Evaluate numerically:

Numerically evaluate a sawtooth with specified range:

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:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix SawtoothWave function using MatrixFunction :

Value at zero:

Values at fixed points:

Evaluate symbolically:

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

Plot the SawtoothWave function:

Visualize scaled SawtoothWave functions:

Visualize SawtoothWave functions with different maximum and minimum values:

Plot SawtoothWave in three dimensions:

Function domain of SawtoothWave :

It is restricted to real inputs:

Function range of SawtoothWave [x]:

SawtoothWave is periodic with period 1:

The area under one period is :

SawtoothWave is not an analytic function:

It has both singularities and discontinuities at the integers:

SawtoothWave [x] is neither nondecreasing nor nonincreasing:

SawtoothWave is not injective:

SawtoothWave [x] is not surjective:

SawtoothWave [x] is non-negative:

SawtoothWave is neither convex nor concave:

First derivative with respect to :

Derivative of the two-argument form with respect to :

The second (and higher) derivatives are zero except at points where the derivative does not exist:

If a==b, SawtoothWave [{a,b},x] is constant and its derivatives are zero everywhere:

Integrals over finite domains:

FourierSeries :

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: