)f(x)=0} . Similarly, the function f {\displaystyle f} that satisfies the equation
f ( x ) 5 + f ( x ) = x {\displaystyle f(x)^{5}+f(x)=x} for all x {\displaystyle x}
is not transcendental, but algebraic, even though it cannot be written as a finite expression involving the basic arithmetic operations. This definition can be extended to functions of several variables.
History 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:
The mathematical notion of continuity as an explicit concept is unknown to Ptolemy. That he, in fact, treats these functions as continuous appears from his unspoken presumption that it is possible to determine a value of the dependent variable corresponding to any value of the independent variable by the simple process of linear interpolation. A revolutionary understanding of these circular functions occurred in the 18th 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 xy = 1 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 exp ( x ) = e x {\displaystyle \exp(x)=e^{x}} . Euler identified it with the infinite series ∑ k = 0 ∞ x k / k ! {\textstyle \sum _{k=0}^{\infty }x^{k}/k!} , where k! denotes the factorial of k. The even and odd terms of this series provide sums denoting cosh(x) and sinh(x), so that e x = cosh x + sinh x . {\displaystyle e^{x}=\cosh x+\sinh x.} These transcendental hyperbolic functions can be converted into circular functions sine and cosine by introducing (−1)k into the series, resulting in alternating series. After Euler, mathematicians view the sine and cosine this way to relate the transcendence to logarithm and exponent functions, often through Euler's formula in complex number arithmetic.
Examples The following functions are transcendental:
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