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

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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 $\exp :\mathbb {C} \to \mathbb {C}$, the complex $\arcsin$ including all branches is the multivalued inverse of function $\sin :\mathbb {C} \to \mathbb {C} .$ 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)