Wikipedia:Reference desk/Archives/Mathematics/2020 August 22

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August 22

Name for symmetric hyperbola conic section

Interactive SVG conic section

Hi Mathmos,

While creating this interactive SVG, I wonder whether the hyperbola where the section plane is parallel to the double cone axis has a special name, analogous to the conic section where they are perpendicular is a circle, and the one where the plane is parallel to the cone's side is a parabola.

Is it a rectangular hyperbola?

Thanks,
cmɢʟee⎆τaʟκ 06:54, 22 August 2020 (UTC)

All hyperbolas are symmetric, with the point of intersection of their asymptotes as the centre of symmetry, whether the cutting plane is parallel to the axis of the cone or not. When the cone axis lies in the cutting plane, the intersection with the cone is formed by two straight lines (which are a pair of generating lines of the cone) – a degenerate hyperbola. As drawn, the angle between these two lines is a right angle. (That angle is also called the "aperture" of the cone.) When the cutting plane is parallel to the cone axis, these two lines are also the asymptotes of the resulting hyperbola. When the angle between the asymptotes is a right angle, the resulting hyperbola is sometimes called a "rectangular hyperbola". (Geometrically speaking this is not a very interesting case, until you try to describe it with analytic geometry.)

So, as drawn, you have a rectangular hyperbola – but mainly because the aperture is a right angle. Make the cone narrower or wider, and see what happens.  --Lambiam 09:04, 22 August 2020 (UTC)

Thanks very much for your detailed reply, @Lambiam: I phrased my question badly: by symmetric, I meant that the hyperbola is centered on the double cone's apex, as in the thumbnail of my diagram or when its SVG is not hovered over. (As I had also rendered the double cone with 90° aperture, I understand that every hyperbola in my diagram is rectangular.) To be more specific, in my diagram, when the double cone is rotated such that its axis is parallel to the cutting plane, can I use a more precise label than "hyperbola"? Thanks again, cmɢʟee⎆τaʟκ 21:24, 22 August 2020 (UTC)

The answer to this question is a simple "no". Whereas the circle is geometrically distinguishable as a special case of an ellipse, the hyperbola generated with a plane parallel to the axis of the cone is geometrically indistinguishable from one generated at any other angle in the range, assuming that you are not given any information about the positioning of the cutting plane relative to the cone, and there is no way to determine from the hyperbola whether the cone's axis is parallel to the cutting plane. At every angle in the range you can generate the identical hyperbola by translation of the cone. —Quondum 23:17, 22 August 2020 (UTC)

 Done Thanks so much, @Quondum:. That's exactly the answer I was looking for. I didn't realise all the hyperbolas from the same aperture angle are identical except for translation. Is the different distance between the dotted lines in the diagram as the double cone rotates purely a scaling effect then? cmɢʟee⎆τaʟκ 00:45, 23 August 2020 (UTC)

Sure thing. Just for clarity, they are not all identical in overall size (scaling): translating the cone towards and away from you scales the overall figure for any given angle of rotation. Aside from scale, they are all identical. —Quondum 01:16, 23 August 2020 (UTC)

A metric for the ordering of matrices?

As in A < B for example. It doesn't seem to be a very useful concept at first glance, but perhaps in some contexts it makes sense? A zero matrix is apparently "less than" an identity matrix. Or is that just complete rubbish? Earl of Arundel (talk) 15:42, 22 August 2020 (UTC)

The usefulness would have to follow from the application, which would inform the precise definition. One ordering that may, perhaps, have some usefulness is the partial ordering by a matrix norm. No potential application comes readily to mind, though.

If, in some application, there is a need for an associative array whose keys are matrices – for example to implement memoization – one may need a total order on the key domain to keep them in sorting order. For that one can use the lexicographic ordering, viewing an m × n matrix as one long vector with the same elements. This does not use the fact that we are dealing specifically with matrices, though, and not just any container type for numbers.  --Lambiam 16:41, 22 August 2020 (UTC)

That might be one way, although one could argue that it doesn't really say anything meaningful with respect to how two matrices may be related. Well, what about the determinant? When zero, we can effectively say that there is no real information content stored in the matrix. Of course we might then ask what the sign of the determinant actually means as well (in terms of ordering), although I suppose tit could just be a matter of convention. Earl of Arundel (talk) 18:07, 22 August 2020 (UTC)

Or maybe even just the absolute value of the determinant. That might make a little more sense. Earl of Arundel (talk) 18:09, 22 August 2020 (UTC)

@Earl of Arundel: There are several matrix norms, but you cannot make them well-ordered in a continuous manner. The determinant can be thought of as the scaling factor by which volume of the parallelepipid spanned by a collection of vectors changes under this transformation (see Jacobian matrix and determinant). Ordering by largest eigenvalue is probably the most useful in practice, but even then we don't say one matrix is "bigger" than another. How much "information" is "stored" in a matrix would be conveyed by how restricted its entries are. If you want a measure of the degree of invertibility of the matrix, then the rank-nullity theorem provides your answer: just order by the matrix's rank.--Jasper Deng (talk) 18:20, 22 August 2020 (UTC)

@Jasper Deng: Very informative answer. Thank you! Earl of Arundel (talk) 18:42, 22 August 2020 (UTC)

counter example for that a real part of holomorphic function is not holomorphic

Can anyone think about a counter example for that a real part of holomorphic function is not holomorphic?--Exx8 (talk) 18:24, 22 August 2020 (UTC)

Please do your own homework.

Welcome to Wikipedia. Your question appears to be a homework question. I apologize if this is a misinterpretation, but it is our aim here not to do people's homework for them, but to merely aid them in doing it themselves. Letting someone else do your homework does not help you learn nearly as much as doing it yourself. Please attempt to solve the problem or answer the question yourself first. If you need help with a specific part of your homework, feel free to tell us where you are stuck and ask for help. If you need help grasping the concept of a problem, by all means let us know.

But see Cauchy–Riemann equations and think about what a holomorphic function with zero imaginary part would necessitate. –Deacon Vorbis (carbon • videos) 18:39, 22 August 2020 (UTC)

This user has been asking questions here since September 2011, so, unless they are a very slow student, this is unlikely to be their "own homework".  --Lambiam 06:31, 23 August 2020 (UTC)

@Exx8: "Holomorphy" is defined only for complex-valued functions, and the real part of any such function is a real-valued function (on the reals), so you need to speak of how to extend that back to the complex domain in order to talk about its "holomorphy".--Jasper Deng (talk) 18:42, 22 August 2020 (UTC)

Are you a type-theorist, or, worse ;), a category-theorist? Mathematicians working in the field of analysis take the usual injections Z ↪ N ↪ Q ↪ R ↪ C for granted and apply them tacitly, so you can talk about z + 1 without having to specify that 1 should be interpreted as a complex number.  --Lambiam 19:06, 22 August 2020 (UTC)

I've never seen that definition used for a real-valued function of a complex variable. Maybe because it's trivial for that case (no nonconstant such functions exist).--Jasper Deng (talk) 19:53, 22 August 2020 (UTC)

Lambiam, I think you have N and Z reversed. 2601:648:8202:96B0:0:0:0:DDAF (talk) 21:37, 22 August 2020 (UTC)

The question is not for an example of a holomorphic function whose real part is not holomorphic – which is how I read the question at first glance – but for a holomorphic function whose real part is holomorphic. Either version should be easy to answer. Fun fact (or not): If f is such a function, then so is Re f, in which the latter expression composes the real-part function with the function f.  --Lambiam 19:06, 22 August 2020 (UTC)

@Lambiam: The section below would seem to completely disprove that: Re f is necessarily harmonic but never holomorphic for nonconstant f.--Jasper Deng (talk) 18:53, 23 August 2020 (UTC)

This is a question of the domain. If the domain is the complex numbers (which I understand from the question), then what Jasper Deng says is true. If the domain is restricted to the reals, then Re f would be an analytic function. But then, we would not use the term holomorphic. —Quondum 19:06, 23 August 2020 (UTC)

Indeed, one would not say "holomorphic"; many real-valued functions that are differentiable or even real analytic aren't holomorphic. The complex conjugate function, whose restriction to the reals is the identity, is a straightforward example (see below), as would be any polynomial function of the complex conjugate.--Jasper Deng (talk) 19:25, 23 August 2020 (UTC)

(ec) Did I claim or imply any such thing? If so, I'm blissfully unaware of it. I am not sure what it means for a function whose domain is $\mathbf {C}$ to be harmonic. To any such real-valued function $f$ we can assign a partner ${\tilde {f}}:\mathbf {R} ^{2}\rightarrow \mathbf {R}$ defined by ${\tilde {f}}(x,y)=f(x+yi)$ and ask whether ${\tilde {f}}$ is harmonic. It is well known that if $f$ is holomorphic, and $g=\mathrm {Re} ~f$, then ${\tilde {g}}$ is harmonic. It seems to me that the Cauchy–Riemann equations imply that $\mathrm {Re} ~f$ is only holomorphic for constant $f$, which then itself is holomorphic, which is also how I interpret the reply offered in the following section. Disclaimer: I do not deny that it is false that I did not claim that this disproves the statement.  --Lambiam 20:21, 23 August 2020 (UTC)

@Lambiam: You said "Fun fact (or not): If f is such a function, then so is Re f, in which the latter expression composes the real-part function with the function f. " What does this mean? Essentially, the only possible counterexamples as desired by the OP are constant functions.--Jasper Deng (talk) 21:29, 23 August 2020 (UTC)

If $f(x+yi)=u(x,y)+iv(x,y)$, then $\mathrm {Re} ~f$ denotes the function $g$ defined by $g(x+yi)=$ $\mathrm {Re} (f(x+yi))=u(x,y)$. Note the hypothetical in the following. If $f$ is a holomorphic function whose real part is holomorphic, then $\mathrm {Re} ~f$ is also a holomorphic function whose real part is holomorphic. So the hypothesis is that $f$ and $\mathrm {Re} ~f$ are both holomorphic, and the claim is that this implies that $\mathrm {Re} ~f$ and $\mathrm {Re} ~\mathrm {Re} ~f$ are both holomorphic. The latter part follows immediately from the idempotence of $\mathrm {Re}$.  --Lambiam 21:58, 23 August 2020 (UTC)

Differentiability of real part, imaginary part, and complex conjugate

Are the functions $a+bi\mapsto a$, $a+bi\mapsto b$, and $a+bi\mapsto a-bi$ differentiable (in complex analysis)? If so, then what are their derivatives?

If ${\frac {d}{dz}}{\bar {z}}=f(z)$, where ${\bar {z}}$ denotes the complex conjugate of z, then it would follow that ${\frac {d}{dz}}\mathrm {Re} (z)={\frac {1}{2}}(1+f(z))$ and ${\frac {d}{dz}}\mathrm {Im} (z)={\frac {1}{2i}}(1-f(z))$, so if the complex conjugate function is differentiable, then so would the real part and the imaginary part functions. GeoffreyT2000 (talk) 22:24, 22 August 2020 (UTC)

No. See the Cauchy-Riemann equations for a criterion for complex differentiability, which these functions fail. –Deacon Vorbis (carbon • videos) 23:17, 22 August 2020 (UTC)