Simplifying expressions in SymPy

The previous post looked at why Mathematica does not simplify the expression Sinh[ArcCosh[x]] the way you might think it should. This post will be a sort of Python analog of the previous post.

SymPy is a Python library that among other things will simplify mathematical expressions. As before, we seek to verify the entries in the table below, this time using SymPy.

$\renewcommand{\arraystretch}{2.2} \begin{array}{c|c|c|c} & \sinh^{-1} & \cosh^{-1} & \tanh^{-1} \\ \hline \sinh & x & \sqrt{x^{2}-1} & \dfrac{x}{\sqrt{1-x^2}} \\ \hline \cosh & \sqrt{x^{2} + 1} & x & \dfrac{1}{\sqrt{1 - x^2}} \\ \hline \tanh & \dfrac{x}{\sqrt{x^{2}+1}} & \dfrac{\sqrt{x^{2}-1}}{x} & x \\ \end{array}$

Here’s the code:

from sympy import *

x = symbols('x')

print( simplify(sinh(asinh(x))) )
print( simplify(sinh(acosh(x))) )
print( simplify(sinh(atanh(x))) )
print( simplify(cosh(asinh(x))) )
print( simplify(cosh(acosh(x))) )
print( simplify(cosh(atanh(x))) )
print( simplify(tanh(asinh(x))) )
print( simplify(tanh(acosh(x))) )
print( simplify(tanh(atanh(x))) )

As before, the results are mostly as we’d expect:

x
sqrt(x - 1)*sqrt(x + 1)
x/sqrt(1 - x**2)
sqrt(x**2 + 1)
x
1/sqrt(1 - x**2)
x/sqrt(x**2 + 1)
sqrt(x - 1)*sqrt(x + 1)/x
x

Also as before, sinh(acosh(x)) and tanh(acosh(x)) return more complicated expressions than in the table above. Why doesn’t

√(x − 1) √(x + 1)

simplify to

√(x² − 1)

as you’d expect? Because the equation

√(x − 1) √(x + 1) = √(x² − 1)

does not hold for all x. See the previous post for the subtleties of defining arccosh and sqrt for complex numbers. The equation above does not hold, for example, when x = −2.

As in Mathematica, you can specify the range of variables in SymPy. If we specify that x ≥ 0 we get the result we expect. The code

x = symbols('x', real=True, nonnegative=True)
print( simplify(sinh(acosh(x))) )

prints

sqrt(x**2 - 1)

as expected.

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