Comparisons

Usage

When using a comparison operator (>, <, ==, >=, <=, !=) with a libcalculus.Function object, a libcalculus.Comparison object is returned. This object is callable, and represents a boolean function - it takes in a value, and returns True if the condition is satisfied at that point (and False if not).

>>> (libcalculus.exp > 3)(1)
False
>>> (libcalculus.exp > 3)(2)
True
>>> (libcalculus.exp > libcalculus.cos)(0)
False

This is especially useful for constructing piecewise functions; for example, to create the function \(x\mapsto\begin{cases} e^x & ;\,x>3 \\\text{sin}\left(x \right ) & ;\,\text{else} \end{cases}\) use:

>>> f = libcalculus.piecewise(libcalculus.identity > 3, libcalculus.exp, libcalculus.sin)
>>> f(3), f(5)
(0.1411200080598672, 148.4131591025766)
>>> f(3j)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "src/Function.pyx", line 36, in libcalculus.Function.__call__
    raise ValueError(f"This function cannot accept input of type {type(x)}.")
ValueError: This function cannot accept input of type <class 'complex'>.

Naturally, complex functions can only accommodate the comparison operators ==, != - so complex piecewise functions based on them are supported. For example, to create the function \(z\mapsto\begin{cases} e^z & ;\,z=3i \\\text{sin}\left(z \right ) & ;\,\text{else} \end{cases}\) use:

>>> g = libcalculus.piecewise(libcalculus.identity == 3j, libcalculus.exp, libcalculus.sin)
>>> g(3.1j)
11.07645103952404j
>>> g(3j)
(-0.9899924966004454+0.1411200080598672j)

Operations

Comparisons support basic logical operations; given two libcalculus.Comparison objects p and q:

Syntax	Mathematical Notation	Notes
~p	\(z\mapsto \neg p\left(z\right)\)	Logical NOT
p | q	\(z\mapsto p\left(z\right) \vee q\left(z\right)\)	Logical OR
p & q	\(z\mapsto p\left(z\right) \wedge q\left(z\right)\)	Logical AND

The corresponding in-place operations are supported as well.