References:
- Book Calculus by Thomas/Finney, 9th Edition
- “4: Transcendental Functions,” Mathematics LibreTexts, Nov. 07, 2013. https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Integral_Calculus/4%3A_Transcendental_Functions (accessed Jan. 28, 2026).
- C. to, “analytic function that does not satisfy a polynomial equation,” Wikipedia.org, Aug. 15, 2003. https://en.wikipedia.org/wiki/Transcendental_function (accessed Jan. 28, 2026).
- “Differentiation of transcendental functions.” Accessed: Jan. 28, 2026. [Online]. Available: https://www.cambridge.edu.au/downloads/education/extra/209/PageProofs/MMCAS%203and4%20on-line/1st%20pages%203and4%20CAS/Chapter%2011.pdf
Transcendental Functions
In mathematics, a transcendental function is a function that does not satisfy a polynomial equation with polynomial coefficients.
To put it simply: it is any function that is not algebraic.
While algebraic functions can be constructed using a finite number of elementary operations (addition, subtraction, multiplication, division, and taking roots), transcendental functions “transcend” (go beyond) these simple algebraic operations.
Key Characteristics
- Cannot be defined by finite algebra: You cannot express a transcendental function using a finite sequence of algebraic operations on the variable $x$.
- Infinite Series: They are often defined using infinite series (like the Taylor series).
- Transcendental Numbers: For many algebraic input values (like integers), these functions often output transcendental numbers (numbers like $\pi$ or $e$ that are not roots of polynomial equations with integer coefficients).
Common Types of Transcendental Functions
These are the most frequently encountered transcendental functions in calculus and analysis:
1. Exponential Functions
Functions where the variable x is in the exponent. Eg. $f(x) = e^x, g(x) = 2^x$
2. Logarithmic Functions
These are the inverses of exponential functions.
- Examples: $f(x)=\ln(x), g(x)=log_{10} (x)$
3. Trigonometric Functions
Functions relating to the angles and sides of triangles, defined via the unit circle or series.
- Examples: $sin(x), cos(x), tan(x)$
- Note: While $sin(x)$ is transcendental, for specific algebraic values of $x$ (like $30°$ or $\frac{\pi}{6}$), the output might be algebraic ($\frac{1}{2}$). However, the function itself is transcendental because this relation does not hold for all $x$.
4. Inverse Trigonometric Functions
- Examples: $arcsin(x), arctan(x)$
5. Hyperbolic Functions
Analogs of trigonometric functions defined using the exponential function.
- Examples: $sinh(x), cosh(x)$
Formal Definition
Formally, a function $f(x)$ is called algebraic if it satisfies a polynomial equation:
where the coefficients $P_i(x)$ are polynomials in $x$ with rational coefficients, and not all $P_i(x)$ are zero.
If a function cannot satisfy such an equation, it is transcendental.