Talk:Hyperbolic functions

message: Hi, I have seen that there is some use of the notation \cosh^2() to indicate \cosh(\cosh()). This should not be done as in some countries \cosh^2() actually means [\cosh()]^2, and this may be a source of confusion. It's evil. Orzetto 09:14, 16 March 2005 (UTC)
replies: text: Is this actually true of non-kiddie level math in those countries: Function composition TomJF 04:09, 12 April 2006 (UTC)

text: For trigonometric functions, function^2() is synonymous with function() \cdot function(). For example, \cos^2(t)+\sin^2(t)=1. Out of curiosity, when would you ever nest trigonometric functions? You'd run into unit problems, wouldn't you? Your basic trigonometric functions (cosine, sine, and tangent) have domains in radians, degrees, or gradians, yet have ranges in unit lengths. The cosine of the cosine of an angle would be meaningless. Sobeita (talk) 02:36, 16 December 2010 (UTC)
replies: text: Almost nine years late, but why not? You might want to find the fixed point of cos(x). Double sharp (talk) 04:24, 9 December 2019 (UTC)

text:   I have replaced the multiplication symbol in your entry with a mid-dot. We often use the star glyph for a multiplication symbol in a plain text, because ASCII did not contain a mid-dot · or a times symbol ×. However, in some contexts the star symbol * can be confused with the convolution operator, so it's better to avoid the star whenever more apropriate symbols are available (like in MathJax/LaTeX). --CiaPan (talk) 18:46, 13 December 2019 (UTC)
thread_title:  Use of exponents on function names

message: There is a move discussion in progress on Talk:Sech (disambiguation) which affects this page. Please participate on that page and not in this talk page section. Thank you.  —RMCD bot 09:06, 18 September 2018 (UTC)
thread_title:  Move discussion in progress 

message: Hi all! I am completely new to editing Wikipedia, and I have an idea for a change. From what I've read, it seems I should propose the change here.
MOTIVATION FOR THE CHANGE
My proposed change is regarding the section "Relationship to the exponential function." When I saw this heading, I was expecting it to contain some mention of the fact that \cosh(x) and \sinh(x) are the even and odd parts of the exponential function, but this is not  directly discussed. I'll explain and provide references, in case this notion is unfamiliar.
It turns out that every f\colon \mathbb{R} \to \mathbb{R} can be uniquely decomposed into the sum of an even function f_\text{e} and an odd function f_\text{o}, which are referred to as the even and odd parts of f. Moreover, these functions are given by 
\begin{align}
f_\text{e}(x) &= \frac{f(x)+f(-x)}{2} \\
f_\text{o}(x) &= \frac{f(x)-f(-x)}{2} \\
\end{align}
Establishing this result is straightforward. To save a little space, I won't do it here. A brief, elementary proof is given in Patrick Honner's blog, and a method of discovering the formulas for f_\text{e} and f_\text{o} is given on Math Stack Exchange. The result is also described in a Wikipedia article.
It's a simple result that can be useful. For example, it can help with integration. See this blog article from John D. Cook for a quick example.
With that background, the connection to hyperbolic functions is immediate. If you already know about this even-odd decomposition, and I tell you that \cosh(x) and \sinh(x) are the even and odd parts of e^x, then you will instantly know how to define them in terms of the exponential function:
\begin{align}
\cosh(x) = \frac{e^x + e^{-x}}{2} \\
\sinh(x) = \frac{e^x - e^{-x}}{2} \\
\end{align}
Currently, the section features the identity e^x = \cosh(x) + \sinh(x), but there is no indication that this is the unique decomposition of e^x into a sum of an even and an odd function, or that this leads directly to the standard definitions of \cosh(x) and \sinh(x).
Below is a proposal for an updated version of the section. I've tried to keep as much of the original language as possible.
PROPOSED LANGUAGE
Hyperbolic cosine and hyperbolic sine can be thought of as the even and odd parts of the exponential function, respectively, in the following sense.
Every f \colon \mathbb{R} \to \mathbb{R}  can be uniquely decomposed into a sum of an even function f_\text{e} and an odd function f_\text{o}, given by the formulas below. [reference to https://en.wikipedia.org/wiki/Even_and_odd_functions#Other_algebraic_properties would go here]
\begin{align}
f_\text{e}(x) &= \frac{f(x)+f(-x)}{2} \\
f_\text{o}(x) &= \frac{f(x)-f(-x)}{2} \\
\end{align}
This can be shown directly from definitions [reference to Honner’s blog would go here]. 
When f(x) is the exponential function, these expressions for f_\text{e} and f_\text{o} are identical to the definitions of hyperbolic cosine and hyperbolic sine. (These are analogous to the expressions for cosine and sine, based on Euler’s formula, as sums of complex exponentials.)
Consequently, we have the following identities:
:e^x = \cosh(x) + \sinh(x)
and similarly
:e^{-x} = \cosh(x) - \sinh(x)
Additionally,
:e^x = \sqrt{\frac{1 + \tanh x}{1 - \tanh x}} = \frac{1 + \tanh \frac{x}{2}}{1 - \tanh \frac{x}{2}}
FEEDBACK
I would appreciate any feedback! If I'm not correctly following the protocol for proposing changes, I would be happy if anyone can point me in the right direction.
Thank you!
Greg at Higher Math Help (talk) 14:19, 30 November 2018 (UTC)
replies: text: By the way, the applicable guideline here is WP:BOLD, which essentially amounts to "if you think it's good, just go ahead and do it" (with some caveats), although asking for feedback first is certainly okay, especially if you want to see how it would be received.  (Also check out WP:RS about using blogs as sources).  Anyway, in this case, I think it's a bit overkill.  A short blurb to note that e^x = \cosh x + \sinh x is the unique decomposition of the exponential function into its odd and even parts would be totally okay to insert (nothing even needs to be linked, because they already are above; see MOS:DUPLINK).  But going into any more detail probably isn't needed.  Also note that almost as much, but not quite as much, is already said under the  section.  –Deacon Vorbis (carbon • videos) 14:58, 30 November 2018 (UTC)
replies: text: Wow, that was quick! Thank you so much for the detailed feedback. I just skimmed through the references you provided. 
#My intent was specifically to emphasize not only the uniqueness, but also that the standard definitions for \cosh(x) and \sinh(x) are just particular instances of the general formulas for f_\text{e} and f_\text{o}. If nothing else, it makes the definitions really easy to remember. Thoughts?
#I'm not sure what you mean about duplicate links. I see even and odd properties mentioned, but I don't see any mention or link for general even-odd decomposition formulas in the current version of the hyperbolic functions article. Did I miss something?
#Regarding the blog as a source, this seems like a tricky issue. I've seen proofs presented on Wikipedia without references before (e.g. I just checked and the Mean Value Theorem is an example of this), so is it better to present the proof with no reference than to provide one with a blog reference? I wouldn't put a proof here, but perhaps the short elementary proof could fit in the article on even and odd functions. I appreciate you helping me learn the ropes!  — Preceding unsigned comment added by Greg at Higher Math Help (talk • contribs) 15:51, 30 November 2018 (UTC)
replies: text: The unique decomposition of a function as the sum of an even and an odd functions was stated in Even and odd functions, but one should know that it was there for finding it. Thus I have restructured this article for making it visible, and I have create the redirects ,  and . Thus your suggestion is now reduced to simply add somewhere in this article:  D.Lazard (talk) 17:56, 30 November 2018 (UTC)
replies: text: Perfect! I like how you structured it, and the proof you provide is concise. I've edited the Definitions section for the hyperbolic functions to include very brief parenthetical remarks linking to your Even–odd decomposition section. I also fixed a couple minor typos in that section and added an indication that f_\text{e} is even and f_\text{o} is odd (it's implied but I think it's more clear this way).  — Preceding unsigned comment added by Greg at Higher Math Help (talk • contribs) 13:23, 1 December 2018 (UTC)
thread_title:  Proposed change to the section "Relationship to the exponential function" 

message: A discussion is taking place to address the redirect :Hypersine. The discussion will occur at Wikipedia:Redirects for discussion/Log/2020 July 3#Hypersine until a consensus is reached, and readers of this page are welcome to contribute to the discussion.  1234qwer1234qwer4 (talk) 17:48, 3 July 2020 (UTC)
thread_title:  "Hypersine" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]]