Wikipedia:Reference desk/Archives/Mathematics/2009 February 22

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February 22

Even and Odd Functions and Their Decomposition

On the Wikipedia entry of Even and odd functions, even function can be written as $f(x)={\frac {1}{2}}[f(x)+f(-x)]\,$ and odd function can be writen as $f(x)={\frac {1}{2}}[f(x)-f(-x)]\,$. How are these formulas dervived in the first place from the definiton of odd and even functions? 142.244.175.223 (talk) 00:14, 22 February 2009 (UTC)

Just apply the definitions - replace f(-x) by the appropriate thing and simplify, you'll find you end up with just f(x) afterwards. --Tango (talk) 01:11, 22 February 2009 (UTC)

Please: Don't use the same notation—the letter ƒ—for two different functions. What the article says is that

$f_{\text{even}}(x)=1/2[f(x)+f(-x)]\,$

where ƒ is one function and ƒ_even is another (and similarly for odd functions). The function called ƒ is in general neither even nor odd. Michael Hardy (talk) 05:28, 22 February 2009 (UTC)

Yes, but I think in the OP they where not different functions... The question was: how is that f even implies f=f_even and f odd implies f=f_odd, and this is explained in Tango's answer. However I think that the OP also means: how are the decomposition formulas derived. An answer is: you want to write a function f(x) as a sum of an even function e(x) plus an odd function o(x):

f(x)=e(x)+o(x).

It turns out that there is exactly one choice for the pair e(x), o(x), because from the definiton of odd and even functions you must also have, for all x

f(-x)=e(-x)+o(-x)=e(x)-o(x)

and from the system of the two you get e(x) and o(x) as in the decomposition formulas. --84.221.198.10 (talk) 09:24, 22 February 2009 (UTC)

So let me get this straight: for even functions we have $f(x)=f(-x)\,$ and we can decompose this into $f(x)=1/2f(-x)+1/2f(-x)\,$ and replacing f(x) with f(-x) we can get since f(x) is an even function $f(x)=1/2f(x)+1/2f(-x)\,$, is this logic even right? 72.53.7.177 (talk) 09:25, 22 February 2009 (UTC)

Unimpeachable. But as Michael Hardy remarks the point is to write in the form "even + odd" functions that are in general neither even nor odd. Example: f(x)=e^x decomposes into the sum of cosh(x) plus sinh(x).--84.221.198.10 (talk) 09:45, 22 February 2009 (UTC)

Indeed. Continuing from where Michael started, any function f(x) can be used to construct an even function

$f_{\text{even}}(x)={\frac {f(x)+f(-x)}{2}}\,$

and an odd function

$f_{\text{odd}}(x)={\frac {f(x)-f(-x)}{2}}\,$

such that

$f(x)=f_{\text{even}}(x)+f_{\text{odd}}(x)$

If f(x) is already an even function then f(x)=f(−x), so we have

$f_{\text{even}}(x)={\frac {f(x)+f(-x)}{2}}={\frac {f(x)+f(x)}{2}}=f(x)$

$f_{\text{odd}}(x)={\frac {f(x)-f(-x)}{2}}={\frac {f(x)-f(x)}{2}}=0$

Gandalf61 (talk) 11:17, 22 February 2009 (UTC)

an odd fuction plus an odd function is odd... That's odd, isn't it? Well, so it's not odd that it's odd. That's odd, isn't it?(...) pma (talk) 13:36, 22 February 2009 (UTC)

Graham's number in terms of the Ackermann function

Can Graham's number be expressed in terms of an output of the Ackermann function with manageable inputs? NeonMerlin 07:08, 22 February 2009 (UTC)

It seems unlikely, since one can be defined in terms of powers of 3, and the other one in terms of powers of 2. Ctourneur (talk) 20:02, 22 February 2009 (UTC)

A more reasonable question would be to try to find k such that A(k) ≤ G ≤ A(k+1). There is (as far as I know) no reason to expect one of these to be an equality. As for what the k is, I have no idea. Staecker (talk) 22:19, 22 February 2009 (UTC)

The value of k there is very, very large: it is certainly much, much larger than 3 -> 3 -> 27 (see Conway chained arrow notation). It is in fact bigger than 3 -> 3 -> (3 -> 3 -> (3 -> 3 -> ( ... (3 -> 3 -> 27) ... ))), where there are 60 sets of parentheses. Eric. 131.215.158.184 (talk) 06:01, 23 February 2009 (UTC)

Closed form of a recursive series

Is there a closed form with a fixed number of terms for the sequence defined by t₀ = 1 and t_n = t_{n - 1} + 2^{t_{n - 1}}? In case anyone's wondering, it originates here. NeonMerlin 08:11, 22 February 2009 (UTC)

I doubt there's anything useful. There's nothing in the Sloane's entry. Algebraist 09:40, 22 February 2009 (UTC)

Fast fourier transform.

Please can anybody explain very popularly in simple way the technique of fast fourier transform?

Try Fast fourier transform? By popular, do you mean simple and feel that article is too complex? Basically the transform analyses the data to determine the amplitude and phase of signals of various frequencies, which information has many uses and for which it is possible to inverse transform back to the original data. -- SGBailey (talk) 07:20, 23 February 2009 (UTC)

Long and Synthetic Division Symbols on Office Word

Hello. How do I get long division symbols and synthetic division symbols on Office Word 2007? I could not find them in Equations Editor. Thanks in advance. --Mayfare (talk) 22:11, 22 February 2009 (UTC)

Regarding long divisions, it is apparently possible with the use of the old friend Math Editor (available through the "Insert Object" feature) (See this online discussion). I couldn't find the appropriate symbol in the new "integrated" editor. Could that be possible? :S
As for the synthetic division, if this is what you mean, then I guess you can implement it building a table and inserting the proper symbols in each cell of the table. Pallida  Mors 14:27, 23 February 2009 (UTC)

They might not exist. In particular, MS Word 2007 appears to use the encoding described in UTN #28 ([[1]]) which mentions on page 14 that an encoding for the long division enclosure is not chosen yet. Note that this is as of nearly 3 years ago though.GromXXVII (talk) 20:52, 23 February 2009 (UTC)