A diversion

Yesterday to an excellent lecture, given by a rather flamboyant French mathematician, in honour of John Nash, the chap about whom they made a film, with Russell Crowe (of all people) impersonating the US mathematician. Cédric Villani being the French mathematician, on this occasion wearing a different insect brooch from that illustrated. I shall return to the lecture in due course, but for the present I notice fractional derivation of a function, something of which I had not previously heard. The extension of derivation as a function of positive integers to derivation as a function of the complex plane.

From where wikipedia takes me to the gamma function, a function of which I have heard in connection with statistics, but I don't think that I ever knew that the gamma function was the extension of the well-known factorial function to the complex plane, the second such extension of the day. This one being quite a reasonable extension, only breaking down, in a reasonably controlled way, at the negative integers. And so earns the title of a meromorphic function, reserved for functions which are not quite well behaved enough to be called regular. Also the Mellin transform of the negative exponential function - a transform named for a Finnish nationalist who backed the wrong horse in the relativity debate back at the beginning of the 20th century. Which all goes to show that the Finns have more to their credit than Sibelius and birch wine. See reference 2.

I wonder if it is possible to prove that the gamma function is the only such extension?

A reminder of how joined-up mathematics is, with all sorts of strange connections all over the shop.

Reference 1: http://cedricvillani.org/.

Reference 2: http://psmv3.blogspot.co.uk/2016/04/twittering.html.