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**AlephZero** When I was coming up, "analysis" just meant Calculus. Thus "complex analysis" is the Calculus of complex-valued functions, etc. "Real analysis" deals in an involved way with the topology of the real numbers in order to rigorously construct the apparatus of Calculus, etc... If one says that Liouville used "analytical methods" to prove the existence of transcendental numbers whereas Cantor used set theoretic arguments, one is referring to the fact that Liouville's proof uses the Mean Value Theorem for Derivitives from Calculus, etc.