Paul Halmos says in his paper Innovation in Mathematics in the Scientific american:
The late John von Neumann liked to cite this example of the relation between technological development and pure mathematics: A hundred and fifty years ago one of the most important problems of applied science-on which develop ment in industrv, commerce and government depended-was the problem of saving lives at sea. The statistics of the losses were frightful. The money and effort expended to solve the problem were frightful too-and sometimes ludicrous. No gadget, however complicated, was too ridiculous to consider-ocean going passenger vessels fitted out like outrigger canoes may have looked funny, but they were worth a try.
While leaders of government and industry were desperately encouraging such crank experiments, mathematicians were developing a tool that was to save more lives than all the crackpot inventors combined dared hope. That tool is what has come to be known as the theory of functions of a complex variable (a variable containing the "imaginary" number $i$, the square root of minus one). Among the many applications of this purely mathematical notion, one of the most fruitful is in the theory of radio communication. From the mathematician Karl Friedrich Gauss to the inventor Guglielmo Marconi it is only a few steps that almost any pair of geniuses such as James Clerk Maxwell and Heinrich Hertz can take in their stride.
I'm asking: what did Halmos have in mind when he stated "mathematicians were developing
a tool that was to save more lives than
all the crackpot inventors combined
dared hope"?
That tool is what has come
to be known as the theory of functions
of a complex variable"?
And how?