Definition:Transcendental Function
Definition
A transcendental function is an analytic function which is not an algebraic function.
That is, it cannot be expressed as a polynomial equation.
Defined by Integral
Definition:Transcendental Function/Defined by Integral
Defined by Differential Equation
Definition:Transcendental Function/Defined by Differential Equation
Also defined as
Some sources define a transcendental function as a real function or complex function which is not an elementary function.
However, the distinction between what is and is not an elementary function is more or less arbitrary, consisting of both algebraic functions and those derived from the exponential function, which itself is not algebraic.
The current school of thought appears to be that this definition: "not an elementary function" is actually considered to be erroneous.
However, the distinction is not considered particularly important nowadays.
As long as it is made clear which definition is being used at the time, that would be adequate.
Examples
Logarithm Functions
The logarithm functions are transcendental.
Trigonometric Functions
The trigonometric functions are transcendental.
Hyperbolic Functions
The hyperbolic functions are transcendental.
Exponential Functions
The exponential functions are transcendental.
Inverse Trigonometric Functions
The inverse trigonometric functions are transcendental.
Inverse Hyperbolic Functions
The inverse hyperbolic functions are transcendental.
Also see
- Results about transcendental functions can be found here.
Sources
- 1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.3$ Functions of a Real Variable: $\text {(g)}$ Transcendental Functions
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $10$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): function (map, mapping)
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): transcendental function
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): function (map, mapping)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): transcendental function