We now come to perhaps the most central theorem in complex analysis (save possibly for the fundamental theorem of calculus), namely Cauchy’s theorem, which allows one to compute a large number of contour integrals even without knowing any explicit antiderivative of. There are many forms and variants of Cauchy’s theorem. To give one such […]

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Academics / Mathematics : The Unapologetic Mathematician

At last we come to the version of Stokes’ theorem that people learn with that name in calculus courses. Ironically, unlike the fundamental theorem and divergence theorem special cases, Stokes’ theorem only works in dimension, where ...

Academics / Mathematics : Terence Tao's Blog

Let be the algebraic closure of, that is to say the field of algebraic numbers. We fix an embedding of into, giving rise to a complex absolute value for algebraic numbers. Let be of degree, so that is irrational. A classical theorem...

Academics / Mathematics : Terence Tao's Blog

Having discussed differentiation of complex mappings in the preceding notes, we now turn to the integration of complex maps. We first briefly review the situation of integration of (suitably regular) real functions of one variable. ...

Academics / Mathematics : Terence Tao's Blog

Next week, I will be teaching Math 246A, the first course in the three-quarter graduate complex analysis sequence. This first course covers much of the same ground as an honours undergraduate complex analysis course, in particular ...

Academics / Mathematics : Terence Tao's Blog

At the core of almost any undergraduate real analysis course are the concepts of differentiation and integration, with these two basic operations being tied together by the fundamental theorem of calculus (and its higher dimensional...

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