Using Recursion to Define a Periodic Function

I'm working my way through Heikki Ruskeepaa's book, Mathematica Navigator and found this elegant one-liner that shows how to define a periodic function with recursion. Note that the base case required to stop the recursion is not a single value but the condition 0 <= t <= 2.

Clear@sawtooth;sawtooth[time_,scale_:1]:=If[0<=time<=2,scale time,sawtooth[scale (time-2)]]
Plot[sawtooth@t,{t,0,10}]

How does the recursion work? If t => 2, sawtooth keeps subtracting 2 from t recursively until t is back in the range 0 <= t <= 2 and computes that value for y. We can see sample values of a function plotted with DiscretePlot.

DiscretePlot[sawtooth@t,{t,0,6,0.2},PlotTheme->"Web"]

Of course a faster way of defining the domain is to use Mod, but there is a lesson here. We don't need the speed of Mod, so either implementation is fine - using Mod for the most compact and fast function, or using an elegant new recursive technique to learn it. This function omits the values at multiples of 2, but is effectively the same function:

Clear@sawtooth2;sawtooth2[time_,scale_:1]:=scale Mod[time,2]

Mathematica has built-in functions to produce various canonical waveforms  such as SawtoothWave. Here are square and triangular wave examples.

Plot[SquareWave[{-50,50},t],{t,0,10},ExclusionsStyle->Dotted,AxesLabel->{"time (mS)","mV"},PlotLabel->"Electric potential at the electrode"]

Plot[TriangleWave[{-40,10},t],{t,0,10},AxesLabel->{"time (mS)","mV"},PlotLabel->"Electric potential seen by the axon"]

Mathematica Plot: Options Overview

Plot Has over 80 Options to Control its Behavior


Here is a Plot of two curves where we use Plot options to add a Frame, tick marks to the left and right y-axes, but to the bottom x-axis only, and to Plot the full -y-range instead of leaving out extreme values.

Plot[Tooltip@{Sin@x, x Sin@x}, {x, 0, 20}, Frame -> True,
 FrameTicks -> {{Automatic, All}, {Automatic, None}}, PlotRange -> Full]

You can ask Mathematica for all the available Options for a function with Options@function. When you do, Mathematica tells you their default values as well.

Options@Plot // Partition[#, 3,3,1,{}] & // TableForm

The Partition syntax may look a little complicated, but is explained in Capturing the Remnant of a Partitioned List. It says: "Partition the list into sublists of length 3, take them in blocks of 3 (i.e. do not overlap them), start the first element of the first list at position 1 of the first sublist (at the beginning), and do not pad the last uneven list (i.e. pad it with the empty set)." While Plot has 57 Options, which divide evenly by 3, the partition syntax will work if Plot has a non-divisible-by-3 number of Options in the future.

I thank a diligent reader, Murta, for suggesting this correction to my original post.

Mathematica Plot: Plotting More than One Function

Using Plot to Plot More than One Function


It is often useful to Plot different function parameter values and compare them. Just give Plot a list of functions with the parameter values you wish to compare and it will Plot them together without your having to use Show. Let us use the function defined in Basic Plotting:


f[x_, t_] := x^t;


You can use Table to automate the parameter sweep, but pre-generate the sweep functions or (for reasons I don't understand) Plot won't automatically color-code them.

In[278]:= sweepFunctionTable = Table[f[x, t], {x, 2, 5}]

Out[278]= {2^t, 3^t, 4^t, 5^t}

In[279]:= Plot[sweepFunctionTable, {t, 0, 5}]

Plot with Tooltip to Identify the Plots

Sometimes you can lose track of which function is which when plotting several function together. Plot helps by color-coding them. But adding Tooltip labels is another technique. Let's compare the three functions Sin x, Cos x, and Sin x + Cos x. You will need to Plot this yourself in a Notebook to mouseover and see the tooltips that label each curve with the three function names ("Sin@x", "Cos@x", "Cos@x + Sin@x").

Plot[Tooltip@{Sin@x, Cos@x, Cos@x + Sin@x}, {x, -3 Pi, 3 Pi}]

Now let's specify that the tick marks should be labeled in radians:

Plot[Tooltip@{Sin@x, Cos@x, Cos@x + Sin@x}, {x, -3 Pi, 3 Pi},
 Ticks -> {{-3 Pi, -2 Pi, -Pi, 0, Pi, 2 Pi, 3 Pi}, Automatic}]