# Who invented the gamma function?

Who was the first person who solved the problem of extending the factorial to non-integer arguments?

Detlef Gronau writes [1]: "As a matter of fact, it was Daniel Bernoulli who gave in 1729 the first representation of an interpolating function of the factorials in form of an infinite product, later known as gamma function."

On the other hand many other places say it was Leonhard Euler. Will the real inventor please stand up?

[1] "Why is the gamma function so as it is?" by Detlef Gronau, Teaching Mathematics and Computer Science, 1/1 (2003), 43-53.

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What makes you so sure there is a sensible answer to this question? Bernoulli and Euler were contemporaries who knew each other. Is the "invention" the discovery of the integral formula, or the posing of the question of extension? Moreover the gamma function is ubiquitous enough it has probably been discovered countless times in different contexts. –  Kevin McGerty Dec 26 '09 at 2:58
"What makes you so sure there is a sensible answer to this question?" I do not claim there is. I ask a question. Do you think that asking a question always has the precondition that there is a sensible answer? What I am looking for is an answer in accordance to the conventions of the historiography of science. Is this not obvious? –  Bruce Arnold Dec 26 '09 at 12:48
Kevin, knowing more about a nonresolvable problem is a sensible answer. Anybody interested in a question wants to know more, even if no resolution is very likely or even makes sense. –  Zoran Skoda Jul 19 '10 at 12:45

I don't have a complete answer. As you say, many sources say that Euler did it, but Gronau gives compelling reason to doubt this. The best source I have found for this issue is "The early history of the factorial function" by Dutka, and for what it's worth I am convinced that Gronau's assessment is a fair one.

First, I'll summarize the usual story. Kline discusses this in chapter 19, section 5 of Mathematical Thought from Ancient to Modern Times (which falls in volume 2 of the paperback printing), and a more thorough source is Davis's article "Leonhard Euler's Integral: A Historical Profile of the Gamma Function". There is agreement in these sources that Euler solved the problem after unsuccessful attempts by Stirling, D. Bernoulli, and Goldbach, and that the first record of Euler's solution appears in outline form in a 1729 letter from Euler to Goldbach. This was expanded in subsequent letters and written up in the article to which Kristal Cantwell links (apparently the article was written in 1729 but not published until 1738). Euler's letters to Goldbach start on the third page of this pdf.

However, Gronau cites a letter by Bernoulli that was written a few days before Euler's and that contains at least a partial solution, possibly contradicting Kline and Davis. Dutka's paper goes into more detail and also claims that Euler's work was influenced by Bernoulli's earlier solution. I could only speculate on what led to the confusion among other authors, and I won't do so here. Perhaps it should be mentioned here (as is done by Gronau and Dutka) that Euler did much more than Bernoulli. For instance, Euler gave the first integral representations of the gamma function.

Edit: Because this answer is accepted and yet incomplete, I want to direct attention to Bruce Arnold's answer below. It contains a link to a copy of the too often neglected letter of D. Bernoulli cited by Gronau and Dutka.

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Davis's article "Leonhard Euler's Integral: A Historical Profile of the Gamma Function" received the Chauvenet Prize and can be downloaded free from maa.org [1]. It says that the problem was ".. bandied unsuccessfully by Daniel Bernoulli..". This is certainly the source for many attributions to Euler found in the literature. However, this paper was written in the year 1959 and Gronau's paper (which references Davis's paper) in the year 2003. Maybe there is sometimes advancement in the history of science? [1] mathdl.maa.org/mathDL/22/… –  Bruce Arnold Dec 26 '09 at 12:21
Yes, I didn't think much about this, but perhaps the letter of Bernoulli hadn't been discovered by modern historians at the time Davis and Kline were writing. However places like maa.org/editorial/euler/… still cite this as fact. –  Jonas Meyer Dec 26 '09 at 12:27

The first person who gave a representation of the so called gamma function was Daniel Bernoulli in a letter to Goldbach from 1729-10-06. The letter can be seen here.

The formula reads in modern notation as given by Gronau in the article cited in the answer:

$x! = \lim_{n\rightarrow \infty}\left(n+1+\frac{x}{2}\right)^{x-1} \prod_{i=1}^n\frac{i+1}{i+x}$

Gronau also observes that "Numerical experiments show that the formula of Bernoulli converges much faster to its limit than that of Euler ...", "that of Euler" refers here to a formula Euler has given in a letter to Goldbach dated 1729-10-13.

Gronau writes: "Euler who, at that time, stayed together with D. Bernoulli in St. Petersburg gave a similar representation of this interpolating function. But then, Euler did much more. He gave further representations by integrals, and formulated interesting theorems on the properties of this function."

Though this justifies the name 'Euler gamma function' Euler's representation was historically only second to Daniel Bernoulli's.

The correspondence between Goldbach, Daniel Bernoulli and Euler which undoubtedly gave birth to the gamma function is well documented in Paul Heinrich Fuss's „Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIeme siècle ..“, St. Pétersbourg, 1843.

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I'm still curious about 2 things: (1) If this correspondence was well documented by Fuss in the 19th century, why did so many 20th century scholars fail to acknowledge Bernoulli's contribution? (2) You are essentially following the source you provided in your question, so what has changed to make you so certain? –  Jonas Meyer Dec 26 '09 at 23:08
@Jonas ->(1) I do not know. ->(2) That I found the letter, saw it with my own eyes and could check that is says what other claim it says. –  Bruce Arnold Dec 26 '09 at 23:37
@Jonas: Clearly your first question deserves a better answer. I think Davis's Chauvenet Prize article makes a serious error in not mentioning Daniel Bernoulli's letter and solution. I just looked it up again: Davis gives only Tome I of Fuss's 'Correspondance' as reference, whereas Daniel's letter is in Tome II. Perhaps Davis missed this important source. Later his mistake was passed down by 'argumentum ad verecundiam' (which is the Latin translation of 'according to the wikipedia' :) –  Bruce Arnold Dec 27 '09 at 23:37

In his first letter to Goldbach (already linked to in Jonas' answer) Euler writes that he communicated his interpolation of the sequence of factorials (or Wallis's hypergeometric series, as it was called back then) to Daniel Bernoulli:

"I communicated this to Mr. Bernoulli, who by his own method arrived at nearly the same final expression" (this is the sentence starting with "communicavi haec . . . " on p. 4).

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Well of course, these three noble men discussed this subject for a long time. Do you think the sentence you cite sheds any new light on the matter resp. doubt that Daniel Bernoulli was the first to come up with the gamma function? Since my Latin is a little bit rusty I might not be able to extract further relevant information from this letter; on the other hand Bernoulli in his letter, which he wrote a week earlier, gives not the slightest indication that he had the "term general pour la suite 1, 1*2, 1*2*3, etc." from Euler. He would certainly have mentioned this if it had been the case. –  Bruce Arnold Dec 10 '10 at 17:35
After Euler communicated his result on the summation of the inverse squares to J. Bernoulli, he found his own proof (similar to Euler's) and even published it without crediting Euler - after all it was his proof!. And yes, I think Euler's letter clearly shows that it was him (and not D.B.) who first came up with the gamma function. –  Franz Lemmermeyer Dec 10 '10 at 18:07

According to the wikipedia article the question was of extending the factorial beyond the integers was first posed in the 1720's by Daniel Bernoulli and Christian Goldbach. It was first solved by Euler in 1729. Here is an english translation of a paper by Euler which contains his solution.

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This does not answer the question in any way. I would like to get answers which rely on scholarly material, like academic and peer-reviewed publications. Wikipedia is not authoritative and unreliable, the lack of fact checking on esoteric topics is notorious. And I would like to get answers which consider Gronau's claim that "it was Daniel Bernoulli who gave in 1729 the first representation [of the gamma function]" seriously. –  Bruce Arnold Dec 25 '09 at 20:09
So, would you say that unless Gronau provides a reference for his assertion, then it is no more authoritative than Wikipedia? –  Gerald Edgar Dec 25 '09 at 20:50
@Gerald: Please do not start a discussion on the value of Wikipedia here and respect the FAQ which says: "Math Overflow is not a discussion forum. There's a place for discussion about mathematics, but it isn't Math Overflow. Blogs and threaded discussion forums are a more appropriate place for discussions." Any comments on the question is welcome. –  Bruce Arnold Dec 25 '09 at 21:12
While I agree that this does not answer the question, I voted it up because of the link to an original source relevant to the issue which was useful for the time when this was the only posted answer. I also agree that this is not the best best place to discuss the value of Wikipedia, but I don't think that Gerald Edgar did start such a discussion. –  Jonas Meyer Dec 27 '09 at 22:09