Transcendental function

Transcendental function

A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation. In other words a transcendental function is a function which "transcends" algebra in the sense that it cannot be expressed in terms of the algebraic operations of addition, multiplication, and root extraction. Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions. Formally, an analytic function ƒ("z") of one real or complex variable "z" is transcendental if it is algebraically independent of that variable.

Algebraic and transcendental functions

The logarithm and the exponential function are examples of transcendental functions. "Transcendental function" is a term often used to describe the trigonometric functions, i.e., sine, cosine, tangent, cotangent, secant, and cosecant, also.

A function that is not transcendental is said to be algebraic. Examples of algebraic functions are rational functions and the square root function.

The operation of taking the indefinite integral of an algebraic function is a source of transcendental functions. For example, the logarithm function arose from the reciprocal function in an effort to find the area of a hyperbolic sector. Thus the hyperbolic angle and the hyperbolic functions sinh, cosh, and tanh are all transcendental.

In differential algebra one studies how integration frequently creates functions algebraically independent of some class taken as 'standard', such as when one takes polynomials with trigonometric functions as variables.

Dimensional analysis

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, log(10 m) is a nonsensical expression (unlike, e.g. log(x meters/y meters) or log(10) m). One could attempt to apply a logarithmic identity to get log(10) + log(m), which highlights the problem: applying a non-algebraic operation to a dimension creates meaningless results.

ome Examples

:f(x) = c^x,

:f(x)=x^x,

:f(x)=x^{x^{-1.

ee also

*Analytic function
*Complex function
*Generalized functions
*Special function


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  • Transcendental function — Transcendental Tran scen*den tal, a. [Cf. F. transcendantal, G. transcendental.] 1. Supereminent; surpassing others; as, transcendental being or qualities. [1913 Webster] 2. (Philos.) In the Kantian system, of or pertaining to that which can be… …   The Collaborative International Dictionary of English

  • transcendental function — Math. a function that is not an algebraic function. [1875 80] * * * transcendental function noun (mathematics) Any function that cannot be defined by a finite number of algebraic operations • • • Main Entry: ↑transcend …   Useful english dictionary

  • transcendental function — Math. a function that is not an algebraic function. [1875 80] * * * In mathematics, a function not expressible as a finite combination of the algebraic operations of addition, subtraction, multiplication, division, raising to a power, and… …   Universalium

  • transcendental function — noun a function which does not satisfy a polynomial equation whose coefficients are themselves polynomials …   Wiktionary

  • Transcendental — Tran scen*den tal, a. [Cf. F. transcendantal, G. transcendental.] 1. Supereminent; surpassing others; as, transcendental being or qualities. [1913 Webster] 2. (Philos.) In the Kantian system, of or pertaining to that which can be determined a… …   The Collaborative International Dictionary of English

  • Transcendental curve — Transcendental Tran scen*den tal, a. [Cf. F. transcendantal, G. transcendental.] 1. Supereminent; surpassing others; as, transcendental being or qualities. [1913 Webster] 2. (Philos.) In the Kantian system, of or pertaining to that which can be… …   The Collaborative International Dictionary of English

  • Transcendental equation — Transcendental Tran scen*den tal, a. [Cf. F. transcendantal, G. transcendental.] 1. Supereminent; surpassing others; as, transcendental being or qualities. [1913 Webster] 2. (Philos.) In the Kantian system, of or pertaining to that which can be… …   The Collaborative International Dictionary of English

  • Transcendental functions — Function Func tion, n. [L. functio, fr. fungi to perform, execute, akin to Skr. bhuj to enjoy, have the use of: cf. F. fonction. Cf. {Defunct}.] 1. The act of executing or performing any duty, office, or calling; performance. In the function of… …   The Collaborative International Dictionary of English

  • Function — Func tion, n. [L. functio, fr. fungi to perform, execute, akin to Skr. bhuj to enjoy, have the use of: cf. F. fonction. Cf. {Defunct}.] 1. The act of executing or performing any duty, office, or calling; performance. In the function of his public …   The Collaborative International Dictionary of English

  • Transcendental — can refer to:In mathematics: * Transcendental number, a class of irrational numbers * Transcendental function, a class of functionsIn philosophy and religion: * Transcendence (philosophy) * Transcendental idealism, a philosophical doctrine… …   Wikipedia

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