Algebraic and Transcendental Functions
For more details on this topic, see elementary function (differential algebra).The logarithm and the exponential function are examples of transcendental functions. Transcendental function is a term often used to describe the trigonometric functions (sine, cosine, tangent, their reciprocals cotangent, secant, and cosecant, the now little-used versine, haversine, and coversine, their analogs the hyperbolic functions and so forth).
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.
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.
Read more about this topic: Transcendental Function
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