Wikipedia:Reference desk/Archives/Mathematics/2024 March 20

= March 20 =

Whether it coincides with a simpler function
Is $y = sin (arcsin x)$ (1) the same function as y=x, if we consider all branches of logarithm (of any real number) and all branches of inverse sine function? Or does (1) remain meaningless for any argument outside the range [-1;1] when we restrict it to real value for both the domain and the image, and (1) will coincide with the identity function only when we regard it as a function that map complex numbers to complex number? Does the logarithm of negative numbers lead to the presence of removable singularities for (1)? (In contrast, the function y=x obviously does not contain any singularity). I was able to prove that $y = arcsin (sin x)$ and $y = sin (arcsin x)$ are not always the same, but I still can't settle the aforementioned problems. 2402:800:63AD:81DB:105D:F4F:3B26:74C5 (talk) 14:36, 20 March 2024 (UTC)


 * A univalued function and a multivalued function $$f:X\to Y,$$ possibly partial, can be represented by a relation $$R_f\subseteq X\times Y.$$ The total identity function $$\text{id}:X\to X$$ corresponds to the identity relation $$I_X=\{(x,x)|x\in X\}\subseteq X\times X.$$ Function composition corresponds to relation composition: $$R_{(g\,\circ f)}=R_f\,;R_g.$$ The multivalued function inverse correspond to relation converse: $$R_{(f^{{-}1})}=(R_f)^\text{T}\!.$$
 * Just like the multivalued complex logarithm is the multivalued inverse of the exponential function $$\exp:\C\to \C$$, the complex $$\arcsin$$ including all branches is the multivalued inverse of function $$\sin:\C\to \C.$$ So
 * $$R_{(\sin\circ\arcsin)}=R_\arcsin\,;R_\sin=(R_\sin)^\text{T};R_\sin.$$
 * Generalizing this from the sine function to an arbitrary (univalued) function $$f$$, we have:
 * $$(x,y)\in (R_f)^\text{T};R_f\Leftrightarrow \exists z:((z,x)\in R_f)\land(z,y)\in R_f)\Leftrightarrow \exists z:(x=f(z)\land y=f(z)).$$
 * Clearly, this implies $$x=y,$$ so the composed relation is the identity relation on the range of $$f,$$ representing the identity function on that range. --Lambiam 18:21, 20 March 2024 (UTC)