Calculus How To

Orthogonal Functions

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Types of Functions >


  1. What is Orthogonality?
  2. Orthogonal Functions
  3. Orthogonal Series

What is orthogonality?

Orthogonality involves right angles. For example, two triangle side lengths at right angles to each other or two independent random variables. The two definition sound unrelated, but they are actually analogous in a mathematical sense:

Two independent random variables X and Y are related by the formula:
Var (X + Y) = Var(X) + Var(Y)
which is analogous to side lengths a and b, related to the hypotenuse in the Pythagorean theorem by √(a2 + b2) [1].

Orthogonal Functions

Orthogonal functions are two functions with an inner product of zero.

Orthogonal functions are particularly useful for finding solutions to partial differential equations like Schrodinger’s equation and Maxwell’s equations.

Orthogonal Functions and Inner Product Example

The inner product of two real-valued functions f and g on the closed interval [a, b] is given by the following definite integral:
inner product

If the number given by the formula is zero, then the functions are orthogonal. You might see slightly different versions of this formula, but the underlying math is the same. For example, the following formula is specifically set up to deal with a steel rod heat conduction problem (Adomatis, n.d.):
orthogonal functions linear algebra

As well as the different function notation, note that the parentheses have been replaced by angled brackets < >. The angled brackets represent the inner product of two functions. It’s technically correct to use them, but I’m leaving them out here so that the following example is accessible to calculus students without any background in set theory or linear algebra.


Example question: Show that the functions f(x) = 1 + x and g(x) = x – x2 are orthogonal on the interval [-2, 2] by calculating the inner product.


Step 1: Set up the definite integral for the inner product:
orthogonal function

Step 2: Solve the definite integral. I used a calculator (integral calculator) to get zero as a solution.


Although the usual definition states that the inner product has to be zero in order for a function to be orthogonal, some functions are (perhaps strangely) orthogonal with themselves. For example, f(x) = cos (nx) is an orthogonal function over the closed interval [-π,π]. When the same function is evaluated using the inner product, it’s orthogonal if the inner product isn’t zero.

Orthogonal Series

The general orthogonal series is represented by:
general orthogonal series

All of the following series are special cases of these equations (WSU, 2020):

The theory of orthogonal series developed in mathematical physics as a result of solutions of problems via the Fourier method. Orthogonal series is one way to solve linear integral equations with symmetric kernels. The theory led to the conception of Hilbert space (Lyusternik & ‘Skii (1965).


[1] Kjos-Hanssen, B. (2019). Statistics for Calculus Students. Retrieved April 28, 2021 from:

Admonatis, R. (n.d.). Chapter 8. Orthogonal Function Sequences. Retrieved February 13, 2020 from:
Bringhurst, R. (2002). The Elements of Typographical Style.
Orthogonal Functions. Retrieved February 13, 2020 from:
L. Lyusternik & A.’Skii (Eds.)(1965). Chapter IV. Orthogonal Series and Orthogonal Systems. Pergamon. Retrieved November 18, 2020 from:
WSU Research Foundation. Introduction to Orthogonal Series. Retrieved November 18, 2020 from:

Stephanie Glen. "Orthogonal Functions" From Calculus for the rest of us!

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