# 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**