@MartinSleziak Hey! It is nice to e-meet you. I actually a community manager here at StackOverflow =) I am glad to make the changes you want =)

Please let me double check what our plan is. As I get it we want to add the following paragraph at the end of the current help center article at /help/searching

> Various ways to search this site (both using the built-in search and using some external tools) are summarized on meta: How to search on this site?

Please let me know what you think @MartinSleziak, @quid

We're talking about this: math.stackexchange.com/help/searching

As I said in my answer on meta, I would prefer to have it somewhere at the beginning (or near the beginning).

@NicolasChabanovsky I am afraid that if goes at the end of that long article, nobody will notice it.

In any case, CMs and moderators should have say in this. I just mentioned what is my opinion on this.

Maybe more mods would notice that this is being discussed if the discussion was in Math Mods' Office.

quid mentioned that they are rather busy right now. I am not sure whether @XanderHenderson or @AlexanderGruber followed this issue too.

I do not see other mods pingable in this room at the moment. (Maybe I overlooked somebody.)

Maybe somebody can suggest better wording...?

Maybe if it goes in the beginning, information that the help-center describes only the built-in search should be added.

So it would start like this:

> Various ways to search this site (both using the built-in search and using some external tools) are summarized on meta: How to search on this site?

> Basic help on the built-in search follows below.

One of the reasons why I think that it would be good to have link to a FAQ post on searching in some relatively prominent place is that searching for formulas is quite difficult - and the FAQ posts describes (among other things) the tools which are suitable for this.

Another reason is that a FAQ post maintained by community is a good thing - it is easier to update and refine than something which is editable only by the SE staff.

To add some context for other users, who did not follow the previous discussion, this is related to the feature request: Should we add external searches in some help pages about search? (And specifically to the suggestion to include the link to this post: How to search on this site?)

Sorry, I'll have to leave.

@MartinSleziak I am totally fine with having it at the beginning. Let us see what others think.

Hi @AlexanderGruber, @XanderHenderson! We would love to hear what you think about that ^^^!

@NicolasChabanovsky thank you for taking care of this. I agree with @MartinSleziak that it would be good to have it early on. I'll drop line to this discussion in our mod room.

Good morning!

@MartinSleziak I have not really been following the issue, but I think I am caught up... the idea is to add a link high up in the help pages which goes to the meta FAQ? Sounds good to me.

One of the biggest problems we have (in my opinion) is that users don't know how to search, and don't know that the built in search is not very good vis-a-vis mathematical formulae.

@JoséCarlosSantos Recently, you rejected a proposed edit here saying that the post does not make the question more comprehensible.... FYI, before rejecting the edit you should have read the heading of the edit which said "comment dissambiguation". the OP of the question did not seem very active. Due to the ambiguity, a fellow user answered the question and had to be downvoted on the grounds of ambiguity

I am very much in favor of adding that information to the help pages.

You can check the deleted answer there as well

@XanderHenderson I have seen that if you directly input your question in the title and body...i.e. say you have to evaluate a limit...then if you put in the funtion, almost all of the time a match results

@AnindyaPrithvi I, also, would have rejected that edit. There is no evidence in the comments that this is what the asker intended. If one has moderator powers and can read deleted comments, then there are a couple of deleted comments attached to the deleted answer which clarify.

An edit reviewer cannot, in general, know that context, and the edit comment "disambiguation" is rather unclear. Some to the effect of "per deleted comments to the deleted answer, I means Z" might have been more clear.

@XanderHenderson check comments of the answer

deleted*

Jean Marie I guess

@AnindyaPrithvi Please finish reading my comments.

I did read the comments. Most of the relevant comments are deleted, and have been attached to a deleted answer.

@XanderHenderson Hmm, this was not known...but I think they can read other posts to the question/answer can't they? I do not know yet but sometimes when I am reviewing questions in the review queue I see that

A normal user cannot see anything there in which the original asker says "I is for integers".

Okay, I'll be more specific about it

In fact, the only comment which references the integers is this one, which is not written by the original asker.

As a moderator, I can see additional comments which make it clear that the asker was talking about the integers, but this is not at all clear from what is visible to the typical user.

I cant see the "this one" comment, but I think I remember it

The intention of the asker is to make an octagon, it can only be done if Z== integers

Perhaps, the issue is now sorted

Thanks

Can we invite smoke detector here?

!!/coffee

> A place to socialize, have fun, share humor and war stories, and more.

Definitely a lot of more

I'm tired... stayed up the whole night so I can go to bed at the right time today and reset my circadian rhythm. Finally taking a break from the searching I've been doing for exact circular functions.

I don't know what happened... feels like only yesterday it was the first of September...

I'm not really a mathematician by the way. I'm just a programmer and a contemplative. I'm looking for circular functions both as a challenge and as a practical function to implement for my project among other things.

@AnindyaPrithvi Like I said, it is not possible for a typical user to see the asker indicating that I stand for the integers. As a moderator, I can see see the relevant comments, but a typical user cannot. As such, I don't think that rejecting the edit was inappropriate. Certainly, it was not so inappropriate that it needed to be brought up here.

@XanderHenderson Then? a meta post? If I had not brought it up here, many other edits would have been semi rejected leaving me uninformed about such things and all I would have thought was the decline was insincere which it wasnt...

And as I said it's sorted, you can delete the comment which tags Jose...the deletion period for me has expired

I feel like I am done with maths :(

Math is just too hard for me

;_;

@Stupidquestioninc Have you seen Eddie Woo's YouTube videos or 3Blue1Brown? Nice places to go if you're on your own.

Maths isn't really hard as there's nothing difficult about manipulating quantities. Your frustration lies in being overwhelmed with new formalisms that you have yet to grasp or understand, and you're not comfortable with it yet (obviously).

@AMDG And Numberphile

@Stupidquestioninc Now, I would help you with your problem except that I am nowhere near qualified to help you.

@Aracanine perhaps, although their presentation of divergent sums is questionable logic at best :)

@Aracanine Like I said, leave a better edit comment.

"As per comments which have been deleted, the asker clearly meant the integers."

Or, even better, leave a comment under the question asking the asker to edit their post themselves to fix the error.

In general, if you are making an edit which it not obviously in line with the asker's intentions, it is better to ask them to make the change. Personally, I read the question and assumed that $\mathbb{I}$ meant the set of irrational numbers, so there is a problematic ambiguity there which needs to be resolved. ;)

Infinite Sum: 1 -1 + 1 - 1 + 1 - 1... logically, if you begin with 1-1, you get zero as the result of 1-1 + 1 - 1. If you begin with 1 + 1, you eventually would get zero as 1 + 1 = 2, then 1 + 1 - 1 + 1 = 2 - 2 = 0... so no matter where you start, logically, I should therefore be beginning at 1, and ending with a result of 0.

@AMDG yes but the way university book are written breaks my spirit

@Stupidquestioninc Find better books.

Haha, I can relate. Communication and socialization are my greatest defects. The only person I can easily and effectively communicate my ideas with without fear of misunderstanding is myself.

@AMDG haha can relate

I have found that the formalisms come naturally if you just practice with the principles themselves and immerse yourself in the topic... as it then follows that you ask, "what notation can I use to represent this mathematical form?" and then the notation and words follow naturally from what you have experienced as a result.

I've been searching for a real-valued, closed-form, arbitrary precision parent for all circular functions (and consequently all waves), and I went in knowing nothing.

I now know significantly more than when I first began.

I started looking at the infinite series of existing circular function definitions and as a result I now understand the essence of these and related functions in general.

As just one little example, ln(x), informally, applies the harmonic series to x as a sort of coefficient to the infinite series of x. It isn't the precise behavior, but this is what I've observed.

Looking at and playing with infinite series, I also now understand why you cannot simply integrate 1/(x^2 + 1) normally to get arctan(x).

Pretty fun stuff messing around in wolfram alpha.

But as for where I'm at now, I found a nice approximation of the circular functions using cosh(x). I'm confident that I can also get a nice error function to apply to it to make it an exact approximation.

Exact insofar as it is still a matter of transcendental functions, but that it is a closed form for which the precision can be chosen arbitrarily based on the number of digits, not the length of the polynomial (so I hope and desire).

If I recall correctly, according to desmos (and its implementation of the circular functions), the maximum error is about 0.00815, so it is on par with Bhaskara I's I suppose, but analytically, I would say it is far more useful compared to just a parabolic approximation.

@AMDG i did not see that $\zeta (-1)$

What now about Riemann zeta function?

but saw many on puzzling quesns

like josephus

@AMDG I remember a relation of 1+2+3... with 1-1+1...

Oh that...

yeah that infamous zeta-1

Yeah I wouldn't really know too much about that, though that 1+2+3... would be related to the harmonic series... I was just mentioning it in regards to Numberphile like one of the sums they mention as being -1/12.

On a different topic, I'm curious to know how gamma (Euler-Mascheroni constant) is going to play a role in the circular functions as well since it is the limiting difference of ln(x) and the harmonic series. Surely there's something yet to be exploited there...

My question though is why no one really appears to be looking for an exact form for sine and cosine. An infinite series approximation is still not computationally ideal, and astronomers need... probably astronomical numbers of digits ;)

Surely, I'm not the only one?

@Stupidquestioninc If you are self-studying, get better books.

If you are in school, the book is not meant to be your only resource.

You have instructors which should be available to answer questions, and all of the internet.

You just have to learn to take advantage of those resources.

I must ask for some feedback: is my theory sound that if I can find an error function by converting my approximation to a simple form like a hemicircle defined by +sqrt(1-x^2), assuming I use exact identities, then the amount of relative error to the true function remains constant and proportional regardless of the number of conversion steps, correct?

Such that said error function denoted as, say, t_err(x), if as the difference of the true end conversion, can be applied to the principal approximation to get a true circular function of zero error?

Once back to the principal approximation with the error function applied, I could then perform any necessary simplification to get an exact function, right?

This here is where I'm at: desmos.com/calculator/kdwobykuwr . I've also asked about this in a recent question I made math SE about mapping Re(f(z)) and Im(f(z)) to real 3D functions and how to obtain them, including the representation of some complex-valued function as a 3D function over the reals which is directly related to sine and cosine of course.

(t_sinp(x) is the principal approximation)

Something like this is sort of what I'm talking about: desmos.com/calculator/vfet0ptqrj

And then I can use my t_sin(x) to go back to a circle... then I create an error function for that and apply it, so on until I get back to the principal approximation t_sinp(x).

@Aracanine Just curious, I thought you used this account to comment (this is what I saw in the transcript), but when I move in the chatroom, you don't even have a MSE account?

@ArcticChar he changed his name on chem SE to aracanine and is talking using chem as a parent account. All his accounts are hidden. So it only shows that specific account.

@ArcticChar Chat exchange is pretty hard to get...sometimes my mathSE account shows, sometimes a deleted chem se profile (reason for rep stuck at 1800)...etc.etc...but "you" has a MSE hiddenaccount

Arghhh, I changed it to math se, now see

@Integrand Hi

morning

Morning, btw for me it's 8pm...

evening

how much coffee is too much coffee

More than four cups a day is too much coffee. That's the rule I take with any one particular thing as such.

Interesting... I just found an even more accurate approximation of the circular functions by replacing e^x with (1/3)^x...

!!/coffee

I suppose the proof, now, is in the infinite series...

@AMDG 1/3^x...isnt close to e^x?

close-ish to e^(-x)

ohh..you wrote e^x above, that's why i thought

yes pretty close as $1/3 \approx 1/e$

using 1/e increases the error (!)

1/3 is the optimum

don't ask why... it just is...

One sec, I'm preparing the demo.

@AMDG What counts as a "cup"?

250mL

So 1L per day max

Oh, okay. I'm fine, then!

I've only had 5 or 6 oz thus far today.

> only

that's not even an 8oz cup lol (which is approx. 250mL)

Yes, I know.

but is it 5/6 oz of uncle crazy's ultra-condensed espresso?

@Integrand Shhh!

Organic or bust

otherwise you better have an excellent reason besides "it tastes good and I like it"

Anyways, brb!

@AMDG I mean, who are you to judge what other people drink?

It is a matter of the well-being of your health. I would rather you not be drinking pesticides...

I feel like that is my business, not yours.

Haha

Also, "organic" encompasses a lot more than "no pesticides".

Yes, it does

And, frankly, given how coffee grows, I am not terribly worried about pesticides. The bean is well protected.

But I really only use it as a heuristic when food shopping.

They still put garbage in food... like canola oil (rapeseed oil)

I think "it tastes good and I like it" is a very good reason to drink whatever coffee one likes.

Oh! Scary! Rapeseed oil!

yes, but I drink coffee that tastes good and doesn't have as many contaminants, so I hope and believe without any way to verify such empirically myself.

@AMDG Good for you. But your statement was "drink organic coffee; if not, you'd better have a reason better than 'it tastes good and I like it'".

What can I say? My standards of quality are higher than most.

No... that isn't what you are asserting.

On the contrary, you shouldn't be so offended.

You are asserting that your standards are, first and foremost, that organic is better.

And that taste is secondary.

Personally, I prefer to first select coffee on the basis of taste.

Yes, because taste is not reasonable in and of itself as it is only a sensation, not something that comes from the intellect. Something just has to taste good in order that I may consume it at all. If I don't like it, then it is literally impossible for me to consume it.

For the record, I typically drink coffee that is on the very hipster end of hipster-dom (in addition to "organic", the adjectives "single origin", "batch roasted", and "fair trade" apply).

But for sure, I'm not saying don't buy the best tasting coffee...

However, I don't claim that my tastes are in any way superior to anyone else's, nor do I desire to dictate what others consume.

If I have the choice, I will pick the one that tastes best, but if it is a matter of health, I take it very seriously.

It is none of my business.

FBI....FREEZE

Yes, but organic is not a matter of taste.

I said FREEZE

hands on your head

lol

Again, I don't care what you drink. That is your decision. But claiming that your choices are superior to others, and that everyone else should make the same choices as you, is offensive.

/me defrosts

!!/water

Heh. I never claimed that my choices are superior to others. That is you speaking.

@AMDG This is you telling others what to do.

@AMDG And this is you claiming that your standards are higher than others.

So, yes, you did claim that your choices are superior.

At most, that would only be implicit, but I'm not very good at communication, so misunderstandings like these happen all the time.

Ah, finally...

desmos.com/calculator/ytm8ir5nt0

The demo is ready

1) why in the world does this work? 2) what is the infinite series of this function and how does it compare to the true circular function of cos(x)? 3) is the error from desmos' approximation, or mine (I'm tempted to say theirs, but it could just be a reeeeaaaallly accurate approximation)?

Hm, but if you drink coffee that is typically "extremely hipster", then what were you offended by? I would like to know. I don't know why we had a mini argument...

I honestly didn't expect your response. I expected banter, not... resistance.

@AMDG It is offensive when one decrees that their choices regarding matters of taste are superior to the choices made by others.

Well I said it in the same way that one might say "PC master race", or at least that was my intent.

I see. This did not translate well in an all-text medium.

Nothing I say translates well in any medium :D

No kidding... while others might say "apple", I might only give a formal description like "red toroid that is the fruit of a tree". I've been told I am verbose; that's probably why: I usually don't have the vocabulary to describe my ideas as the results of contemplation compared to everyday, as well as common, experiences.

So where do I even begin to make a proof that this is or is not a true circular function? (half of a period of one at least.)

wolframalpha.com/input/…

I still haven't gotten to retrofitting the old t_arcsin yet either, but that's a minor detail. The question is which of these two functions is the more accurate? How do I test for this?

If I didn't have wolframalpha, I honestly would have no clue about how to get the infinite series of these functions among other things.

I don't even know how I find this stuff honestly... it started with trying to approximate the exponential logarithm using the real exponential... then I just tried putting the function of (half) a circle as the x value for complex cosine... and now I'm here.

@AMDG I honestly cannot discern what your goal is.

The sine and cosine functions are transcendental functions which can be defined in a number of ways.

Personally, I like to define them to be the unique solutions to $s'' + s = 0$, $c'' + c = 0$, $s(0) = c'(0) = 0$, $c(0) = s'(0) = 1$, or something similar.

To find a closed-form, real-valued, arbitrary precision parent of the circular functions. This is probably about as close to that as I have ever gotten with what little I have available to me.

What is a "parent" of a function?

the function which generates all the others including itself.

And why do you want to do this?

"Generates" how?

Because that is the desired quality of circular functions, or more generally, of all wave functions.

Like I said, I have a high standard. :)

@AMDG This doesn't make any sense to me.

Generates? well, I can use cosine to generate sine, for example. That would be a generation.

HOW? How does sine "generate" cosine?

You just translate it by pi/2.

This is a rather odd notion of "generating".

Functions generate values. In this sense, one function is generating another which also consists of values. All functions are implicit data sets.

Therefore, all functions are generators of a particular set of values, hence what I mean by generation.

This is not really the right way to think of a function.

Hm... then I extend it to all relations

A function is a set. Specifically, a function with domain $X$ and codomain $Y$ is a subset of the Cartesian product $X\times Y$ with $(x_1, y), (x_2, y) \in ff$ if and only if $x_1 = x_2$.

Can a plane not be considered a set?

Your notion of "generates" seems to have something to do with the ability to transform one set into another (e.g. by a horizontal translation).

But it still isn't clear.

Please define the phrase "$f$ generates $g$".

Well, if we must go to the ultimate generality...

Don't just give examples; define the term.

All things in existence are origins. An origin is defined as a cause of existence. An origin causes one or more causes of existence which are also origins. It may therefore be said that all things generate something and are therefore generators of a particular kind of order; and each new origin is itself a unique order. It then follows that there is a hierarchy of orders created by the generation of origins from one origin to another.

The limitation of each origin with regards to what it is capable of generating is that an origin may only produce anything that is less than or equal to itself in its order.

(As such, I find mathematics and its notion of function incredibly limited by comparison :) )

Therefore, when I say "parent that generates the circular functions", I mean something which is capable of producing all circular functions, all waves, and consequently, all functions and constants therefore because at that point, you have all that is sufficient to define the constants as well as non-periodic functions and... everything less than it, as per the limitations of an origin.

(It does not, however, necessarily need to generate all infinite sums, both convergent and divergent)

It just has to be sufficiently general such that you can map between all geometries and all norms.

I'd say that the generator of sine and cosine more or less fits that description since you can just use varying perspectives of just sine or cosine to produce every other function. Fundamentally, a curve could be defined via angular momentum alone, and a plane can be described in R^3 using an arbitrary curve, but with waves like sine or cosine, you can define all of those things.

Do you get me?

Distance is effectively a universal metric (which I sadly am not competent enough to prove to you), therefore, as the circular functions relate 1-norm to points in a given space, the circular functions can be used to define just about anything since a point is two norms, each a distance away from the respective axis. "Everything is made of waves."

brb

I'm back. I'm going to go for a little while and do other things. Talk to you later!

@AMDG Until you can write down a definition of what it means for a function to "generate" or be "generated by" another function, you aren't doing mathematics, and I can't help you.

Hey, I'm about to leave and get in the car, but... in the meantime, I still would like to know the accuracy of the approximation I have found. I don't know what you mean by "you aren't doing mathematics". You might consider what I'm doing as "generalized mathematics", or working with relations and algebra that describe the unique set of points in a given space like a plane rather than the unique set of y coordinates which defines a function strictly speaking, with functions merely as a subset...

...of all relations.

so wherever you see "function", just replace it with "relation", and wherever you see "generate" or "generated by" [another relation], let generation be the act of transforming or converting one input object (a point, a norm, a polygon, a polyhedron, etc.) into a unique output object. (I'll clarify what "unique output object" means when I get back.)

@AMDG @XanderHenderson Is this the right place?

@AMDG I am a mathematician. I am qualified to answer questions about mathematics. If this is not what you are doing, then I cannot help you. The first step in doing mathematics is clearly defining your terms.

At this point, I have no idea what you mean by "generates" or "is generated by". A good first step would be giving precise definitions of these words.

@Safdar There wasn't much else going on, so no reason not to relax and do some math.

@MartinSleziak, @XanderHenderson, @quid, thank you a lot for your help! I've updated the help center article. Please let me know if you think we should adjust the sentence or do something else.

@NicolasChabanovsky Thanks a lot for doing that. The wording you've chosen seems very nice to me. (Clearly, you have experience with this sort of things.)

I have accepted your answer on meta. (If somebody raises some objections, I might consider unaccepting - but I don't find it very likely.)

@MartinSleziak Woohooo! Thanks!

@NicolasChabanovsky can we know how many Moderators are there on the whole SE sites?

My guess would be around 700

@XanderHenderson I wouldn't know how to define these things formally. I couldn't do so to save my life.

@DeadGuy Hey! Catija gave me a cool link that I was not aware of: stackexchange.com/about/moderators?by=users

I mean I don't know how best to explain it informally either: taking an input and converting to receive an input by a function is one kind of generation. The other would be taking a function as the input and putting out a function as the output. Such a generation may be described by processes such as integration and differentiation, but it extends to all behaviors which modify a function in some way to make it an entirely unique function.

To get the number one needs to "Find on page" (or the like) for a phrase "show details". It seems currently there are 561 moderators on Stack Exchange sites.

@XanderHenderson does that help to define "generation"? I should say more accurately that a given function implicitly generates another, and by that I mean that I can use one function to compute the output of a different function by applying some constraint to the function.

Consider this function, z(a,x,b,y), found here: desmos.com/calculator/z8l7kyskro

z generates all parabolas as well as all exponential functions.

I would consider z to be the parent function of all parabolas and exponential functions.

x^2 can be a parent function of all lines passing through the origin assuming the domain of x is all reals since a line may be defined as y = mx+b. y=x^2 understood as a line in slope-intercept form is then y = xx + 0, where the slope of the line is equal to x. Since I can write it in this style, then I can say that x^2 generates all lines y=mx; generation does not, however, require one to explicitly describe details like the rate at which the slope changes for each generated line.

I could just as easily have a function y = kx + 0 for which I can specify an arbitrary slope, k, and it, too, is a parent function of all lines through the origin.

@NicolasChabanovsky Thank you so much. It looks good.