Talk:Exponential function

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Contents 1 Notation 2 Coherent Graphs 3 My edits of Aug 6, 2004 4 information should be reordered 5 very strange 6 Comment moved from article 7 Title change 8 sin−1x is ambiguous 9 Only with base of e? 10 How to compute exp(x) in computers 11 Computing exp(x) in computers 12 incorrect formula 13 Link to proof for article that ex is the only non-zero function that is its own derivative?

[edit] Notation

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

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You don't really need the b; the most general form is ka^x. 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 a^x, so I started the article with exp(x) and then came to a^x as soon as possible. --AxelBoldt

You can define a^x without defining exp(x). For positive a, you define a⁰ = 1, aⁿ⁺¹ = a×aⁿ for natural n, and a⁻ⁿ=1/aⁿ. Then you define a^1/n to be the unique positive real x such as xⁿ = a, and a^m/n=(a^1/n)^m. Since any real x can be approximated arbitrarily well by a fraction m/n, and a^m/n is monotonous, you just need to require that a^x be continuous. --Army1987 12:00, 19 September 2006 (UTC)

I've changed the opening paragraphs to introduce the form ka^x even sooner. - dcljr 12:50, 6 Aug 2004 (UTC)

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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).

[edit] Coherent Graphs

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)

[edit] My edits of Aug 6, 2004

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 . 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)

[edit] information should be reordered

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)

[edit] very strange

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 (talk • contribs) 23:10, 18 September 2006 (UTC2)

[edit] Comment moved from article

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

[edit] Title change

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)

[edit] sin⁻¹x is ambiguous

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)

[edit] Only with base of e?

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

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

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.

[edit] Computing exp(x) in computers

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 = smbⁿ 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(sm2ⁿ) = log(s) + log(m) + nlog(2)

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

Thus n is found as the integer such that

We then find log(m) as the value log(m) = x − nlog(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:

Thus, the result y = exp(x) = m2ⁿ 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) + isin(y)) = e^xcos(y) + ie^xsin(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:

θ = arctan2(y,x)

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

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

Thus, a^b = exp(bloga) = exp((u + vi)(log(r) + θi)) = exp(ulog(r) − vθ + (uθ + vlog(r))i)

If we define p = ulog(r) − vθ and q = uθ + vlog(r) (both p and q are real values) we then get:

a^b = exp(p + qi) = exp(p)exp(qi) = exp(p)(cos(q) + isin(q))

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

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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 aⁿ 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)

[edit] incorrect formula

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).

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

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 such that for all x₁,x₂ in the reals . 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)