The factorial is one of the simplest operations in mathematics and crops up everywhere from combinatorics to analyisis. As a function, it is normally defined over the integers but, as often in mathematics, it is useful to look for ways in which to extend where certain functions or sequences make sense. For the factorial, if we are to be able to define it over the real (and later complex) numbers, this means looking for a more general formula which would remain valid for these while having some useful properties such as differentiability or convexity. In other words, we need to find a meaningful way to "join the dots".

As Euler, about 20 years old at the time, was appointed at the academy of St Petersburg, where he held his first important post as a professor, some of his colleagues there were working on exactly these types of problems, which they called "interpolation of sequences". As an example, Jakob Bernoulli had found an interesting formula which expressed the sequence in terms of which would still work if we replaced the integer by a real number; for example, if , we get the well known . Even so, two sequences, in particular, still bothered them. One was the so called harmonic series (which Euler managed to show was essentially equivalent to the natural logarithm) and the other was none other than the factorial . All this soon piqued Euler's interest and, starting from the clever observation that managed to create the function, effectively solved the problem (and much more!). It turns out, in fact, that the Gamma function is not only the sole sensible way of extending the factorial but gives rise to an incredible amount of fascinating mathematics.

So, if you've ever wondered what would be equal to, join us this Thursday at 7:30 PM in MS.04 as Dave McCormick takes us on a wild ride through analysis to show us what the Gamma function really is and some of its cool properties, after which we interpolate our way to the pub.