Last month saw the death of Alex Craik, emeritus professor of mathematics at St Andrews. He spent most of his career here, working on fluid dynamics, and (latterly) on History of Mathematics.
I didn’t know him very well, though he came in to the School from time to time, and had spoken at various events including Research Days.
But what I will mainly remember him for is a paper in the American Mathematical Monthly 112 (2005), 119-130, entitled “Prehistory of Faà di Bruno’s formula”.
This formula gives the nth derivative of a composite function f(g(x)), in an analogous way to Leibniz’s formula for the nth derivative of a product. It is particularly relevant to discrete mathematics, for two reasons: first, it applies to general formal power series over commutative rings with identity, no issues of convergence need to be considered; and second, the coefficients that arise are slightly disguised Stirling numbers (much as Leibniz’s formula involves binomial coefficients).
Francesco Faà di Bruno (1825-1888) is perhaps the only mathematician to be also a Roman Catholic saint. He was a student of Cauchy and friend of Hermite. He returned to Turin and devoted himself to work among the poor, for which he was canonised in 1988.
In his paper, Alex Craik describes several authors who had anticipated the formula to varying extents, and awards the palm of discovery to Arbogast in 1800. I recommend the paper to you.
However, even if Faà di Bruno was not the first to come up with it, it seems that his contribution was sufficient that the naming is not entirely unjust.
Rest in peace, Alex, and thank you for this insight.
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