Calculus How To

Sawtooth Function (Wave)

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sawtooth function

The sawtooth function, named after it’s saw-like appearance, is a relatively simple discontinuous function, defined as f (t) = t for the initial period (from -π to π in the above image).

This periodic function then repeats (as shown by the first and last lines on the above image). The additional periods are defined by a periodic extension of f (t):
f (t + kT) = f (t).

In function notation, the sawtooth can be defined as:
sawtooth wave definition


Graphing the Sawtooth Function

The function is challenging to graph, but can be represented by a linear combination of sine functions.

Some mathematical software have built in functions for the sawtooth. For example, in Mathematica, the function is:
Plot[SawtoothWave[x],{x,0,1}].

Fourier Series of the Sawtooth Wave

fourier series of the sawtooth wave

The sawtooth function can be represented by a Fourier series. The graph shows three terms; more are typically used.


The Fourier series representation is:
sawtooth fourier series

The Mathematica code (Jordan, n.d.) is:
L = 10
a=2*L/Pi
one=a*Sin[Pi x/L]
two=one‐(a/2)*Sin[2 Pi x/L]
three=two + (a/3)*Sin[3 Pi x/L]
Plot[{one, two, three},{x,‐L,L}]

Discontinuous Parts

The function has a jump discontinuity after every period. These jumps are called the function’s points of discontinuity (Edwards & Penney, 2002).

Alternative Definition for the Sawtooth Function

triangle wave function

The triangle wave function.


The term “sawtooth function” is also sometimes also used as another name for the triangle wave function (e.g. Trott, 2004).

References

Beerends, R. et al. (2003), Fourier and Laplace Transforms. Cambridge University Press.
Jordan, K. Fourier. Retrieved December 22, 2019 from: http://www.pitt.edu/~jordan/chem1000-s18/fourier.pdf
Spanier, J. and Oldham, K. B. An Atlas of Functions. Washington, DC: Hemisphere, p. 74, 1987.
Trott, M. (2004). The Mathematica GuideBook for Programming. New York: Springer-Verlag
Edwards, C. & Penney, D. (2007). Calculus, Early Transcendentals 7th Edition. Pearson.

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Stephanie Glen. "Sawtooth Function (Wave)" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/types-of-functions/sawtooth-function-wave/
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