TRANSCENDENTAL FUNCTION

Definition: TRANSCENDENTAL FUNCTION

TRANSCENDENTAL FUNCTION

1. (Math.) See under Function . Syn: Transcendental , Empirical . Usage: These terms, with the corresponding nouns, transcendentalism and empiricism, are of comparatively recent origin. Empirical refers to knowledge which is gained by the experience of actual phenomena, without reference to the principles or laws to which they are to be referred, or by which they are to be explained. Transcendental has reference to those beliefs or principles which are not derived from experience, and yet are absolutely necessary to make experience possible or useful. Such, in the better sense of the term, is the transcendental philosophy, or transcendentalism. Each of these words is also used in a bad sense, empiricism applying to that one-sided view of knowledge which neglects or loses sight of the truths or principles referred to above, and trusts to experience alone; transcendentalism, to the opposite extreme, which, in its deprecation of experience, loses sight of the relations which facts and phenomena sustain to principles, and hence to a kind of philosophy, or a use of language, which is vague, obscure, fantastic, or extravagant.

Source: Webster's Revised Unabridged Dictionary (1913)

Specialty Definition: Transcendental function

A transcendental function is a function which does not satisfy a polynomial equation whose coefficients are themselves polynomials. Saying it more technically, a function of one variable is transcendental if it is algebraically independent of that variable.

The logarithm and the exponential function are examples of transcendental functions.

A function which 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 a function is a prolific source of transcendental functions, in the way that the logarithm function arises from the reciprocal function. In differential algebra it is studied how integration frequently creates functions algebraically independent of some class taken as 'standard', such as it created by taking polynomials with trigonometric functions.

Modern Translation: TRANSCENDENTAL FUNCTION

Language		Translations for "TRANSCENDENTAL FUNCTION"; alternative meanings/domain in parentheses.
Japanese Kanji		超越関数 . (various references)

Japanese Katakana		ちょうえつか�"すう. (various references)

Pig Latin		anscendentaltray unctionfay
Source: compiled by the editor from various translation references.