SymPy Basics

Question

I'm just starting to play with SymPy , and I'm a little surprised by some of its behavior, for example, these are not the results that I would expect:

>>> import sympy as s >>> (-1)**sI == sE**(-1* s.pi) False >>> sI**sI == s.exp(-s.pi/2) False 

Why do they return False and is there a way to make it convert one way of writing a complex number to another?

Answer 1

From the FAQ :

Why does SymPy say that two equal expressions are not equal?

The equality operator (==) checks to see if the expressions are identical in shape and not mathematically equivalent.

To make equality testing useful in basic cases, SymPy tries to rewrite mathematically equivalent expressions in canonical form when evaluating them. For example, SymPy evaluates both x + x and - (- 2 * x) to 2 * x, and x * x - x ** 2.

The simplest example, when the default transformations cannot generate the canonical form, are nonlinear polynomials, which can be represented in both factorized and expanded form. Although mathematically mathematically a(1+b) = a+ab , SymPy gives:

 >>> bool(a*(1+b) == a + a*b) False 

Similarly, SymPy cannot detect that the difference is zero:

  >>> bool(a*(1+b) - (a+a*b) == 0) False 

If you want to determine the mathematical equivalence of nontrivial expressions, you must apply a more advanced simplification procedure to both sides of the equation. In the case of polynomials, expressions can be rewritten in canonical form, completely decomposing them. This is done using the .expand() method in SymPy:

 >>> A, B = a*(1+b), a + a*b >>> bool(A.expand() == B.expand()) True >>> (A - B).expand() 0 

If .expand() does not help, try simplify() , trigsimp() , etc., which try to perform more complex conversions. For instance,

 >>> trigsimp(cos(x)**2 + sin(x)**2) == 1 True 

Answer 2

Because they are not equal. Try the following:

sE ** (sI * s.pi) == sI * sI