Talk:Exponential function

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Matrix[edit]

So this article doesn't have almost anything about exponentiation to a matrix. Maybe we should expand it a little bit and possibly include an example. It is right now merged with the section about Banach spaces, which I don't really understand. Banach algebra seems to be a much more abstract concept. Matrix exponentiation has a lot of application in systems engineering, circuit design, biology, climate science and economy so I think it deserves an example. Anybody agree? In my opinion, the optimal thing would be to have an example of both e to a 2x2 Matrix and e to a 3x3 Matrix. Just to illustrate how much more complicated it gets with a larger matrix.

You're looking for Matrix exponential. ᛭ LokiClock (talk) 01:23, 19 April 2013 (UTC)

Notation[edit]

Aren't there multiple forms of the exponential function? Specifically, isn't the most general form kabx ? Admittedly, the differences can be incorporated in to b thus: kebxln(a), but wouldn't the first form be more clear to the laymen?


You don't really need the b; the most general form is kax. I mention these in the "science" paragraph; they are indeed the most useful to the "laymen". However they cannot be defined without exp(x) and in mathematics, exp(x) is tremendously more important than ax, so I started the article with exp(x) and then came to ax as soon as possible. --AxelBoldt

You can define ax without defining exp(x). For positive a, you define a0 = 1, an+1 = a×an for natural n, and an=1/an. Then you define a1/n to be the unique positive real x such as xn = a, and am/n=(a1/n)m. Since any real x can be approximated arbitrarily well by a fraction m/n, and am/n is monotonous, you just need to require that ax be continuous. --Army1987 12:00, 19 September 2006 (UTC)
I've changed the opening paragraphs to introduce the form kax even sooner. - dcljr 12:50, 6 Aug 2004 (UTC)

It shouldn't read: The graph of ex does not ever touch the x axis, although it comes very close.

but rather: The graph of ex does not ever touch the x axis, although it comes arbitrarily close (in a limit sense).

Coherent Graphs[edit]

Many of the math articles have graphs in smooth brown, using the same design all over. (Other colors also exist, like in the article Taylor expansion.) How are these graphs created? Interesting to anyone starting a new maths article. -- Sverdrup 22:40, 9 Dec 2003 (UTC)

The graphs were written in Java, and copied from the screen. The code is at User:Cyp/Java. Κσυπ Cyp   01:27, 10 Dec 2003 (UTC)

My edits of Aug 6, 2004[edit]

I just made a major edit to this article. Mostly small changes, but some major.

  • Added a section heading earlier in the article so the table of contents will (most likely) be "above the fold".
  • Rearranged things to put off non-real variables as long as possible. I think this will benefit readers who are less mathematically experienced.
  • Right-aligned the graph again (see page history). What's wrong with having it right-alinged?
  • Consistently replaced <i>italics</i> with ''wikitalics''.
  • Included multiple interpretations of {d \over dx} e^x = e^x. I know they're redundant, but that's the point: to say it in ways that might be more familiar to the reader. A picture would be particularly helpful here, I think.
  • Removed mention of "linear ordinary differential equations" in the similarly named section; if you think it's crucial, add it back (see page history).
  • Removed parenthetical remark in "Banach algebras" section:
if xy = yx (we should add the general formula involving commutators here.)

After much struggle, I decided not to change anything else of substance in that section or the next one on Lie algebras. I don't trust my understanding of these things (or lack thereof). Speaking of the "Lie groups" section, someone should probably try to make it more clear.

Oh, and BTW: Do we really need the same properties listed 3 times? I know we're talking about different mathematical objects at different places in the article, but still, I find it kinda redundant. Couldn't we name or number the properties and refer to them that way?

- dcljr 12:43, 6 Aug 2004 (UTC)

information should be reordered[edit]

IMHO,

  • the very first paragraph should be as short as possible for several reasons
    • editing it makes it necessary to edit the whole page, which may become impossible at some point. So it should only contain things that are absolutely necessary and which will (almost) certainly never need any modification.
    • it lessens the usefulness of the "Contents" table, which should come before any detailed information, except for a minimalistic explanation of "what is this page about" and "what is found elsewhere".
  • there are too many details about the graph of the exp function in the 1st paragraph (postitive with explanation, increasing with explanation), and still its not complete (convexity, asymptotics, ...). The picture itself is enough on top of the article, as it contains all that information.
  • It is strange to have the (body of the) article start with "Properties" instead of an (even informal) definition.
  • It is even more strange to have the "Properties" to start with a generalization of the exp function.

MFH 15:07, 8 Apr 2005 (UTC)

very strange[edit]

It is taught in China that y = a to the x-th power is the standard definition of exponential function, while y = e to the x-th power is just a particular case of exponential functions. Got confused when try to translate this article to zh.wikipedia.org.

See exponentiation Bo Jacoby 16:36, 23 October 2005 (UTC)
But this is the article about THE exponential fucntion. AN exponential fuction is y=a^x (which can be derived from THE exponential function by imputting x=ln(a)x). I would therefore agree with China's definition of AN exponential function.--Hypergeometric2F1[a,b,c,x] 11:03, 20 December 2005 (UTC)

This article reads like a mathematical textbook for degree students, not an encyclopedia article. What is its intended audience? People who are studying mathmetics will surely have the literature to tell them what Exponential is, and won't be consulting wikipedia. —The preceding unsigned comment was added by 82.46.29.168 (talkcontribs) 23:10, 18 September 2006 (UTC2)

Comment moved from article[edit]

Moved from top of article [The definition below is incomplete and not rigorous (Paulo Eneas, SP, Brazil)] (Enchanter 22:09, 26 November 2005 (UTC))

Title change[edit]

This article should be named "The Exponential Function" or at least there should be a disambiguation page differentiating this from "AN Exponential Function" f(x)=a^x, to avoid possible confusion.--Hypergeometric2F1[a,b,c,x] 11:06, 20 December 2005 (UTC)

I disagree with the move to "The exponential function". Maybe something more can be said about the general a^x function in addition to what already is in here. Oleg Alexandrov (talk) 21:34, 21 December 2005 (UTC)
If the article were to be moved it should be named the natural exponential function in accord with the term "natural logarithm" (a logarithm can be based on any exponential function in the form a^x, e^x is special because of his "natural" properties). --Friðrik Bragi Dýrfjörð 16:30, 27 April 2006 (UTC)

sin−1x is ambiguous[edit]

There is no need for a shorthand of this kind for reciprocal trigonometric functions since they each have their own name and abbreviation already: (sin x)-1 is normally just written as csc x.

Actually, also the arcsine has its own abbreviaton, i.e. arcsin x. And sin−1x is sometimes used to mean 1/sin x. Then we have cscx and arcsinx, which are both unambiguous, whereas sin−1x is ambiguous. See Trigonometric functions#Inverse function. I personally never use the notation sin−1x to avoid misunderstandings. But the current wording implies that it always refers to the arcsine. (And by the way, I've seen the notation fnx for any functions denotated by a string of lowercase latin characters, e.g. log2x to mean (logx)2 rather than log logx. --Army1987 20:41, 10 March 2006 (UTC)

Only with base of e?[edit]

I was under the impression that exponential functions included all functions in the form of f(x) = abx, not necessarily with a base of Euler's number. For example, y = 5(6x). DroEsperanto 18:00, 15 November 2006 (UTC)

How to compute exp(x) in computers[edit]

Shouldn't we add something about how computers compute e^x?

Something like:

exp(x) is typically computed using a floating point x and giving a floating point result y. A floating point value y is a structure or tuple of 3 values - sign, exponent and mantissa. exp(x) always gives a positive result (exp(x) >= 0 for all x) so sign is always 0 or positive. Thus, we only have to find the mantissa and the exponent.

y = mantissa * 2^exponent.

log(y) = log(mantissa) + exponent * log(2)

However, log(y) is also log(exp(x)) = x so this means x = log(mantissa) + exponent*log(2).

Thus, dividing x with log(2) - a fixed constant - and getting an integer quotient exponent and a fractional part U where 0 <= U < log(2) can therefore be done. We then next compute exp(U) to find the mantissa.

Since U is small (0 <= U < log(2)) this mantissa = exp(U) can be found by using a series computation as described on the page already. Thus, we find mantissa and can compute the result y by simply putting these parts together in a floating point number.

Computing exp(x) in computers[edit]

When computing y = exp(x) we can first consider how floating point values are represented in computers. A floating point value y is represented as y = smb^n Here s is +1 for positive values and -1 for negative values. For y = exp(x) we always have y >= 0 and so s is always +1. b is typically 2 for most computers.

If y = exp(x) we also have log(y) = x and so we get:

x = \log(y) = \log(sm2^n) = \log(s) + \log(m) + n\log(2)

log(s) = 0 since s = +1.

Thus n is found as the integer such that

n <= \frac{x}{\log(2)} < n+1

We then find log(m) as the value \log(m) = x - n\log(2).

Since log(m) is guaranteed to be in the range 0 <= log(m) < \log(2) we know that log(m) is small enough so that we can use the previously indicated series: m = \exp(\log(m)) = 1 + \log(m) (1 + \log(m)(\frac12 + \log(m) (...)))

Thus, the result y = exp(x) = m2^n where m is found by the series and n is found earlier by the division of x/log(2).

For complex variables z = x + yi it is simply an excersize in the identity:

e^{x + yi} = e^xe^{yi} = e^x(\cos(y) + i \sin(y)) = e^x\cos(y) + i e^x\sin(y)

A general a^x function is defined as exp(x \log(a)) and a^b where both a and b being complex is thus deifned in terms of exp(b \log(a)) where both a and b is complex and log(a) for complex a yield a complex value and the complex multiplication yield a complex result which is then in turn as deifned above.

To outline it:

a = x + yi b = u + vi

Need to compute log(a) first, convert a to polar co-ordinates:

r = \sqrt{x^2 + y^2} \theta = \arctan2(y,x)

Note that both the square root and the arctan2 here have real values as arguments.

The above gives us a = re^{{\theta}i} and so \log(a) = \log(r) + i\theta where we only use the principal value of the multi-valued function.

Thus, a^b = exp(b \log a) = \exp((u + vi)(\log(r) + {\theta}i)) = exp(u\log(r) - v\theta + (u\theta +v\log(r))i)

If we define p = u\log(r) - v\theta and q = u\theta + v\log(r) (both p and q are real values) we then get:

a^b = \exp(p + qi) = \exp(p)\exp(qi) = \exp(p)(\cos(q) + i\sin(q))

Thus, a^b for complex a and complex b can be defined.


I plan to add the above to the regular page in 3 days - comments are appreciated.

salte 13:06, 4 December 2006 (UTC)

Added several sections. I didn't copy it exactly as I wrote it above since I split the description into several sections. I also added a description of the algorithm for a^n for positive integers n as well as a short comment on how to expand it for all integers n. I also placed a description of exp(z) for complex z below the description of complex exponential and a description of how to compute a^b for complex a and complex b. Hope people find this useful.

salte 10:04, 8 December 2006 (UTC)

incorrect formula[edit]

Factorial signs (!) are missing in the denominators in the formula in 'Exponential_function#Computing exp(x) for real x'. The correct formulas is in the subsection just above on 'numerical value'. Bo Jacoby 14:57, 8 February 2007 (UTC).

Link to proof for article that e^x is the only non-zero function that is its own derivative?[edit]

The article correctly states that "e^x is its own derivative. It is the only function with that property (other than the constant function f(x)=0)." However I noticed it doesn't provide a proof or a link to a proof of that uniqueness. It would be nice if someone could provide a footnote reference at that sentence that links to a published proof that e^x is the only non-zero function that is its own derivative. Dugwiki 22:54, 4 April 2007 (UTC)

How about just pointing to the Picard-Lindelöf theorem? - Fredrik Johansson 23:06, 4 April 2007 (UTC)
No, I don't think that quite works because the function e^x isn't Lipschitz continuous as required by the theorem. In order for e^x to be Lipschitz continuous, there would have to exist a constant K \ge 0 such that for all x_1, x_2 in the reals |e^{x_1} - e^{x_2}| \le K|x_1 - x_2|. That isn't true, though. Note that f(t)=0 is the only one of the two functions that is Lipschitz continuous, and so is the unique Lipschitz continuous solution implied by the PLT if f(x)=f'(x) and f(0)=0. Dugwiki 15:48, 5 April 2007 (UTC)
In fact, I just realized that all functions of the form Ke^x for constant K have this trait. So any constant multiple works. Dugwiki 15:53, 5 April 2007 (UTC)
The exponential function in this case is the "y" for which the equation is being solved. The function f in the case of the differential equation y' = y is simply the identity function with respect to the second argument, f(t,y) = y, which is Lipschitz. As you say, there are infinitely many solutions; to get uniqueness per the Picard theorem, you need to specify the initial value y(0) = 1. Fredrik Johansson 16:19, 5 April 2007 (UTC)
Ah, thanks, that clears it up for me. I was misreading the theorem, basically thinking that "y" was the Lipschitz continuous function. So you're correct, and in fact PLT says that the function y=Ke^t is the unique function that solves the differential equation y'(t) = y(t), y(0)=K (with the identity function f(t,y(t)) = y(t)). I'll include that reference in the article as well. Dugwiki 17:31, 5 April 2007 (UTC)

vague lede[edit]

The lede of this article doesn't tell us what is an exponential function. The lede statement appears at the top of the first section, while the lede includes several statements that tell us facts about exponential functions, but not what is an exponential function.

The rambling lede:
  1. It is one of the most important functions in mathematics. (about the function, not a general description of the function)
  2. ... is written as... (about representing the function, not a general description of the function)
  3. (or) this can be written in the form... (about representing the function, not descriptive of the entire function)
  4. ...function is nearly flat ... (about aspects of the function, not a general description of the function)
  5. ... the graph of y=ex is always positive... (about aspects of the function, not a general description of the function)
  6. Sometimes, the term exponential function is more generally used for functions of the form kax, where a, called the base, is any positive real number not equal to one. (how the function is alternately used)
  7. ...This article will focus initially on the exponential function with base e, Euler's number. ( about the article, not about the subject)
  8. In general, the variable x can be any real or complex number, or even an entirely different kind of mathematical object; see the formal definition below. (about aspects of the function, not a general description of the function)

Only in the first section do we get the simple answer, couched in a rhetorical apology for not having told us sooner.

  • Most simply, exponential functions multiply at a constant rate.

This sentence belongs at the top, but I don't plan to fix it by entering a correct lede sentence: The exponential functions is an (arithmetic? algebraic? mathematic?) function that multiplies values at a constant rate. Last week, or last month, the style would have been "In mathematics, the exponential function is" but this project is so disorganized, there is no way to know if that is the preferred style today. Doubtless there are two groups somewhere in this project that both claim to have reached a consensus, each around contradicting styles. People get so harassed for adding content to Wikipedia articles someone else owns, I would rather expose the weakness and let it be. If you don't want to be associated with weak, vague text, you fix it. Arithmawhiz 18:48, 18 July 2007 (UTC)

Well put. I have not worked on exponential function, but on exponentiation, where you will find a definition , see Exponentiation#Powers_of_e. Bo Jacoby 16:55, 5 August 2007 (UTC).

Continued fraction for  e^x [edit]

A generalized continued fraction for  e^x can be constructed by examining the simple continued fractions for  e^{1/n} \,\!. and  e^{2/n} \,\!. formulas found in http://en.wikipedia.org/wiki/Continued_fraction#Regular_patterns_in_continued_fractions.

 e^{1/n} = [1; n-1,1,1, 3n-1,1,1, 5n-1,1,1, 7n-1,1,1, ..., (2k+1)n-1,1,1, ...] \,\!.

Taking the 1st, 4th, 7th, and 10th convergents for each value of n, a pattern develops:

 e^{1/1}: {1 \over 1} {3 \over 1} {19 \over 7} {193 \over 71}... =  1 + {2 \over 1+} {1 \over 6+} {1 \over 10+} {1 \over 14+} ... {1 \over 4k+2+} ...
 e^{1/2}: {1 \over 1} {5 \over 3} {61 \over 37} {1225 \over 743}... =  1 + {2 \over 3+} {1 \over 12+} {1 \over 20+} {1 \over 28+} ... {1 \over 8k+4+} ...
 e^{1/3}: {1 \over 1} {7 \over 5} {127 \over 91} {3817 \over 2735}... =  1 + {2 \over 5+} {1 \over 18+} {1 \over 30+} {1 \over 42+} ... {1 \over 12k+6+} ...

Leading to  e^{1/n} = 1 + {2 \over 2n-1+} {1 \over 6n+} {1 \over 10n+} {1 \over 14n+} ... {1 \over 2n(2k+1)+} ...

and, for m = 2n,  e^{2/m} = 1 + {2 \over m-1+} {1 \over 3m+} {1 \over 5m+} {1 \over 7m+} ... {1 \over m(2k+1)+} ...

 e^{2/m} = [1; (m-1)/2, 6m, (5m-1)/2, 1, 1, \dots, 3k+(m-1)/2, (12k+6)m, 3k+(5m-1)/2, 1, 1, ... ] \,\!

Here, take the 1st, 3rd, 6th, 8th and 11th convergents for each odd value of m, and

multiply the numerator and denominator of the 2nd and 7th convergents for another pattern:

 e^{2/1}: {1 \over 1} {2 \over 0} {7 \over 1} {37 \over 5} {266 \over 36} {2431 \over 329} {27007 \over 3655} ... = 1 + {2 \over 0+} {1 \over 3+} {1 \over 5+} {1 \over 7+} ... {1 \over 2k+1+} ...
 e^{2/3}: {1 \over 1} {4 \over 2} {37 \over 19} {559 \over 287} {11776 \over 6046} ... = 1 + {2 \over 2+} {1 \over 9+} {1 \over 15+} {1 \over 21+} ... {1 \over 6k+3+} ...
 e^{2/5}: {1 \over 1} {6 \over 4} {91 \over 61} {2281 \over 1529} {79926 \over 53576} ... = 1 + {2 \over 4+} {1 \over 15+} {1 \over 25+} {1 \over 35+} ... {1 \over 10k+5+} ...

Leading to  e^{2/m} = 1 + {2 \over m-1+} {1 \over 3m+} {1 \over 5m+} {1 \over 7m+} ... {1 \over m(2k+1)+} ... - what we had for m even!

Now, please notice a pattern for the continued fractions of  e^{x/3} \,\!., based on  e^{2/m} \,\!.:

 e^{1/3} = e^{2/6} = 1 + {2 \over 5+} {1 \over 18+} {1 \over 30+} {1 \over 42+} {1 \over 54+} ... = 1 + {2 \over 5+} {1 \over 18+} {1 \over 30+} {1 \over 42+} {1 \over 54+} ...
 e^{2/3} = e^{2/3} = 1 + {2 \over 2+} {1 \over 9+} {1 \over 15+} {1 \over 21+} {1 \over 27+} ... = 1 + {4 \over 4+} {4 \over 18+} {4 \over 30+} {4 \over 42+} {4 \over 54+} ...
 e^{3/3} = e^{2/2} = 1 + {2 \over 1+} {1 \over 6+} {1 \over 10+} {1 \over 14+} {1 \over 18+} ... = 1 + {6 \over 3+} {9 \over 18+} {9 \over 30+} {9 \over 42+} {9 \over 54+} ...

From the above comes a similar pattern for  e^x \,\!.:

 e^{1} = e^{2/2} = 1 + {2 \over 1+} {1 \over 6+} {1 \over 10+} {1 \over 14+} ... = 1 + {2 \over 1+} {1 \over 6+} {1 \over 10+} {1 \over 14+} ...
 e^{2} = e^{2/1} = 1 + {2 \over 0+} {1 \over 3+} {1 \over 5+} {1 \over 7+} ... = 1 + {4 \over 0+} {4 \over 6+} {4 \over 10+} {4 \over 14+} ...
 e^{3} = e^{2/(2/3)} = 1 + {2 \over -1/3+} {1 \over 2+} {1 \over 10/3+} {1 \over 14/3+} ... = 1 + {6 \over -1+} {9 \over 6+} {9 \over 10+} {9 \over 14+} ...

Leading to  e^x = 1 + {2x \over 2-x+} {x^2 \over 6+} {x^2 \over 10+} {x^2 \over 14+} ... {x^2 \over 4k+2+} ...

Special cases for e^x:

For x=1, e^1 = e^(1/1) changes from [1;0,1,1,2,1,1,4,1,1,...,2k-2,1,1,...] to [2;1,2,1,1,4,1,1,...,2k,1,1,...].

This changes  e^1 = 1 + {2 \over 1+} {1 \over 6+} {1 \over 10+} {1 \over 14+} {1 \over 18+}... to  2 + {1\over 1+} {2 \over 5+} {1 \over 10+} {1 \over 14+} {1 \over 18+}...

For x=2, e^2 = e^(2/1) changes from [1; 0,6,2,1,1, 3,18,5,1,1, ...,3k,12k+6,3k+2,1,1,...] to [7; 2,1,1,3,18, 5,1,1,6,30, ...,3k-1,1,1,3k,12k+6,...].

This changes the cumbersome  e^2 = 1 + {2 \over 0+} {1 \over 3+} {1 \over 5+} {1 \over 7+} {1 \over 9+} {1 \over 11+}... to  7 + {2 \over 5+} {1 \over 7+} {1 \over 9+} {1 \over 11+}...

Glenn L (talk) 03:53, 28 February 2008 (UTC)

Update: Glenn L (talk) 04:16, 11 May 2009 (UTC)

Update #2: These results for e^x are now reflected on the main page. -- Glenn L (talk) 05:12, 11 February 2013 (UTC)

Division properties[edit]

The division properties should be listed along with the other properties; I'm just not sure how to write them the way the others are written.

(a/b)^n = (a^n)/(b^n)

(a^n)/(a^m) = a^(n-m)

Fuzzform (talk) 04:17, 18 December 2007 (UTC)

Correctness of complex a^b[edit]

In section Computation of \,a^b where both a and b are complex there is a formula

\,a^b = (e^{\log(r) + {\theta}i})^{u + vi} = e^{(\log(r) + {\theta}i)(u + vi)}

That generally seems to be wrong. It may be correct in this context (though I think it's not) but in general for complex a, b c it's not true that: (a^b)^c = a^{bc} (see http://mathworld.wolfram.com/ExponentLaws.html) —Preceding unsigned comment added by Findepi (talkcontribs) 19:43, 1 February 2008 (UTC)

The first equals sign here just means "write a and b like this". The second equals sign is essentially a definition. The text could (and should) make this a lot clearer than it is, though.
In general much of the section seems to be focused on nitty-gritty numerical issues that I find to be of limited relevance of this article. For example "Watch out for potential overflow though and possibly scale down the x and y prior to computing x²+y² by a suitable power of 2". Even if this is relevant information about the exponential function in general (which I disupte), it is written like an instruction manual, which Wikipedia is not.
I'm inclined to think the section should either be removed (as irrelevant) or rewritten from scratch to be much shorter and focus on the mathematics of complex powers, rather than the minutiae of implementing them on a computer. –Henning Makholm 02:19, 27 April 2008 (UTC)

I would say instead that complex exponentiation is multiple valued, and that you might end up on a different branch. If you take the branch cuts into account, then you should find that the rule still works. If you expect the principle value, then you might find that it doesn't. More specifically, instead of \,a^b = (e^{\log(r) + {\theta}i})^{u + vi} = e^{(\log(r) + {\theta}i)(u + vi)} you might use \,a^b = (e^{\log(r) + {\theta}i + 2m\pi i})^{u + vi} = e^{(\log(r) + {\theta}i)(u + vi)+ 2n\pi i} Gah4 (talk) 05:36, 21 July 2008 (UTC)

The power ab is well defined when a is a positive real, or when b is an integer, but in the general case it is multivalued and thus ill-defined. The expression ab is not used in practice for nonpositive a and noninteger b, so the section is not really useful, and I support Henning Makholm's point of view. The case is treated in the article on exponentiation where I think it belongs, but the treatment there is not satisfactory either. The formula of Gah4 above should probably read \,a^b = (e^{\log(r) + {\theta}i + 2m\pi i})^{u + vi} = e^{(\log(r) + {\theta}i+ 2m\pi i)(u + vi)}. Bo Jacoby (talk) 08:07, 21 July 2008 (UTC).

incorrect formula in the complex a^b[edit]

Somebody please check if "The result a^b is thus p + qi" should be changed to "The result a^b is thus e^(p + qi)". —Preceding unsigned comment added by 194.67.106.32 (talk) 17:25, 3 March 2008 (UTC)

Image of e^z[edit]

The image in the "On the complex plane" shows some unsightly (and misleading) zig-zag artifacts between the real-part values of -2.5 and 1, roughly. Or is it just my monitor that is out of calibration? –Henning Makholm 02:43, 27 April 2008 (UTC)

I didn't see any artifacts, but what I did find is that if you stare near the right side of the image you can see a black to gray gradient, or a vertical white-on-gray square grating, with a rainbow glowing off the edge of the image, and the original image burned into your eyes. I think it's just that the part where the curves transition from thin to thick is close to flat. But I can't tell if it's actually curved at all, because it's a color gradient. I hate complex color plots sometimes. ᛭ LokiClock (talk) 04:01, 9 March 2012 (UTC)

Delete computational part[edit]

I propose to delete the bits about computing the value of ex, i.e. Exponential_function#Numerical value and Exponential_function#Computing exp(x) for real x. Neither of them is particularly noteworthy and exp is normally evaluated using much better approximations using one polynomial dividing another. The only other type of evaluation which might be interesting is one doing a bit at a time. Dmcq (talk) 21:07, 8 November 2008 (UTC)

Slight muddle?[edit]

The article says:

The exponential function is written as an exponentiation because it obeys the basic exponentiation identities, that is:
\,e^1 = e,\ e^{x+y} =e^x \cdot e^y\ .
It is the unique continuous function satisfying these identities for real number exponents.

The function b^x, for any base b, satisfies these identities (well, it doesn't literally satisfy b^1 = e, but that's hardly the point), so "the" exponential function, as defined in the introduction, with base-e, is not unique in this respect as is claimed. I think this part has got mixed up in a confusion about whether the term "exponential function" is always base-e or can apply to any base. —Preceding unsigned comment added by 81.152.169.35 (talk) 21:09, 5 January 2009 (UTC)

  • There's another problem here actually. The form of the identities to be satisfied presupposes that we already know the answer. It's like saying "2 is the only number that satisfies 2 + 1 = 3", rather than "x = 2 is the only solution to x + 1 = 3". The identities need to be written as, for example, f(x + y) = f(x)f(y), where f is an arbitrary function, and then we can say the exponential function is the only function that works.
  • I think the "unique continuous function" line should just be eliminated. To get the unique base here it's probably easiest to require that f'(x) = f(x) for all x in R, but then you don't need the exponentiation identities, you just need to say it's nonzero. Dcoetzee 00:01, 6 January 2009 (UTC)
I agree. I'm tempted to rewrite this section to change the emphasis. I propose we structure it something like this:
1. The exponential function (we don't yet know its form) satisfies f'(x) = f(x). It can be defined and calculated without recourse to any notion of raising to non-integer powers -- e.g. using the series expansion explained later.
2. It turns out that the function so defined can be written as some number, e, raised to the power of its argument, in the sense that it satisfies the identities familiar from the way integer powers work.
3. Given this property, the exponential function and its inverse can then be conveniently used to define what it means to raise any (positive) real number to any real power using a^b = exp(b*log(a)) -- possibly we would mention this is an alternative to raising to integer powers and taking roots (for rational b) or some sort of limit argument (for irrational b). —Preceding unsigned comment added by 81.152.169.35 (talk) 00:20, 6 January 2009 (UTC)
I disagree with most of the above and have put in a bit of explanation. The e1 is the exponential function and e on its own is e. The continuous on the real numbers part is important, it is possible to satisfy the two equations and still get something which isn't the exponential function. This bit is in an overview, you can go and stick in pedantry later on in the article but the intro and overview should deal with the main description without going and trying to prove everything and make it all boilerplated. This bit is necessary otherwise one is just writing ex without explaining why it is written that way. Dmcq (talk) 00:51, 6 January 2009 (UTC)
I pretty much agree with you - I only wanted to remove the last bit saying "it is the unique continuous function..." You don't need a unique characterization of the function to understand why it's written with exponentiation notation. Dcoetzee 01:57, 6 January 2009 (UTC)
There are still a number of problems with this passage. It still says "It is the unique continuous function satisfying these identities for real number exponents" which is either wrong (since exponentiation to other bases satisfies the same identities), or trivially and uninterestingly true if it's supposed to mean the only function that literally satisfies e^1 = e. The other presentational problem that I mentioned is still present. If we say "the only function that satsifies..." then the equation(s) presented must be in terms of a general function f(x), otherwise we are presupposing that the answer is already known. The way it is now simply does not make sense. Additionally, it has not previously been defined what it means to raise a real number to a real power. Later a definition is given in terms of exp and log, but this cannot be applied to the definition of e^x itself -- it's circular reasoning. Either we can say exp(x) defined without reference to raising to the power of a real number satisfies the expected power laws, therefore we use the function to extend the basic concent of integer powers to all real numbers, or we can have already defined what a^b means for real a and b, so we already know what e^x means. Either is fine, but the article is currently muddled about which approach it is taking. 81.152.169.35 (talk) 03:25, 6 January 2009 (UTC)
There are discontinuous functions satisfying those identities. For instance one can define exp2(q)=eq for all the rational numbers q but keep extending exp2 to the other reals by the axiom of choice for any numbers not yet defined by rational combinations of the ones already defined, for instance one could have exp2(π)=23 and exp(√2)=13. It is unnecessary to say it is the unique continuous function satisfying the identities as this bit is simply explaining why the exponential notation is used but this objection would almost certainly be raised against what is written there if the line was removed.
I don't see that it would matter too much if this section described it in terms of exponentiation to a real power since it is just an overview, but defining it that way would lead to circular reasoning and people would confuse describing and defining. The text however doesn't describe it in terms of exponentiation to a real power, it simply says it is written as an exponentiation because it satisfies the exponentiation identity and that doesn't involve any circular reasoning.
I'll add a bit saying that exponentiation to a real number can be defined using the exponential function, perhaps that will make it more obvious. Dmcq (talk) 09:29, 6 January 2009 (UTC)

Overview and motivation[edit]

I've reinstated the overview and motivation section and deleted the bit moved from the original section to the formal definition section.

I think it is silly to launch straight into the formal definition without even saying why it is called the exponential function and overviewing its main features. The section was deleted before with a comment 'first section hopelessly circular'. This sounds like an argument against it because it was not formally derived but instead described what was to follow. That is part of what overview is about.

I think a little could be removed from the leader as it is long for a leader and there is duplication with the overview section. However wikipedia is supposed to give the information first and jumping straight into formal definition is to write a textbook rather than an encyclopaedia. And actually good textbooks try to be readable, making them unreadable is some sort of conceit I feel. Dmcq (talk) 18:23, 28 July 2009 (UTC)

The Overview section has been deleted again, this time with the comment 'This section is a tradgedy and cannot be allowed to be the first section of the page. It requires major editing'. I have posted a note on the edditor's page to be a bit more specific but perhaps someone else can comment on whether the overview section is a useful thing or whether you think the article should start straight with the mathematical definition? Or if that editor doesn't come back what they meant? Dmcq (talk) 12:44, 5 August 2009 (UTC)
To be fair, User:ObsessiveMathsFreak didn't actually delete the Overview and motivation section - he renamed it to History and Properties and moved it to later in the article, after Derivatives and differential equations. But I agree that he hasn't explained what he doesn't like about this section, and it is not clear to me why he wants to move it. Gandalf61 (talk) 12:56, 5 August 2009 (UTC)

You have stated that it is silly to launch straight into the formal definition without giving an overview of the function first. While this may seem like a good idea, in mathematics it is generally a very, very bad one.

The overview in question begins by listing underived properties of a poorly defined function, and then discusses, without proof, its application to first order odes without even mentioning the derivative property of the function. It then finishes by convolving the discussion with general exponentiation and mentions applications without even stating why it is applied. It's all over the place! For anyone who knows nothing about the exponential function, the entire section is less than useless as it will only serve to confuse. There's no information coming in.

The Bernoulli definition is technically correct but it is exceedingly difficult to prove that it even makes sense, let alone present such a proof in an introductory article. The Taylor series proof is precise, robust and can be used to prove the differentiation formula with the minimum of fuss.

It is better to begin with an appropriate definition and proceed from there. Most articles take that view. Whatever useful information exists in the section is best placed in the introduction at the top of the page. When people scroll past the contents, they expect to get to the meat as quickly as possible, and as mathematical definitions go the Taylor series definition of the exponential function is about as pithy an introduction as you will ever come across.

Unless there are any objections here, I will try to make a major edit on this section in 24 hours. The edit will probably involve taking the disparate pieces of the section and putting them in more appropriate place in the article.ObsessiveMathsFreak (talk) 13:09, 5 August 2009 (UTC)

I most certainly do object. Wikipedia is not a textbook. Please see the Mathematics manual of style in particular the section at the start about the structure of articles. Putting in an informal introductory section is actively encouraged. Starting with a mathematical definition and developing logically until you finally get to something that might be readable if you don't know half the answer already is not.
As to the original definition derived from continuous compound interest it is used in the Exponentiation article under 'Powers of e' and has a very short proof there that it satisfies the exponential identity. It may not be ultra rigorous but it was good enough to kick start the subject. You don't get that so easily from the series or differential equation. By the way the one of the points of an 'Overview' is to be all over the place and not be specific. Dmcq (talk) 13:32, 5 August 2009 (UTC)
I wipe my arse with the Mathematics manual of style!! Are you seriously suggesting that a rambling tirade about the advanced properties of a function that has not even been defined is of use to anyone? Read the section! There is already sufficient introductory material at the top of the page without this section utterly confusing matters. How can the identity exp(x+y)=exp(x)*exp(y) be honestly introduced when the reader has no idea what the function exp(x) even is? The discussion of first order ODEs is even worse as the reader has no idea whatsoever how the function behaves under differentiation, much less of its application to the equations shown. This is no way to present mathematics.
The limit definition is entirely inappropriate for any introductory article whatsoever. The short "proof" given in the Exponentiation article is essentially meaningless, with powers being passed through limits to infinity without the slightest regard for whether the series even converges or not! This very question was the reason behind Bernoulli's original investigation into the number! Given its esoteric nature, I doubt a sufficiently rigorous proof even exists on Wikipedia at all.
The "origins and motivations" section begins with Bernoulli's outdated definition and proceeds to list a series of largely unrelated items which will be of no use to anyone. You cannot have any serious discussion about a mathematical quantity before you have defined it, at least in some fashion. The origins section does not even do that.
The single best way of introducing the exponential function is by its Taylor series. The series is absolutely convergent and moreover can be directly used in later definitions such as the exponentiation of complex numbers or square matrices. It's short, sweet, to the point and the derivative formula can be seen immediately using it. The post contents material should open with it and a better place can be found for most of the material in the current train wreck of a fist section.ObsessiveMathsFreak (talk) 15:55, 5 August 2009 (UTC)
I have raised the matter of your attitude to WP:MSM at Wikipedia talk:WikiProject Mathematics#Manual of style disagreement. Do you have the same problem with WP:NOTTEXTBOOK? Dmcq (talk) 16:27, 5 August 2009 (UTC)
Hang them all is my opinion. You can raise that too if you like.ObsessiveMathsFreak (talk) 17:03, 5 August 2009 (UTC)
This is supposed to be an encyclopaedia and the exponential function is one people who aren't mathematicians can come across quite easily. Compound interest and the continuous version is something people are altogether too much familiar with as opposed to series produced from a hat. If we cannot deal with this little bit of maths which is one of the most familiar to uneducated peasants working their fields in Bangladesh then we've an awful lot of problems. Dmcq (talk) 16:41, 5 August 2009 (UTC)
Fine. If you can present a short paragraph or two linking compound interest to the Bernoulli limit I'll gladly drop my objections to the entire section. I'd try to write one myself, but I was never any good at compound interest myself.ObsessiveMathsFreak (talk) 17:03, 5 August 2009 (UTC)
I'll be away till tomorrow so will try if no-one beats me to it. Continuously compounded interest refers to the section in compound interest about it. Dmcq (talk) 17:30, 5 August 2009 (UTC)
Hi Dmcq, I beat you to it! --WardenWalk (talk) 08:02, 6 August 2009 (UTC)

I am posting here after seeing the message at WT:WPM. There is no reason why we need to start with the series definition, or any other definition. A well-written introductory section can be appropriate for an article where we want to maximize accessibility (another example is First-order logic). On the other hand, we are not writing an axiomatic treatment nor a textbook, so it is not a significant issue if we talk about something before defining it. — Carl (CBM · talk) 00:07, 6 August 2009 (UTC)

Exactly. Paul August 14:03, 6 August 2009 (UTC)
I agree as well. Although I don't think that the introduction as it stands is well written. Another problem is what definition is most appropriate. As a physicist the definition as the inverse of the ln or equivalently the solution to the differential equation df/dt = f is sufficient for most of what we do. Sometimes the (Bernoilli?) form of the compound interest pops up but the vast majority of the time exp(x) comes from a differential equation of some sort. The taylor series definition is used in physics to generalization of exp(x) to include complex arguments and arguments with operators. (It can be used to prove Eulers theorem-at least it is a good enough proof for physicists.) I have no clue which definition is more technically correct, though; from a physicist perspective that is usually unimportant. TStein (talk) 19:56, 7 August 2009 (UTC)
All the definitions are correct, and we should present them all. Because we are writing an overview, not a textbook, there is no reason why we need to choose one specific definition here as "the" definition. That would only be needed if we wanted to prove things, which I believe would be out of scope for this particular article. — Carl (CBM · talk) 12:13, 8 August 2009 (UTC)
There most certainly is a reason we should choose one definition and stick with it. Presenting people with half a dozen definitions, all totally different, will serve only to completely confuse anyone who reads the article. They'll have to choose, probably randomly, between them all. More likely, they will simply leave disgusted, dismissing the exponential function as unfathomable. ObsessiveMathsFreak (talk) 04:32, 9 August 2009 (UTC)

The lede[edit]

A recent editor replaced the introductory sentences with his/her own giving no explanation except for "use sensible definition in lede", even though the previous version had a citation of a text on the subject authored by well-respected experts, and the replacement had none. So that there is no misunderstanding, I wanted to explain why the approach in that text is sensible before doing a partial revert of the edit to reinstate the citation. The real issue is how to explain what e is. If one says that it is approximately 2.718281828..., that does not specify it uniquely. If one says that it is the base of the natural logarithm, that does specify it, but it is not ideal in that it is not self-contained: it presumes that the natural logarithm has already been defined, and that is a topic of the same level of difficulty as the exponential function itself. Saying that e is the number such that e^x is its own derivative specifies e precisely (and as a bonus, highlights the reason the exponential function is important in the first place). --WardenWalk (talk) 07:24, 6 August 2009 (UTC)

Diagram of slope of function[edit]

There is a diagram under Exponential function#Derivatives and differential equations showing a base of 1 and the tangent to the function. Unfortunately this could be misleading for people as they might think the vertical line was at 0 and get totally the wrong idea. Anyone like to try their hand at a better diagram? Dmcq (talk) 08:47, 9 August 2009 (UTC)

I totally agree. In fact, I independently came to the same conclusion, and made an edit to its caption before noticing your comment here. Anyway, I too hope that someone will produce a better diagram. It could also be improved by removing the θ and Tan θ, which are irrelevant to what the diagram is intended to show. --WardenWalk (talk) 14:33, 15 August 2009 (UTC)

Numerical computation missing[edit]

Information about the numerical computation of exp(x) is missing. There once was a section about the topic, but is has apparently been removed due to lack of quality.--84.189.68.236 (talk) 18:54, 21 August 2009 (UTC)

Yes the section was rubbish. There's lots of ways of doing it accurately, I'm not sure it is notable though. If you want something like that probably your best bet is just to refer to gnu libc for a good version and perhaps the IBM version for exactly rounded values. Dmcq (talk)

I don't agree that because a section is poorly written that therefore the subject has no place it the article. That seems EXACTLY opposite the Wikipedia philosophy of improving articles rather than tearing them apart. I find it curious Dmcq, that you refer the above user to a different source of information. Are you suggesting that efficient computation is not a notable part of a function?? Risible. You. Are. Wrong. I am not qualified to determine whether or not the fp algorithm outlined in the two (very poorly written, I agree) sections 11 & 12 here on this talk page are accurate, otherwise I'd put a small portion of them back in, a few lines at most. Many functions, numbers, etc., if not most, have a computation section in Wikipedia. Dmcq you made a bad error in judgment. IMHO.72.172.1.28 (talk) 14:31, 14 September 2013 (UTC)

e^-x[edit]

e^-x —Preceding unsigned comment added by 202.44.111.74 (talk) 16:10, 8 November 2009 (UTC)

(?_?)   :-\   Dmcq (talk) 17:19, 8 November 2009 (UTC)

$$$$$$$$$
$l |    $
$ \|    $
$  |` - $
$$$$$$$$$

There you go! ᛭ LokiClock (talk) 23:00, 25 November 2013 (UTC)

Too technical?[edit]

Someone stuck on a too technical tag at the top last month. Personally I feel it addresses that issue fairly well and much better than most other articles at its difficulty level and have removed the tag. No specific issue was mentioned. Any specific issues people see with it? Dmcq (talk) 11:46, 8 March 2011 (UTC)

formal definition all wrong[edit]

we should define a function exp(x)

infinite series or differential equation

  • then* show it is e^x for some e

otherwise it is fallacious

double definition —Preceding unsigned comment added by 70.189.170.229 (talk) 13:42, 13 May 2011 (UTC)

Have you looked at Characterizations of the exponential function? Dmcq (talk) 15:07, 13 May 2011 (UTC)

Exponential function and human understanding[edit]

I am wondering if there is also space in this article to discuss the suggested "inability of human beings to understand the exponential function" (Albert Bartlett). There is also a lecture on this topic from back in 2004 titled Arithmetic, Population and Energy. --spitzl (talk) 21:53, 13 June 2011 (UTC)

It's just a bit of rhetoric. There's an article Exponential growth where it might go as some sort of remark somewhere I guess. Dmcq (talk) 21:58, 13 June 2011 (UTC)

general exponential[edit]

I've removed some stuff about a to the power of x in the derivatives section as the article is about the exponential function and there's other articles exponential growth and exponentiation about such things. However I think there should probably be a duplication here of the definition of ax using the exponential function. There is a short section of what I'm thinking of for the complex version of but that's too late. The alternatives are to have another section like it for reals or othewise move that up and talk about complex numbers before getting to dealing with them in general here. Dmcq (talk) 23:00, 6 October 2011 (UTC)

Why was computation of e^x removed?[edit]

I agree that the section as written here ( see Talk page sections 11 How to compute exp(x) in computers AND 12 Computing exp(x) in computers ) is opaque, poorly written, verbose, etc. (ie. terrible). However, why isn't there a short section in the article detailing efficient algorithms for computing e^x? There should be.72.172.1.28 (talk) 14:24, 14 September 2013 (UTC)

Non-standard analysis[edit]

I think the text about doing a non-standard analysis version of differentiation is an undue digression in this article. Wikipedia isn't about loads of proofs. I think what perhaps could be done in that line is to give the non-standard analysis definition of e^x but any example should be move to the non-standard analysis article which has no examples in it. Dmcq (talk) 09:17, 30 April 2014 (UTC)