too rapidly. The final chapters are devoted to the Gleason-Kahane-Żelazko Theo-
rem (in a unital Banach algebra, a subspace of codimension 1 which contains no
invertible elements is a maximal ideal) and the Fatou-Julia-Baker Theorem (the
Julia set of a rational function of degree at least 2 or a nonlinear entire function is
the closure of the repelling periodic points). We end on a high note, with a proof
of the Prime Number Theorem. A coda deals very briefly with two unusual appli-
cations: one to fluid dynamics (the design of shockless airfoils for partly supersonic
flows), and the other to statistical mechanics (the stochastic Loewner evolution).
To a certain extent, the choice of topics is canonical; but, inevitably, it has
also been influenced by our own research interests. Some of the material has been
adapted from [FA]. Our title echoes that of a paper by the second
Although this book has been in the planning stages for some time, the actual
writing was done during the Spring and Summer of 2010, while the second author
was on sabbatical from Bar-Ilan University. He thanks the Courant Institute of
Mathematical Sciences of New York University for its hospitality during part of this
period and acknowledges the support of Israel Science Foundation Grant 395/07.
Finally, it is a pleasure to acknowledge valuable input from a number of friends
and colleagues. Charles Horowitz read the initial draft and made many useful com-
ments. David Armitage, Walter Bergweiler, Alex Eremenko, Aimo Hinkkanen, and
Tony O’Farrell all offered perceptive remarks and helpful advice on subsequent ver-
sions. Special thanks to Miriam Beller for her expert preparation of the manuscript.
Peter D. Lax Lawrence Zalcman
New York, NY Jerusalem, Israel
Zalcman, Real proofs of complex theorems (and vice versa), Amer. Math. Monthly
81 (1974), 115-137.