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Further Transcendental Functions

A transcendental function is a function that is not algebraic.

Learning Objective

  • Identify a transcendental function as one that cannot be expressed as the finite sequence of an algebraic operation


Key Points

    • Transcendental functions cannot be expressed as a solution of a polynomial equation whose coefficients are themselves polynomials with rational coefficients.
    • Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions.
    • Transcendental functions can be an easy-to-spot source of dimensional errors.

Terms

  • trigonometric function

    any function of an angle expressed as the ratio of two of the sides of a right triangle that has that angle, or various other functions that subtract 1 from this value or subtract this value from 1 (such as the versed sine)

  • polynomial

    an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power

  • exponential function

    any function in which an independent variable is in the form of an exponent; they are the inverse functions of logarithms


Full Text

A transcendental function is a function that is not algebraic. Such a function cannot be expressed as a solution of a polynomial equation whose coefficients are themselves polynomials with rational coefficients. Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions.

Trigonometric Functions

Top panel: Trigonometric function sinθ for selected angles $\theta$, $\pi - \theta$, $\pi + \theta$, and $2\pi - \theta$ in the four quadrants. Bottom panel: Graph of sine function versus angle.

Formally, an analytic function $ƒ(z)$ of the real or complex variables $z_1, \cdots ,z_n$ is transcendental if $z_1, \cdots ,z_n$, $ƒ(z)$ are algebraically independent, i.e., if $ƒ$ is transcendental over the field $C(z_1, \cdots ,z_n)$.

A transcendental function is a function that "transcends" algebra in the sense that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, power, and root extraction.

The following functions are transcendental:

$f_1(x) = x^{\Pi }$

$f_2(x) = c^{x }, x \neq 0, 1$

$f_3(x) = x^{x}$

$f_4(x) = x^{\frac{1}{x}}$

$f_5(x) = \log_{c}(x)$

$f_6(x) = \sin(x)$

Note that, for $ƒ_2$ in particular, if we set $c$ equal to $e$, the base of the natural logarithm, then we find that $e^x$ is a transcendental function. Similarly, if we set $c$ equal to $e$ in ƒ5, then we find that $\ln(x)$, the natural logarithm, is a transcendental function.

In dimensional analysis, transcendental functions are notable because they make sense only when their argument is dimensionless (possibly after algebraic reduction). Because of this, transcendental functions can be an easy-to-spot source of dimensional errors. For example, $\log(5 \text{ meters})$ is a nonsensical expression, unlike $\log \left ( \frac{5 \text{ meters}}{3 \text{ meters}} \right )$ or $\log(3)\text{ meters}$. One could attempt to apply a logarithmic identity to get $\log(10) + \log(m)$, which highlights the problem: applying a non-algebraic operation to a dimension creates meaningless results.

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