Certain formulas I really enjoy looking at like the EulerMaclaurin formula or the Leibniz integral rule. What's your favorite equation, formula, identity or inequality?

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$e^{\pi i} + 1 = 0$ 


Stokes' Theorem 


Trivial as this is, it has amazed me for decades: $(1+2+3+...+n)^2=(1^3+2^3+3^3+...+n^3)$ 


There are many, but here is one. $d^2=0$ 


There's lots to choose from. RiemannRoch and various other formulas from cohomology are pretty neat. But I think I'll go with $$\sum\limits_{n=1}^{\infty} n^{s} = \prod\limits_{p \text{ prime}} \left( 1  p^{s}\right)^{1}$$ 


1+2+3+4+5+... = 1/12 Once suitably regularised of course :) 


Mine is definitely $$1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}+\cdots=\frac{\pi^2}{6},$$ an amazing relation between integers and pi. 


I'm currently obsessed with the identity $\det (\mathbf{I}  \mathbf{A}t)^{1} = \exp \text{tr } \log (\mathbf{I}  \mathbf{A}t)^{1}$. It's straightforward to prove algebraically, but its combinatorial meaning is very interesting. 


$$ \frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left \frac{\tan t+\sqrt{7}}{\tan t\sqrt{7}}\right\\ dt = \sum_{n\geq 1} \left(\frac n7\right)\frac{1}{n^2}, $$ where $\left(\frac n7\right)$ denotes the Legendre symbol. Not really my favorite identity, but it has the interesting feature that it is a conjecture! It is a rare example of a conjectured explicit identity between real numbers that can be checked to arbitrary accuracy. This identity has been verified to over 20,000 decimal places. See J. M. Borwein and D. H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A K Peters, Natick, MA, 2004 (pages 9091). 


$V  E + F = 2$ Euler's characteristic for connected planar graphs. 


1/(1z) = (1+z)(1+z^2)(1+z^4)(1+z^8)... Both sides as formal power series work out to 1 + z + z^2 + z^3 + ..., where all the coefficients are 1. This is an analytic version of the fact that every positive integer can be written in exactly one way as a sum of distinct powers of two, i. e. that binary expansions are unique. 


For a triangle with angles a, b, c $$\tan a + \tan b + \tan c = (\tan a) (\tan b) (\tan c)$$ 


$196884 = 196883 + 1$ 


I always thought this one was really funny: $1 = 0!$ 


It's too hard to pick just one formula, so here's another: the CauchySchwarz inequality:
Simple, yet incredibly useful. It has many nice generalizations (like Holder's inequality), but here's a cute generalization to three vectors in a real inner product space:
There are corresponding inequalities for 4 vectors, 5 vectors, etc., but they get unwieldy after this one. All of the inequalities, including CauchySchwarz, are actually just generalizations of the 1dimensional inequality:
or rather, instantiations of it in the 2^{nd}, 3^{rd}, etc. exterior powers of the vector space. 


Given a square matrix $M \in SO_n$ decomposed as illustrated with square blocks $A,D$ and rectangular blocks $B,C,$ $$M = \left( \begin{array}{cc} A & B \\\ C & D \end{array} \right) ,$$ then $\det A = \det D.$ What this says is that, in Riemannian geometry with an orientable manifold, the Hodge star operator is an isometry, a fact that has relevance for Poincare duality. http://en.wikipedia.org/wiki/Hodge_duality http://en.wikipedia.org/wiki/Poincar%C3%A9_duality But the proof is a single line: $$ \left( \begin{array}{cc} A & B \\\ 0 & I \end{array} \right) \left( \begin{array}{cc} A^t & C^t \\\ B^t & D^t \end{array} \right) = \left( \begin{array}{cc} I & 0 \\\ B^t & D^t \end{array} \right). $$ 


I think that Weyl's character formula is pretty awesome! It's a generating function for the dimensions of the weight spaces in a finite dimensional irreducible highest weight module of a semisimple Lie algebra. 


$2^n>n $ 


GaussBonnet, even though I am not a geometer. 


Euclid, Elements, Book1 Prop 47: Ἐν τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν τὴν ὀρθὴν γωνίαν περιεχουσῶν πλευρῶν τετραγώνοις. That is, In rightangled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. 


It has to be the ergodic theorem, $$\frac{1}{n}\sum_{k=0}^{n1}f(T^kx) \to \int f\:d\mu,\;\;\mu\text{a.e.}\;x,$$ the central principle which holds together pretty much my entire research existence. 


The same statement for number fields essentially describes the Galois theory. Now the idea that those should be somehow unified was one of the reasons in the development of abstract schemes, a very fruitful topic that is studied in the amazing area of mathematics called the abstract algebraic geometry. Also, note that "actions on sets" is very close to "representations on vector spaces" and this moves us in the direction of representation theory. Now you see, this simple line actually somehow relates number theory and representation theory. How exactly? Well, if I knew, I would write about that, but I'm just starting to learn about those things. (Of course, one of the specific relations hinted here should be the Langlands conjectures, since we're so close to having Lfunctions and representations here!) 


RiemannRoch, and its generalizations: GrothendieckHirzebruchRiemannRoch AtiyahSinger (which is also a generalization of GaussBonnet) Is it cheating to put all of these in a single answer? :) 


E[X+Y]=E[X]+E[Y] for any 2 random varibles X and Y 


It may be trivial, but I've always found $\sqrt{\pi}=\int_{\infty}^{\infty}e^{x^{2}}dx$ to be particularly beautiful. 


My favorite is the KoikeNortonZagier product identity for the jfunction (which classifies complex elliptic curves): j(p)  j(q) = p^{1} \prod_{m>0,n>1} (1p^{m}q^{n})^{c(mn)}, where j(q)744 = \sum_{n >2} c(n) q^{n} = q^{1} + 196884q + 21493760q^{2} + ... The left side is a difference of power series pure in p and q, so all of the mixed terms on the right cancel out. This yields infinitely many identities relating the coefficients of j. It is also the Weyl denominator formula for the monster Lie algebra. 


$ D_A\star F = 0 $ YangMills 


Addendum to $e^{i \pi}$Benjamin Peirce apparently liked this mathematical synonym for the additive inverse of $1$ so much that he introduced three special symbols for $e, i, \pi$ — ones that enable $e^{i \pi}$ to be written in a single cursive ligature, like so:



$(A\lambda _1) (A\lambda _2) \ldots = 0$, the CayleyHamilton theorem. 


Var[X+Y]=Var[X]+Var[Y] for any two independent random variables X and Y, which is the statistics equivalent of the Pythagorean Theorem. 

