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Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the first publication in the Perspectives in Logic series, Pour-El and Richards present the first graduate-level treatment of computable analysis within the tradition of classical mathematical reasoning. The book focuses on the computability or noncomputability of standard processes in analysis and physics. Topics include classical analysis, Hilbert and Banach spaces, bounded and unbounded linear operators, eigenvalues, eigenvectors, and equations of mathematical physics. The work is self-contained, and although it is intended primarily for logicians and analysts, it should also be of interest to researchers and graduate students in physics and computer science.

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We give a rough statement of the main result. Let D be a compact subset of R3 x W. The propagation u ( z , y, z , t ) of a wave can be noncomputable in any neighborhood of any point of D even though the initial conditions which determine the wave propagation uniquely are computable. A precise statement of the result appears below. Mathematics Subject Classification: 03D80, 03F60. 1 1 2" implies If ( . )f(y)l 5 -. I . -YI L 2"o computable sequence of functions is defined in a similar way just as was done for computable sequences of real numbers. proofs can be found in [5].The following basic facts about computable functions will be used later. Their 1. Let {fn} be a computable sequence of functions. Suppose {fn} converges to f effectivelyi. e. suppose there is a recursive function e from N to N such that, for all 1 z E I, k 2 e ( n ) implies 1fk(z)f(x)I 5 -. Then f is a computable function. 2" 2 . The integrals' f(z)dx is a computable real number iff is a computable function3 . Iff is a Cz function and is computable, then its derivative f' is also computable. and both a and b are computable reals.

We present a mathematical definition of an analog generable function of a real variable. This definition is formulated in terms of a simultaneous set of nonlinear differential equations possessing a "domain of generation." (The latter concept is explained in the text.) Our definition includes functions generated by existing general-purpose analog computers. Using it we prove two theorems which provide a characterization of analog generable functions in terms of solutions of algebraic differential polynomials. The characterization has two consequences. First we show that there are entire functions which are computable (in the sense of recursive analysis) but which cannot be generated by any analog computer in any interval-e.g. l/r(x) and 2^-i (x°/n^')). Second we note that the class of analog generable functions is very large: it includes special functions which arise as solutions to algebraic differential polynomials. Although not all computable functions are analog generable, a kind of converse holds. For entire functions,/(x) = X"o b,x', the theorem takes the following form. If f(x) is analog generable on some closed, bounded interval then there is a finite number of bk such that, on every closed bounded interval, f(x) is computable relative to these bk. A somewhat similar theorem holds if/is not entire. Although the results are stated and proved for functions of a real variable, they hold with minor modifications for functions of a complex variable.