Hallo,
the command "validExponential" from package EFSTRUX (Elementary Function
Structure Package) applies Risch's Structure theorem for Elementary
functions. The theorem checks if a given exponential or logarithm function
is algebraic over a given elementary extension field.
exp(2*log(x)) = x^2 is algebraic over the field \mathbb{C}(x) and algebraic
over the field \mathbb{C}(x,e^x). But FriCAS says it isn't:
f:=2*log(x)
g:=exp(f)
K:=kernels([x,exp(x)])
validExponential(K,f,x)
[image: \label{eq4}\verb#"failed"#]
Where I'm wrong?
Is Risch's structure theorem applicable here? Does Risch's structure
theorem have an error in this case?
Thanks.
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