gives a sawtooth wave that varies from 0 to 1 with unit period.


gives a sawtooth wave that varies from min to max with unit period.



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Basic Examples  (3)

Evaluate numerically:

Plot over a subset of the reals:

SawtoothWave is a piecewise function over finite domains:

Scope  (25)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate with custom heights:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate efficiently at high precision:

SawtoothWave threads over lists in the last argument:

Specific Values  (4)

Value at zero:

Values at fixed points:

Evaluate symbolically:

Find a value of x for which SawtoothWave[{2,-3},x]=1 :

Visualization  (4)

Plot the SawtoothWave function:

Visualize scaled SawtoothWave functions:

Visualize SawtoothWave functions with different maximum and minimum values:

Plot SawtoothWave in three dimensions:

Function Properties  (4)

Function domain of SawtoothWave:

It is restricted to real inputs:

Function range of SawtoothWave:

SawtoothWave is periodic with period 1:

The area under one period is :

Differentiation and Integration  (4)

First derivative with respect to :

Derivative of the two-argument form with respect to :

The second and all higher derivatives are zero:

Integrals over finite domains:

Series Expansions  (5)


Since SawtoothWave is odd except for a constant, FourierTrigSeries gives a simpler result:

The two results are equivalent:

FourierCosSeries of a scaled SawtoothWave:

Taylor series at a smooth point:

Series expansion at a singular point:

Taylor expansion at a generic point:

Applications  (2)

Fourier decomposition of sawtooth wave signal:

Sawtooth wave sound sample:

Properties & Relations  (3)

Use FunctionExpand to expand SawtoothWave in terms of elementary functions:

Use PiecewiseExpand to obtain piecewise representation on an interval:


Possible Issues  (1)

SawtoothWave is not defined for complex arguments:

Wolfram Research (2008), SawtoothWave, Wolfram Language function,


Wolfram Research (2008), SawtoothWave, Wolfram Language function,


@misc{reference.wolfram_2021_sawtoothwave, author="Wolfram Research", title="{SawtoothWave}", year="2008", howpublished="\url{}", note=[Accessed: 22-September-2021 ]}


@online{reference.wolfram_2021_sawtoothwave, organization={Wolfram Research}, title={SawtoothWave}, year={2008}, url={}, note=[Accessed: 22-September-2021 ]}


Wolfram Language. 2008. "SawtoothWave." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2008). SawtoothWave. Wolfram Language & System Documentation Center. Retrieved from