BTW, (too ?) simple check :
sage: all(map(lambda
u,v:bool(u(v(x)._sympy_().rewrite("log")._sage_()).exponentialize().simplify_full()==x),
....: (sin, cos, tan, csc, sec, cot, sinh, cosh, tanh, csch, sech,
coth),
....: (arcsin, arccos, arctan, arccsc, arcsec, arccot, arcsinh,
arccosh, arctanh, arccsch, arcsech, arccoth)))
True
Le samedi 2 juillet 2022 à 10:32:30 UTC+2, Emmanuel Charpentier a écrit :
> The SR.exponentialize method implements :
>
> sage: [u(x)==u(x)._sympy_().rewrite("exp")._sage_()
> ....: for u in (sin, cos, tan, csc, sec, cot, sinh, cosh, tanh, csch, sech,
> coth)]
> [sin(x) == -1/2*I*e^(I*x) + 1/2*I*e^(-I*x),
> cos(x) == 1/2*e^(I*x) + 1/2*e^(-I*x),
> tan(x) == -I*(e^(I*x) - e^(-I*x))/(e^(I*x) + e^(-I*x)),
> csc(x) == 2*I/(e^(I*x) - e^(-I*x)),
> sec(x) == 2/(e^(I*x) + e^(-I*x)),
> cot(x) == I*(e^(I*x) + e^(-I*x))/(e^(I*x) - e^(-I*x)),
> sinh(x) == -1/2*e^(-x) + 1/2*e^x,
> cosh(x) == 1/2*e^(-x) + 1/2*e^x,
> tanh(x) == -(e^(-x) - e^x)/(e^(-x) + e^x),
> csch(x) == -2/(e^(-x) - e^x),
> sech(x) == 2/(e^(-x) + e^x),
> coth(x) == -(e^(-x) + e^x)/(e^(-x) - e^x)]
>
> However, we also have :
>
> sage: [u(x)==u(x)._sympy_().rewrite("log")._sage_()
> ....: for u in (arcsin, arccos, arctan, arccsc, arcsec, arccot, arcsinh,
> arccosh, arctanh, arccsch, arcsech, arccoth)]
> [arcsin(x) == -I*log(I*x + sqrt(-x^2 + 1)),
> arccos(x) == 1/2*pi + I*log(I*x + sqrt(-x^2 + 1)),
> arctan(x) == -1/2*I*log(I*x + 1) + 1/2*I*log(-I*x + 1),
> arccsc(x) == -I*log(sqrt(-1/x^2 + 1) + I/x),
> arcsec(x) == 1/2*pi + I*log(sqrt(-1/x^2 + 1) + I/x),
> arccot(x) == -1/2*I*log(I/x + 1) + 1/2*I*log(-I/x + 1),
> arcsinh(x) == log(x + sqrt(x^2 + 1)),
> arccosh(x) == log(sqrt(x + 1)*sqrt(x - 1) + x),
> arctanh(x) == 1/2*log(x + 1) - 1/2*log(-x + 1),
> arccsch(x) == log(sqrt(1/x^2 + 1) + 1/x),
> arcsech(x) == log(sqrt(1/x + 1)*sqrt(1/x - 1) + 1/x),
> arccoth(x) == 1/2*log(1/x + 1) - 1/2*log(-1/x + 1)]
>
> Hence two questions :
>
> -
>
> Is it worth implementing it (for the benefit of
> high-school/undergrads/engineering math) ?
> -
>
> Should it be implemented by SR.exponentialize or by a distinct method
> ? (My preference is for the former, notwithstanding the apparent
> discrepancy in names…).
>
> Your advice is welcome.
>
>
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