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The equation $xsin x=a$ can be computed numerically, but are there methods to lớn solve it algebraically?

Note: A similar equation in that form, $xe^x=a$ can be solved using lambert W function.


I think there are two things that shouldn"t be confused here.

1 : algebraic solution

First there is the notion of algebraic functions, which can be defined as roots of polynomials with rational coefficients.

But this is a very limited set of functions, for instance $exp,ln, sin,...$ would not be considered.

If we consider the problem $f(x)=a$ for $ainfirmitebg.combb Q$ for a certain function $f$, we can wonder whether $x$ should be an algebraic number or not.

The explicit form would be to lớn find an algebraic function $phi$ such that $x=phi(a)$.

The implicit form would be khổng lồ find a polynomial $P$ with rational coefficients such that $P(x)=a$ (at least for some $a$).


2 : algebraic combination of known functions

And then, there is the different problem of finding a closed formula for $x=f^-1(a)$ in term of an algebraic combination of known functions (meaning using $+,-, imes,/$ & powers lớn a rational).

But this statement is quite fuzzy, since it all depends of your initial set of so-called known functions. Does it includes $ln,exp$, does it includes trig functions, reciprocal of trig functions, etc...

Before Lambert W function was invented, there was no closed formula according lớn this rule for $xe^x=a$. So if I define today the Ziomek function $Z$ such that $Z(a)$ is solution of $xsin(x)=a$, now I have a closed formula for $x$.

Of course, it only becomes khổng lồ be interesting when this $Z$ function can be used for other kind of equations, if we can find interesting properties for it, if we can develop it in nguồn series, or find algorithms khổng lồ compute it quickly and accurately, or if it simply has any kind of theoretic interest.

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This is the reason why our limited mix of known functions has been extended to erf, Bessel, Gamma, polylogs và other creatures there were initially not even considered.