Piecewise Functions in Python’s sympy

Your Daily Dose of Scientific Python

In many applications, we are faced with functions that are defined piecewise. For instance, consider the function

sympy offers an easy and intuitive way to work with functions like that: the Piecewise class. For each case in the piecewise function, we define a pair (2-tuple) with the function on the subdomain (e.g. 3−𝑥²) and a condition specifying the subdomain (e.g. |𝑥|<1). For the example above, we would write

We can easily plot piecewise functions using plot_piecewise from the sympy plotting backends spb:

Note that plot_piecewise not only marks the transition points between two cases but also discontinuities (if present) are displayed nicely.

We can work on piecewise-defined functions like normal functions. For example, we differentiate them:

We can expand them in a power series, which keeps the piecewise structure:

Or we can integrate them. For a definite integral, we just say

For an indefinite integral, we can use

and each case is handled separately.

Often, we don’t define piecewise functions ourselves, but they are output of some operation. Then we often want to get access to the different parts of the function. This can be obtained by the method as_expr_set_pairs:

Evaluating piecewise function work just like normal expressions, using subs. It behaves as one would intuitively expect. Substituting 𝑥=0 just returns the single function value for the subdomain in which 𝑥=0 lies:

However, if you substitute 𝑥 for another symbol, say 𝑦, we retain the piecewise function structure:

Finally, what if you want to multiply a piecewise function by some expression? The intuitive approach doesn’t work:

Use the function piecewise_fold instead! Here, we get

That covers the typical use cases of piecewise functions. Have fun playing!

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