A transcendental function is a function that cannot be expressed algebraically, i.e. as algebraic function. For example, sin x (see sine) cannot be expressed in algebraic terms and hence, if f(x) = sin x, f(x) is a transcendental function. Transcendental functions are represented by transcendental curves.

 

The following transcendental functions are used in elementary mathematics.

 

Exponential functions. For example, y = a ^x, y = e ^x, y = e ^x.

 

Logarithmic functions (inverse functions of the exponential functions). For example, y = log x, y = ln x, y = log (x ² – 1).

 

Trigonometric functions, also known as circular functions. These are: y = sin x, y = cos x, y = tan x, y = sec x, y = cosec x, and y = cot x.

 

Inverse trigonometric functions: y = sin^–1 x, y = cos^-1 x, y = tan^–1 x, y = sec^–1 x, y = cosec^–1 x, and y = cot^–1 x.

 

Hyperbolic functions: y = sinh x, y = cosh x, y = tanh x, y = sech x, y = cosech x, and y = coth x.

 

Inverse hyperbolic functions: y = sinh^–1 x, y = cosh^–1 x, y = tanh^–1 x, y = sech^–1 x, y = cosech^–1 x, and y = coth^–1 x.