Talk:Gamma function

Euler's definition as an infinite product

The reference for Euler's infinite product says that the limit goes to n!, not 1. 104.187.53.82 (talk) 16:21, 6 November 2023 (UTC)

I believe that you are talking about two different expressions. Both of these are true:

$$\lim _{n\to \infty }{\frac {n!\,\left(n+1\right)^{z}}{(n+z)!}}=1$$

$$\lim _{n\to \infty }{\frac {1}{z}}\prod _{k=1}^{n}\left[{\frac {1}{1+{\frac {z}{k}}}}\left(1+{\frac {1}{k}}\right)^{z}\right]=\Gamma (z)$$ —Quantling (talk | contribs) 16:41, 6 November 2023 (UTC)

log vs ln

I don't care much about which, but the article should be consistent. Do we want to use "log" or "ln" to indicate a natural logarithm? We could use one or the other throughout. Or, if we mostly go with "ln", we could nonetheless use "log" in those fewer cases where the expression works regardless of the base of the logarithm. What do you prefer? —Quantling (talk | contribs) 17:06, 10 June 2024 (UTC)

I've made it consistent. Since I had to choose one, I chose log, with a notation on first use that it is the natural logarithm. I think this is the more common style for mathematical analysis and analytic number theory, the main subfields of mathematics for this topic. —David Eppstein (talk) 18:18, 10 June 2024 (UTC)

Equation appears wrong

The equation immediately following the words "Laplace transform" appears wrong.

The number log(π) has a positive sign in the previous equation, on the right of the equals sign.

So when it is brought to the left of the equals sign as in the equation following "Laplace transform", it should have a negative sign.

But it does not.

If this observation is correct, I hope someone familiar with this subject can fix this.

For clarity, I believe the issue referenced here has been resolved. The following is what this comment meant to say:

As of the revision at the time, the equality before "Laplace transform" had the term ${\frac {1}{2}}\log(2\pi ).$ Let's say that you then move this term to the left but forget to flip the sign. From there, assume that you make no more mistakes. On the right, move everything but the integral to the left. Then, apply the equality $z=\log(e^{z})$ to the $z$ term, and apply properties of logarithms to put everything into a single log. This would give you the wrong equality seen in this old revision. The equality has since been corrected.

The comment noted that the old revision's mistake could have arisen from the specific faulty line of logic I described, though this was not clear to me the first time I read the comment. In fact, I was extremely confused. I leave this reply to clarify what exactly is going on here, just to prevent future confusion. ISaveNewspapers (talk) 12:45, 8 February 2025 (UTC)

t?

What is t? This is crucial really, but I have never found a definition for it. Is it a constant? If not, how would you go about working it out on a scientific calculator with standard trig and logarithmic functions? Koro Neil (talk) 00:55, 3 August 2024 (UTC)

You mean the t in the first displayed equation and in the infobox? It's the variable of integration. Obviously. That's what the "dt" part at the end of the integral denotes. If you don't know what a variable of integration is, this article may not be for you. —David Eppstein (talk) 01:35, 3 August 2024 (UTC)

Yes, it's so obvious. To you. Not so much to the other person, it would appear. I guess that wasn't obvious. ISaveNewspapers (talk) 13:01, 8 February 2025 (UTC)

Missing 0! in image

In the Generalized factorial function more infos image in the Motivation section, the point 0!=1 is missing. As much as I'd like to know how to fix this, I don't, so can someone else fix it instead? ISaveNewspapers (talk) 07:12, 30 December 2024 (UTC)

The most likely reason why the 0!=1 point is missing from the image is because the y values of the discrete points come from the original definition of the factorial function, $x!=\prod _{i=1}^{x}i$, which doesn’t work for $x\leq 0$. Also, if you want to fix the image, you can just use an image editor, such as Photopea for example, to add the 0!=1 point into the image, save it, upload it to this website, and then edit the page to replace the current image you mentioned with that. 107.9.41.132 (talk) 23:33, 24 January 2025 (UTC)

When you say the original definition, are you talking about the definition used by the first person ever to study factorials? I don't think we know enough even to confidently say anything about that. Regardless, I don't think the creator of the image specifically preferred that version over the one we use today; it seems far more likely that they just forgot about 0!=1.

And yeah, I could probably use an image editor to cobble together a new version of the image that gets the point across adequately. However, it'd almost definitely end up looking sloppy, which I don't want. The original is an SVG image with highlightable text, and I would want that to be maintained in any updated version. It should be as if it were made by the original creator.

Looking deeper, it appears that the image was created at least partly with the help of Mathematica, a software system for math stuff. I might try to learn how to use that if I want to get this done. However, I won't make any guarantees because I have other things to do. ISaveNewspapers (talk) 13:34, 8 February 2025 (UTC)