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 Sep22 awarded Nice Answer Aug21 awarded Revival May30 awarded Yearling Jul4 comment Evaluating definite integrals There are certainly real integrals which are much more easily computed with contour integration in the complex plane (for example, $\int_{-\infty}^\infty \frac{1}{1+x^n}$) but as the other comments suggest, I'm not sure that this methodology precludes being a technique of "real analysis." Jul4 awarded Critic Jun23 comment Is there any direct application of Gödel's Theorems outside of logic? Philosophers have certainly tried to use it. Jun23 comment Studying quantum mechanics without physics background You should probably feel comfortable with basic material before proceeding to Quantum Mechanics at the graduate level. Math in QM includes a lot of linear algebra, as well as some more advanced material, like the representation theory of Lie Groups. Even if you have a handle on the math, physics isn't always formal, and you will certainly need to have a well-tuned intuition. It sounds like you mostly have the background, just brush up on the basics. Jun19 comment Funny identities Sophomore's Dream? Jun18 answered What is a good book to study linear algebra? Jun11 comment Set {1,1} = Set {1}, origin of this convention Computer scientists tend to think of sets this way. Jun11 comment Question about flat modules and exact sequences I agree with Serkan's answer, but it's worth noting that in your example, $\mbox{Ker}(f) = 0$ and $\mbox{Im}(g) = C,$ just by exactness. So the equivalence you desired actually holds. Jun7 awarded Commentator Jun7 comment Conjugacy classes and irreducible representations of GL_2(q) A good reference for this is Fulton and Harris. They have a very complete analysis of the irreps of $GL_2(\mathbb{F}_q).$ Jun7 comment Is there really no way to integrate $e^{-x^2}$? Sure, just use polar coordinates. That's what Gauss did. Jun6 comment Winding number on a simply connected region A simply connected open set is, by definition, is a set in which every closed chain is homologous to a point. I.e., an open set in which, by definition, every closed chain has winding number 0. There's not necessarily an intuitive reason, but the motivation for the definition is that these are exactly the open sets in which you can define the logarithm. Jun6 answered Application residue theorem for improper integrals Jun3 comment Order of Essential Singularity No. The term order only applies to removable singularities. Your functions both have "infinite order" poles; $e^{1/z^3}$ certainly does not have a singularity of order 3 at 0. Jun3 awarded Supporter Jun3 comment Applications of monads in general topology? As Sl2 said, adjoint pairs give rise to monads (and conversely, every monad gives rise to an adjoint pair of functors). These objects are ubiquitous in topology. The modern approach to homotopy theory relies on a categorical framework, and many theorems are framed in this language. Jun3 comment How to prove there are an infinite number of squarefree numbers of the form $2^p-1$? Well, if you believe that there are infinitely many Mersenne primes, you automatically believe this fact. My guess would be that this is a hard fact.