Talk:List of integrals of exponential functions

Useful citation to simplify integrals

Many of these integrals can be calculated online on the wolfram mathworld website: http://integrals.wolfram.com/index.jsp Can this be added as a citation? — Preceding unsigned comment added by 87.211.188.107 (talk) 20:27, 29 June 2011 (UTC)

Removal of apparently incorrect integral

Removed this, since it is incorrect:

$\int _{-\infty }^{\infty }e^{-{x^{2}}/{a^{2}}}\cos bx\,dx=a{\sqrt {\pi }}(\sin {a^{2}b^{2} \over 4}+\cos {a^{2}b^{2} \over 4})$

Try numerically with $a=b=1$. Should probably be ${a{\sqrt {\pi }}}\exp(-({ab \over {2}})^{2})$ for $a>0$. TB

GENERAL REMARK: it is better to write x{sqrt(c)} than (sqrt(c))x, because it is prone to misunderstanding if the sqrt sign just does not stretch to include the x, or just does.

Admittedly, this remark would be clearer if I had the "real" sqrt symbol available here, but you understand what I mean, I hope. My remark applies to the integral of the exponential of minus x squared, which results in the error function. Alex Vermeulen, The Netherlands Jordaan12 (talk) 15:35, 19 February 2010 (UTC)

What is n?

This integral doesn't make much sense; what's n?

$\int e^{x^{2}}\,dx=e^{x^{2}}\left(\sum _{r=1}^{n}{\frac {1}{2^{n}x^{2n-1}}}\right)+{\frac {2n-1}{2^{n}}}\int {\frac {e^{x^{2}}\;dx}{x^{2n}}}$

Ed Sanville 00:23, 30 May 2005 (UTC)

I guess it should hold with any n. I think this is obtained by integrating by parts n times. So, I don't know the details, but I have a feeling this formula is correct as it stands. Oleg Alexandrov 22:04, 1 Jun 2005 (UTC)

Yeah, it does appear to be the n-th iteration of integration by parts. I think it's mathematical common sense that if nothing is said about n then you get to pick your favourite one, but for the people with actual common sense, we should probably include some comment which tells them that. I know this problem has come up regarding this specific integral among my friends. I just don't think I'm the right person to put that blurb in, for fear of confusing them further. Dusik 16:58, 22 September 2005 (UTC)

The formula is in fact obtained by repeated application of partial integration but is not correct as it stands (wrong coefficients). The (presumably) correct formula can or could be derived from the asymptotic expansion of the error function and is (to be checked once more)

$\int e^{x^{2}}\,dx=e^{x^{2}}\left(\sum _{j=0}^{n-1}{\frac {2j\,!}{j!}}\,{\frac {1}{2^{2j+1}x^{2j+1}}}\right)+{\frac {(2n-1)\,!}{(n-1)!2^{2n-1}}}\int {\frac {e^{x^{2}}}{x^{2n}}}\;dx\quad {\mbox{ for }}n>0$

I thought that the above integral was not convergent. Is the solution exact?

In the case of the errorfunction (integral :$\int _{0}^{t}e^{-x^{2}}\,dx$) the expansion has the same coefficients but now with alternating sign. The main application of this integral identity is to give an approximation formula (for large t) for the error function values or the complementary (erfc-) function. Here one has to choose a suitable n (cut-off parameter) depending on t because the full series diverges for every x, it is only an asymptotic series. --212.18.24.11 09:14, 23 September 2005 (UTC)

EXACTLY THE SAME QUESTION by Alex Vermeulen, The Netherlands. What's n? I would greatly appreciate an answer.Jordaan12 (talk) 15:01, 19 February 2010 (UTC)

As Oleg Alexandrov said, any positive integer n should work for the corrected version of the formula. You may pick any value which is convenient for the region in which you want to approximate the integral. I would suggest that you try values of n near to x². JRSpriggs (talk) 20:22, 16 March 2010 (UTC)

Nicer integral

ive found a nicer looking solving of the integral

$\int x^{n}e^{cx}\;dx={\frac {1}{c}}x^{n}e^{cx}-{\frac {n}{c}}\int x^{n-1}e^{cx}dx$

wich to me looks nasty. i have a easier method. should i add it?

Yes please. The solution is still an integral. Aren't there any serial represantation of this integral. Or something nicer —Preceding unsigned comment added by 88.229.24.154 (talk) 13:26, 19 December 2008 (UTC)

Inaccurate Integral

$\int _{-\infty }^{\infty }x^{2}e^{-ax^{2}}\,dx={\frac {1}{2}}{\sqrt {\pi \over a^{3}}}$

is incorrect. That's the answer if the limits were

$\int _{0}^{\infty }$

I've changed it to the correct:

$\int _{-\infty }^{\infty }x^{2}e^{-ax^{2}}\,dx={\sqrt {\pi \over a^{3}}}$

Melink14 18:43, 27 February 2007 (UTC)

I lied. It was correct as before as far as I can tell. If you can verify please do. Melink14 19:21, 27 February 2007 (UTC)

Needs proofs

The page needs proofs. That is, for every identity a link to a proof should be included, perhaps by linking to a page which contains proofs directly, like:

<complicated integral> = <nice value> (proof)

Or:

<complicated integral> = <nice value> (proof)

Or perhaps using <ref>:

<complicated integral> = <nice value> [1]

To me most of them look doable, but I assume that to a layman this looks just like your average unreferenced article. Shinobu 23:27, 1 June 2007 (UTC)

I am a new user to wikipedia, but I think I can so this, at least for some of these integrals. Might take a while though. If anyone wants to help out, that would be awesome. Going3killu (talk) 00:55, 14 December 2009 (UTC)

I personally disagree. I found the inline proofs with the 'show' and 'hide' rather distracting from what the article was talking about. I'm not going to remove them (I'm new and wouldn't want to hurt anything) but if others agree, maybe the proofs should be removed.Chessofnerd (talk) 02:05, 27 January 2011 (UTC)

I agree with you, and I think that so does MOS:MATH ("... don't include them when they serve only to establish the correctness of a result").—Emil J. 11:52, 27 January 2011 (UTC)

one more please

can someone please verify this, and add to the table?

$\int {x \over \sigma {\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2}/2\sigma ^{2}}}\;dx={\frac {1}{2}}{\mbox{erf}}\,{\frac {x-\mu }{\sigma {\sqrt {2}}}}-{\frac {\sigma }{\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2}/2\sigma ^{2}}}$

Jackzhp 22:52, 8 August 2007 (UTC)

What does $\sigma$ signify?. Going3killu (talk) 00:08, 14 December 2009 (UTC)

When talking about the Normal distribution, σ refers to the standard deviation. JRSpriggs (talk) 02:58, 14 December 2009 (UTC)

beware

89.181.138.253 is polluting this article. —Preceding unsigned comment added by 136.152.170.195 (talk) 01:12, 24 September 2007 (UTC)

Complex Constants

Some of this intergrals work with complex values of the constants, it would be useful if it can be noted in wich cases. 200.145.46.252 (talk) 00:04, 29 August 2008 (UTC)

Gamma function with two (??) arguments?

The formula disputed by Diffequa (talk · contribs) contains $\Gamma (n+1,-\ln x)\,.$ I have never seen a Gamma function with two arguments. What does its mean? JRSpriggs (talk) 06:59, 3 July 2009 (UTC)

To partly answer my own question, it may refer to the Gamma distribution. On the other hand, that seems to need three arguments, not just two. JRSpriggs (talk) 00:42, 25 November 2009 (UTC)

Better still, it may be the upper incomplete gamma function defined by

$\Gamma (s,x)=\int _{x}^{\infty }t^{s-1}\,e^{-t}\,dt.\,\!$

This would imply that

$\Gamma (n+1,-\ln x)=\int _{-\ln x}^{\infty }t^{n}\,e^{-t}\,dt\,.$

If $t=-\ln u\,,$ then

$\Gamma (n+1,-\ln x)=\int _{0}^{x}{(-\ln u)}^{n}\,du\,$

where 0≤x≤1. JRSpriggs (talk) 01:21, 25 November 2009 (UTC)

In the works

Working on a proofs page. Please just give me a while to get it up and going. Also, any feedback would be awesome. I might need help. Thanks. Going3killu (talk) 03:08, 14 December 2009 (UTC)

Just in case you're not aware, there is an ongoing discussion of the value of proofs in Wikipedia at Wikipedia talk:WikiProject Mathematics/Proofs. Also keep in mind that proofs as separate articles must still conform to Wikipedia notability and verifiability policies.--RDBury (talk) 15:09, 14 December 2009 (UTC)

Yes, I have been made aware of the problems associated with making a "proof page", and have scrapped that idea. Instead, I have decided to opt for a different approach, which is demonstrated on the second integral on the list. could you take a look at that, and let me know if that seems acceptable. Thanks.Going3killu (talk) 02:50, 16 December 2009 (UTC)

c != 0 should be added at the top level

I was doing some programming and c=0 had to be considered :) — Preceding unsigned comment added by Rrogers314 (talk • contribs) 15:49, 15 May 2015 (UTC)