Sympy: How To Simplify Logarithm Of Product Into Sum Of Logarithms?

To motivate the question, sympy.concrete has some efficient tools to manipulate symbolic sums. In order to apply these tools to symbolic products, one has to take a logarithm. Howe

Solution 1:

Assuming that there is no standard function with desired behaviour yet, I wrote my own, mimicking the behaviour of

sp.expand_log(expr, force=True)

This code recursively goes over expression trying to locate patterns log(product) and replaces them by sum(log). This also supports multi-index summation.

Code.

defconcrete_expand_log(expr, first_call = True):
    import sympy as sp
    if first_call:
        expr = sp.expand_log(expr, force=True)
    func = expr.func
    args = expr.args
    if args == ():
        return expr
    if func == sp.log:
        if args[0].func == sp.concrete.products.Product:
            Prod = args[0]
            term = Prod.args[0]
            indices = Prod.args[1:]
            return sp.Sum(sp.log(term), *indices)
    return func(*map(lambda x:concrete_expand_log(x, False), args))

Example.

import sympy as sp
from IPython.display import display
sp.init_printing() # display math as latex
z = sp.Symbol('z')
j,k,n = sp.symbols('j,k,n')
Prod = sp.Product( (z + sp.sqrt(1-4*j*z**2))**(-1), (j,0,k))
expr = sp.log(z**(n-k) * (1 - sp.sqrt((1 - 4*(k+2)*z**2)/(1-4*(k+1)*z**2)) ) * Prod)
display(expr)

display(concrete_expand_log(expr))