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In mathematics and computer science, computable analysis is the study of mathematical analysis from the perspective of computability theory. It is concerned with the parts of real analysis and functional analysis that can be carried out in a computable manner. The field is closely related to constructive analysis and numerical analysis.
A notable result is that integration (in the sense of the Riemann integral) is computable.[1] This might be considered surprising as an integral is (loosely speaking) an infinite sum. While this result could be explained by the fact that every computable function from to is uniformly continuous, the notable thing is that the modulus of continuity can always be computed without being explicitly given. A similarly surprising fact is that differentiation of complex functions is also computable,[2] while the same result is false for real functions.[3]
The above motivating results have no counterpart in Bishop's constructive analysis. Instead, it is the stronger form of constructive analysis developed by Brouwer that provides a counterpart in constructive logic.
^See Simpson, Alex K. (1998), Brim, Luboš; Gruska, Jozef; Zlatuška, Jiří (eds.), "Lazy functional algorithms for exact real functionals", Mathematical Foundations of Computer Science 1998, Lecture Notes in Computer Science, vol. 1450, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 456–464, doi:10.1007/bfb0055795, ISBN 978-3-540-64827-7
^by Cauchy's integral formula.
^because discontinuous operators are automatically uncomputable.
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