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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)[reply]

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)[reply]

March 22

Sin, cos and ellipses

In this book [1] (linked to the right page), left column towards the bottom. I'm having a problem with the "hence."

I understand that:

A: the point M has the coordinates (x,y), which is also (sin φ, cos φ), no matter how you slide the straight KL, for all φ.

B: the formula for the ellipse. ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.$

I just don't get how B follows from A. Maybe I'm missing a concept that the authors take for granted. But shouldn't the text have explained something in between? Something like why the ellipse matches the Pythagorean trigonometric identity? Or why the set of all possibles values of M is $\sin ^{2}\theta +\cos ^{2}\theta =1$ ? Why is that when you have the coordinates (sin, cos) you add them and equal to 1 to produce an ellipse? Grapesofmath (talk) 17:27, 22 March 2024 (UTC)[reply]

I think you've misread the text: the point M (x, y) is actually (a sin φ, b cos φ). So x/a = sin φ and y/b = cos φ. Substituting into the identity

\sin ^{2}\phi +\cos ^{2}\phi =1

(which is true for all φ) gives the formula in (B). AndrewWTaylor (talk) 18:09, 22 March 2024 (UTC)[reply]

More precisely, the text identifies the point M (x; y) by "y = b sin φ and x = a cos φ". The resulting equation is the same. --Lambiam 19:32, 22 March 2024 (UTC)[reply]

Thanks for the feedback. But shouldn't the text be explicit here and explain that A applied to Pythagorean trigonometric identity result in B? Isn't this a jump too big in the train of thought? Grapesofmath (talk) 23:59, 22 March 2024 (UTC)[reply]

It's not immediately obvious, but it's not that hard either given the diagram. The y = b sin φ come from the lower right triangle and x = a cos φ comes from the upper left triangle. Once you have those equations, the equation (1) in the text follows as explained above. I think as a reader you're meant to figure out this kind of detail yourself. The alternative would be that the text becomes long-winded and pedantic. It's also a better learning experience for the reader if they have to think about the text as they're reading it and fill in some missing steps. A lot depends on the intended audience as well; apparently the book is meant for people with a certain basic knowledge of geometry, perhaps with some experience writing proofs. (That kind of information is often given in an introduction, but this is the introduction.) --RDBury (talk) 04:17, 23 March 2024 (UTC)[reply]

April 2

Is a stretched circular arc similar to a circular arc of a different circle?

Consider a circular segment, such as the one bounded by circular arc S and chord C in the diagram below. Assume that θ is "small". If we stretch (scale) the circular segment horizontally, will the "stretched" circular arc still be (approximately?) similar to a circular arc on a circle with a different R? If that's the case, will the (approximate?) similarity get worse as the scaling factor increases? Is there a simple formula that can characterize how similar the stretched circular arc is to a true circular arc?

134.242.92.97 (talk) 03:26, 2 April 2024 (UTC)[reply]

By 'stretching', do you mean a non-uniform scaling, such that the size is enlarged along the chord while preserved along sagitta...? --CiaPan (talk) 14:48, 2 April 2024 (UTC)[reply]

Yes (scaling along the direction of the chord but not along the direction of the sagitta). --134.242.92.97 (talk) 16:25, 2 April 2024 (UTC)[reply]

When you scale a circle, the resulting figure is an ellipse. An elliptical segment is not similar to a circular one in geometric sense. Of course the elliptic segment can be approximated with a circular one, so in a common speech they can be called 'similar', but I suppose that's not an appropriate use of the word here, at math ref.desk. --CiaPan (talk) 17:36, 2 April 2024 (UTC)[reply]

For which n, A269252(n) is -1?

[2] proves that A269254(110) and A269252(34) are both -1, and there is a sequence for the n such that A269254(n) = -1: A333859, but there seems to be no OEIS sequence for the n such that A269252(n) = -1, and A269254 and A269252 are similar sequence, so for which n, A269252(n) is -1? 118.170.19.90 (talk) 09:41, 2 April 2024 (UTC)[reply]