What's your favorite equation, formula, identity or inequality? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-20T04:19:32Z http://mathoverflow.net/feeds/question/3134 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality What's your favorite equation, formula, identity or inequality? Kim Greene 2009-10-28T20:19:50Z 2010-08-22T16:14:57Z <p>Certain formulas I really enjoy looking at like the <a href="http://en.wikipedia.org/wiki/Euler_Maclaurin" rel="nofollow">Euler-Maclaurin formula</a> or the <a href="http://en.wikipedia.org/wiki/Leibniz_integral_rule" rel="nofollow">Leibniz integral rule</a>. What's your favorite equation, formula, identity or inequality?</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/3137#3137 Answer by Sonia Balagopalan for What's your favorite equation, formula, identity or inequality? Sonia Balagopalan 2009-10-28T20:28:09Z 2009-11-08T19:57:01Z <p>$e^{\pi i} + 1 = 0$</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/3138#3138 Answer by Ilya Nikokoshev for What's your favorite equation, formula, identity or inequality? Ilya Nikokoshev 2009-10-28T20:35:57Z 2009-10-28T21:32:37Z <blockquote> <p>For X a based smooth manifold, the category of finite covers over X is equivalent to the category of actions of the fundamental group of X on based finite sets:</p> </blockquote> <pre><code> \pi-sets === et/X </code></pre> <p>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 <strong>abstract algebraic geometry</strong>. Also, note that "actions on sets" is very close to "representations on vector spaces" and this moves us in the direction of representation theory.</p> <p>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.</p> <p>(Of course, one of the specific relations hinted here should be the Langlands conjectures, since we're <em>so</em> close to having L-functions and representations here!)</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/3145#3145 Answer by Proportional for What's your favorite equation, formula, identity or inequality? Proportional 2009-10-28T21:23:44Z 2009-10-28T21:23:44Z <p><a href="http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D47" rel="nofollow">Euclid, Elements, Book1 Prop 47</a>:</p> <p>Ἐν τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν τὴν ὀρθὴν γωνίαν περιεχουσῶν πλευρῶν τετραγώνοις. </p> <p>That is, </p> <p>In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/3148#3148 Answer by S. Carnahan for What's your favorite equation, formula, identity or inequality? S. Carnahan 2009-10-28T21:39:41Z 2009-10-28T21:39:41Z <p>My favorite is the Koike-Norton-Zagier product identity for the j-function (which classifies complex elliptic curves):</p> <p>j(p) - j(q) = p<sup>-1</sup> \prod<sub>m>0,n>-1</sub> (1-p<sup>m</sup>q<sup>n</sup>)<sup>c(mn)</sup>,</p> <p>where j(q)-744 = \sum<sub>n >-2</sub> c(n) q<sup>n</sup> = q<sup>-1</sup> + 196884q + 21493760q<sup>2</sup> + ... 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.</p> <p>It is also the Weyl denominator formula for the monster Lie algebra.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/3149#3149 Answer by Sammy Black for What's your favorite equation, formula, identity or inequality? Sammy Black 2009-10-28T21:43:38Z 2009-10-28T21:43:38Z <p>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.</p> <p><img src="http://upload.wikimedia.org/math/8/d/f/8df922b7028262e8ec0910790394127a.png" alt="alt text" /></p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/3153#3153 Answer by Kevin Lin for What's your favorite equation, formula, identity or inequality? Kevin Lin 2009-10-28T22:10:47Z 2009-10-28T22:15:58Z <p><a href="http://en.wikipedia.org/wiki/Riemann%E2%80%93Roch%5Ftheorem" rel="nofollow">Riemann-Roch</a>, and its generalizations:</p> <p><a href="http://en.wikipedia.org/wiki/Hirzebruch%E2%80%93Riemann%E2%80%93Roch%5Ftheorem" rel="nofollow">Hirzebruch-Riemann-Roch</a></p> <p><a href="http://en.wikipedia.org/wiki/Grothendieck%E2%80%93Hirzebruch%E2%80%93Riemann%E2%80%93Roch%5Ftheorem" rel="nofollow">Grothendieck-Hirzebruch-Riemann-Roch</a></p> <p><a href="http://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer%5Findex%5Ftheorem" rel="nofollow">Atiyah-Singer</a> (which is also a generalization of <a href="http://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet%5Ftheorem" rel="nofollow">Gauss-Bonnet</a>)</p> <p>Is it cheating to put all of these in a single answer? :-)</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/3158#3158 Answer by Sam Derbyshire for What's your favorite equation, formula, identity or inequality? Sam Derbyshire 2009-10-28T22:42:27Z 2009-10-28T22:42:27Z <p>Stokes' Theorem <br /> <img src="http://upload.wikimedia.org/math/9/8/d/98da50d6b2b3b0abc3de5994d98c5562.png" alt="Stokes' Theorem" /></p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/3164#3164 Answer by Michael Lugo for What's your favorite equation, formula, identity or inequality? Michael Lugo 2009-10-28T23:09:57Z 2009-10-28T23:09:57Z <p>1/(1-z) = (1+z)(1+z^2)(1+z^4)(1+z^8)...</p> <p>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.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/3172#3172 Answer by Jonathan Kariv for What's your favorite equation, formula, identity or inequality? Jonathan Kariv 2009-10-29T00:08:00Z 2009-10-29T00:08:00Z <p>E[X+Y]=E[X]+E[Y] for any 2 random varibles X and Y</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/3182#3182 Answer by Qiaochu Yuan for What's your favorite equation, formula, identity or inequality? Qiaochu Yuan 2009-10-29T00:42:06Z 2009-11-14T23:10:44Z <p>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.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/3214#3214 Answer by Darsh Ranjan for What's your favorite equation, formula, identity or inequality? Darsh Ranjan 2009-10-29T04:44:21Z 2010-06-02T18:46:28Z <p>$\pi = 2 \times 1/\sqrt(1/2) \times 1/\sqrt((1+\sqrt(1/2))/2) \times 1/\sqrt((1+\sqrt((1+\sqrt(1/2))/2))/2) \times \ldots$</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/3264#3264 Answer by Anna Varvak for What's your favorite equation, formula, identity or inequality? Anna Varvak 2009-10-29T14:36:27Z 2009-10-29T14:36:27Z <p>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.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/3350#3350 Answer by Dan Piponi for What's your favorite equation, formula, identity or inequality? Dan Piponi 2009-10-29T22:06:58Z 2009-10-29T22:06:58Z <p>1+2+3+4+5+... = -1/12</p> <p>Once suitably <a href="http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7_%C2%B7_%C2%B7" rel="nofollow">regularised</a> of course :-)</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/3355#3355 Answer by George Lowther for What's your favorite equation, formula, identity or inequality? George Lowther 2009-10-29T23:32:31Z 2009-10-29T23:32:31Z <p>There's lots to choose from. Riemann-Roch and various other formulas from cohomology are pretty neat. But I think I'll go with</p> <p>&Sigma;<sub>n=1</sub><sup>&infin;</sup>&nbsp;n<sup>-s</sup>&nbsp;=&nbsp;&Pi;<sub>p prime</sub> (1-p<sup>-s</sup>)<sup>-1</sup>.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/3361#3361 Answer by Darsh Ranjan for What's your favorite equation, formula, identity or inequality? Darsh Ranjan 2009-10-30T00:58:23Z 2009-10-30T00:58:23Z <p>It's too hard to pick just one formula, so here's another: the Cauchy-Schwarz inequality: </p> <blockquote> <p>||x|| ||y|| >= |(x.y)|, with equality iff x&amp;y are parallel.</p> </blockquote> <p>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: </p> <blockquote> <p>||x||<sup>2</sup> ||y||<sup>2</sup> ||z||<sup>2</sup> + 2(x.y)(y.z)(z.x) >= ||x||<sup>2</sup>(y.z)<sup>2</sup> + ||y||<sup>2</sup>(z.x)<sup>2</sup> + ||z||<sup>2</sup>(x.y)<sup>2</sup>, with equality iff one of x,y,z is in the span of the others. </p> </blockquote> <p>There are corresponding inequalities for 4 vectors, 5 vectors, etc., but they get unwieldy after this one. <em>All</em> of the inequalities, including Cauchy-Schwarz, are actually just generalizations of the 1-dimensional inequality:</p> <blockquote> <p>||x|| >= 0, with equality iff x = 0,</p> </blockquote> <p>or rather, <b>instantiations</b> of it in the 2<sup>nd</sup>, 3<sup>rd</sup>, etc. exterior powers of the vector space. </p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/4668#4668 Answer by Csar Lozano Huerta for What's your favorite equation, formula, identity or inequality? Csar Lozano Huerta 2009-11-08T19:55:16Z 2009-12-18T15:43:55Z <p>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.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/4680#4680 Answer by MBN for What's your favorite equation, formula, identity or inequality? MBN 2009-11-08T21:38:22Z 2009-11-08T21:38:22Z <p>There are many, but here is one.</p> <p>$d^2=0$</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/5588#5588 Answer by unknown (google) for What's your favorite equation, formula, identity or inequality? unknown (google) 2009-11-14T22:52:21Z 2009-11-14T22:52:21Z <p>The Newton iteration for finding the inverse, X, of a matrix A:</p> <p>X<sub>i+1</sub> = 2 * X<sub>i</sub> - X<sub>i</sub> * A * X<sub>i</sub></p> <p>Completely impractical and yet so beautiful. The first time I saw a Newton iteration working I thought it was "magical".</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/5640#5640 Answer by Dan for What's your favorite equation, formula, identity or inequality? Dan 2009-11-15T19:13:59Z 2009-11-15T19:13:59Z <p>For a triangle with angles a, b, c $$\tan a + \tan b + \tan c = (\tan a) (\tan b) (\tan c)$$</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/5680#5680 Answer by Jon Awbrey for What's your favorite equation, formula, identity or inequality? Jon Awbrey 2009-11-16T07:20:32Z 2009-12-18T15:30:09Z <h2>Addendum to $e^{i \pi}$</h2> <p><a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Peirce%5FBenjamin.html" rel="nofollow">Benjamin Peirce</a> apparently liked this mathematical synonym for the additive inverse of $1$ so much that he introduced three special symbols for $e, i, \pi$ &mdash; ones that enable $e^{i \pi}$ to be written in a single cursive ligature, like so:</p> <p><img src="http://mywikibiz.com/images/8/86/Benjamin%5FPeirce%5F--%5Fe%5E%28i%5Fpi%29.png" alt="Benjamin Peirce's script for e^(i pi)" title="" /></p> <ul> <li>Benjamin Peirce (1870/1882), <i><a href="http://www.archive.org/details/linearassociati00peirgoog" rel="nofollow">Linear Associative Algebra</a></i>, § 15, p. 5.</li> </ul> <p><img src="http://mywikibiz.com/images/2/28/Benjamin%5FPeirce%5F--%5FLAA%5F%C2%A7%5F15.jpg" alt="Benjamin Peirce, LAA, § 15, p. 5." title="" /></p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/5687#5687 Answer by Ian Morris for What's your favorite equation, formula, identity or inequality? Ian Morris 2009-11-16T11:05:54Z 2009-11-16T11:05:54Z <p>It has to be the ergodic theorem, $$\frac{1}{n}\sum_{k=0}^{n-1}f(T^kx) \to \int f\:d\mu,\;\;\mu\text{-a.e.}\;x,$$ the central principle which holds together pretty much my entire research existence.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/5690#5690 Answer by sushil for What's your favorite equation, formula, identity or inequality? sushil 2009-11-16T11:51:53Z 2009-11-16T11:51:53Z <p>polynomially convex hull of K = plurisubharmonic hull of K , where K is compact subset of C^n. For n>1, the equality is very interesting.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/8217#8217 Answer by Federico Ramponi for What's your favorite equation, formula, identity or inequality? Federico Ramponi 2009-12-08T17:47:50Z 2009-12-08T17:47:50Z <p>$V - E + F = 2$</p> <p>Euler's characteristic for connected planar graphs.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/9655#9655 Answer by Jonas Meyer for What's your favorite equation, formula, identity or inequality? Jonas Meyer 2009-12-24T04:25:31Z 2009-12-24T05:20:42Z <p>$2^n>n$ </p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/11772#11772 Answer by norondion for What's your favorite equation, formula, identity or inequality? norondion 2010-01-14T19:15:29Z 2010-07-15T11:18:21Z <p>The isogeny theorem: $\mathrm{Hom}_K(A,A')$ </p> <p>$= \mathrm{Hom}_{G_K}(T_\ell(A),T_\ell(A'))$</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/11780#11780 Answer by Carsten Schultz for What's your favorite equation, formula, identity or inequality? Carsten Schultz 2010-01-14T21:41:25Z 2010-01-14T21:41:25Z <p>Gauss-Bonnet, even though I am not a geometer.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/11782#11782 Answer by Alexander Noll for What's your favorite equation, formula, identity or inequality? Alexander Noll 2010-01-14T21:52:09Z 2010-01-14T21:52:09Z <p>$D_A\star F = 0$</p> <p>Yang-Mills</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/11809#11809 Answer by François G. Dorais for What's your favorite equation, formula, identity or inequality? François G. Dorais 2010-01-15T01:18:45Z 2010-01-15T01:18:45Z <p>I always thought this one was really funny: $1 = 0!$</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/11894#11894 Answer by luc_martineau_luc for What's your favorite equation, formula, identity or inequality? luc_martineau_luc 2010-01-15T18:24:29Z 2010-01-15T18:24:29Z <p>I'm surprised that nobody said</p> <p>$e=mc^2$</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/14650#14650 Answer by Yaakov Baruch for What's your favorite equation, formula, identity or inequality? Yaakov Baruch 2010-02-08T14:14:11Z 2010-02-08T14:14:11Z <p>Trivial as this is, it has amazed me for decades:</p> <p>$(1+2+3+...+n)^2=(1^3+2^3+3^3+...+n^3)$</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/17112#17112 Answer by Liran Shaul for What's your favorite equation, formula, identity or inequality? Liran Shaul 2010-03-04T17:19:02Z 2010-03-04T17:19:02Z <p>Lately, I really like the Greenlees-May duality: $RHom_A(R\Gamma_{\mathfrak{a}}M,N) \cong RHom_A(M,L\Lambda_{\mathfrak{a}}N)$ which holds for any pair of complexes over a noetherian ring.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/17139#17139 Answer by Will Jagy for What's your favorite equation, formula, identity or inequality? Will Jagy 2010-03-04T22:03:11Z 2010-03-05T02:21:39Z <p>I just cannot get this thing to make the 2 by 2 matrices of letters I want. Wait, fixed it myself. There is a thread in Meta about Latex/jsMath inconsistencies, one known problem is backslash being interpreted as an escape. So where I intended double backslash I just put three backslashes and that works for now. If it fails later I will switch to four or five backslashes.</p> <p>Given a square matrix $M \in SO_n$ decomposed as illustrated with square blocks $A,D$ and rectangular blocks $B,C,$</p> <p>$$M = \left( \begin{array}{cc} A &amp; B \\<br> C &amp; D \end{array} \right) ,$$</p> <p>then $\det A = \det D.$</p> <p>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.</p> <p><a href="http://en.wikipedia.org/wiki/Hodge_duality" rel="nofollow">http://en.wikipedia.org/wiki/Hodge_duality</a></p> <p><a href="http://en.wikipedia.org/wiki/Poincar%C3%A9_duality" rel="nofollow">http://en.wikipedia.org/wiki/Poincar%C3%A9_duality</a></p> <p>But the proof is a single line:</p> <p>$$\left( \begin{array}{cc} A &amp; B \\ 0 &amp; I \end{array} \right) \left( \begin{array}{cc} A^t &amp; C^t \\ B^t &amp; D^t \end{array} \right) = \left( \begin{array}{cc} I &amp; 0 \\ B^t &amp; D^t \end{array} \right).$$</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/17150#17150 Answer by Justin Curry for What's your favorite equation, formula, identity or inequality? Justin Curry 2010-03-05T01:34:11Z 2010-03-05T01:34:11Z <p>I learned Quantum Mechanics and Linear Algebra in tandem, so Schrodinger's linear time-independent equation has always had a special place in my heart. It shows that eigenvalues and eigenvectors are fundamental to our description of atomic physics. Also treating observables as operators was a great conceptual revolution.</p> <blockquote> <p>$H\psi=E\psi$</p> </blockquote> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/17151#17151 Answer by Andrea Ferretti for What's your favorite equation, formula, identity or inequality? Andrea Ferretti 2010-03-05T01:37:21Z 2010-03-05T01:37:21Z <p>Well, of course my favorite is Stokes theorem (it used to be the background of my mobile in the old days where you still manually designed monochromatic backgrounds pixel by pixel), but that is already suggested. And so are many others. So I'll go for Kontsevich formula for the number $N_d$ of rational curves through $3d-1$ generic points in the plane:</p> <p>$N_d + \sum_{\stackrel{d_A, d_B \geq 1}{d_A + d_B = d}} \binom{3d - 4}{3 d_A - 1} N_{d_A} N_{d_B} d_A^3 d_B = \sum_{\stackrel{d_A, d_B \geq 1}{d_A + d_B = d}} \binom{3d - 4}{3 d_A - 2} N_{d_A} N_{d_B} d_A^2 d_B^2$</p> <p>Although I admit this looks ugly until you see the proof. Then it becomes so neat!</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/22516#22516 Answer by Sunni for What's your favorite equation, formula, identity or inequality? Sunni 2010-04-25T17:25:50Z 2010-04-25T17:25:50Z <p>Ky Fan's inequality seems rather beautiful. The most beatiful proof can be found here <a href="http://files.ele-math.com/articles/jmi-01-07.pdf" rel="nofollow">http://files.ele-math.com/articles/jmi-01-07.pdf</a></p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/22525#22525 Answer by Harry Altman for What's your favorite equation, formula, identity or inequality? Harry Altman 2010-04-25T19:09:12Z 2010-04-25T19:09:12Z <p>I think this fits the original question's request for something nice-looking: $\binom{2n}{n}=(-4)^n\binom{-1/2}{n}$</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/26855#26855 Answer by hypercube for What's your favorite equation, formula, identity or inequality? hypercube 2010-06-02T19:22:27Z 2010-06-02T19:22:27Z <p>$\sum_{i=1}^m \sum_{j=1}^n a_{ij} = \sum_{j=1}^n \sum_{i=1}^m a_{ij}$</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/26858#26858 Answer by Lucas K. for What's your favorite equation, formula, identity or inequality? Lucas K. 2010-06-02T19:47:53Z 2010-06-02T19:47:53Z <p>Bayes equations:</p> <p>P(A|B) = P(A∩B)/P(B)</p> <p>It is the basis of conditional probability.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/31960#31960 Answer by Jesse Madnick for What's your favorite equation, formula, identity or inequality? Jesse Madnick 2010-07-15T05:00:55Z 2010-07-15T05:00:55Z <p>The Gauss Formula from Riemannian geometry:</p> <p>$\overline{\nabla}_XY = \nabla_XY + \text{II}(X,Y)$</p> <p>It may just be a decomposition into tangential and normal parts, but I find it very aesthetically pleasing. (It's also not completely immediate that the tangential part of the ambient connection should actually be the intrinsic connection.)</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/31973#31973 Answer by Mark Schwarzmann for What's your favorite equation, formula, identity or inequality? Mark Schwarzmann 2010-07-15T07:29:07Z 2010-07-15T07:29:07Z <p>The Spectral theorem for normal operators on a Hilbert space:</p> <p>$T = \int_{\sigma (T)} \lambda dP(\lambda)$</p> <p>where $\sigma (T)$ is the spectrum of $T$ and $P$ is a regular projection-valued measure supported on $\sigma (T)$.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/31975#31975 Answer by muad for What's your favorite equation, formula, identity or inequality? muad 2010-07-15T08:01:47Z 2010-07-15T08:01:47Z <p>$196884 = 196883 + 1$</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/31979#31979 Answer by Bo Peng for What's your favorite equation, formula, identity or inequality? Bo Peng 2010-07-15T08:51:57Z 2010-07-15T08:51:57Z <p>$\prod_{n=1}^{\infty} (1-x^n) = \sum_{k=-\infty}^{\infty} (-1)^k x^{k(3k-1)/2}$</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/31983#31983 Answer by Hans Lundmark for What's your favorite equation, formula, identity or inequality? Hans Lundmark 2010-07-15T09:10:49Z 2010-07-15T09:10:49Z <p>I have a soft spot for Heine's formula from the theory of orthogonal polynomials (since the proof is such a pretty calculation):</p> <p>If $\mu$ is a measure with finite moments $\beta_k=\int x^k d\mu(x)$, then</p> <p>$$\det(\beta_{i+j})_{i,j=0,\ldots,k-1} = \frac{1}{k!} \int \cdots \int \Delta(x_1,\ldots,x_k)^2 d\mu(x_1) \cdots d\mu(x_k)$$</p> <p>where $\Delta$ is the Vandermonde determinant.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/31986#31986 Answer by David Corwin for What's your favorite equation, formula, identity or inequality? David Corwin 2010-07-15T10:57:13Z 2010-07-15T10:57:13Z <p>How about $\displaystyle \sigma_7(n)=\sigma_3(n)+120\sum_{k=1}^{n-1} \sigma_3(k) \sigma_3(n-k)$? This is an utterly shocking result, and the only known proof uses complex analysis.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/32001#32001 Answer by Jeff Strom for What's your favorite equation, formula, identity or inequality? Jeff Strom 2010-07-15T14:01:03Z 2010-07-15T14:01:03Z <p>I'm a fan of $\Omega SU \simeq BU$.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/32002#32002 Answer by Robert Bell for What's your favorite equation, formula, identity or inequality? Robert Bell 2010-07-15T14:02:45Z 2010-07-15T14:02:45Z <p>The braid relation is probably my favorite equation, algebraically capturing the Reidemeister III move as $x y x = y x y$. Although to a younger person, I still find that suggesting that 5 is not prime is reliably charming revelation: $5 = (2 + i)(2 - i)$.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/32004#32004 Answer by Pooja for What's your favorite equation, formula, identity or inequality? Pooja 2010-07-15T14:12:35Z 2010-08-20T16:30:15Z <p>$\sin^2 A + \cos^2 A = 1$ </p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/32015#32015 Answer by Steve Flammia for What's your favorite equation, formula, identity or inequality? Steve Flammia 2010-07-15T15:27:43Z 2010-07-15T15:27:43Z <p>The Euler-Lagrange equations, $$\frac{\partial L}{\partial q_j} = \frac{d}{dt}\frac{\partial L}{\partial \dot{q}_j}$$</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/32018#32018 Answer by Blue for What's your favorite equation, formula, identity or inequality? Blue 2010-07-15T15:37:38Z 2010-07-15T15:37:38Z <p>The Pythagorean Theorem for Right-Corner Tetrahedra[*]:</p> <blockquote> <p>Euclidean: $A^2 + B^2 + C^2 = D^2$<br /><br /> Hyperbolic: $\cos\frac{A}{2} \cos\frac{B}{2} \cos\frac{C}{2} \; - \; \sin\frac{A}{2} \sin\frac{B}{2} \sin\frac{C}{2} = \cos\frac{D}{2}$<br /><br /> Spherical: $\cos\frac{A}{2} \cos\frac{B}{2} \cos\frac{C}{2} \; + \; \sin\frac{A}{2} \sin\frac{B}{2} \sin\frac{C}{2} = \cos\frac{D}{2}$</p> </blockquote> <p>where $A$, $B$, $C$ are the areas of the "leg-faces" and $D$ is the area of the "hypotenuse-face".</p> <p>For right-corner simplices in higher Euclidean dimensions, we have that the sum of the squares of the <i>content</i> of leg-simplices equals the square of the <i>content</i> of the hypotenuse-simplex. (I don't happen to know the non-Euclidean counterparts of this generalization. Perhaps this makes for a good MO question!)</p> <p>As generalizations of the Pythagorean Theorem for Triangles, I always found these (Euclidean) results to be more satisfying than the diagonal-of-a-box/distance formulas: instead of dealing only with segments, we have that, as the dimension of the ambient space goes up, so does the dimension of the objects involved in the relations.</p> <p>[*] Edges meeting at the "right corner" are mutually orthogonal.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/32025#32025 Answer by David Corwin for What's your favorite equation, formula, identity or inequality? David Corwin 2010-07-15T16:03:29Z 2010-07-15T16:03:29Z <p>The formula $\displaystyle \int_{-\infty}^{\infty} \frac{\cos(x)}{x^2+1} dx = \frac{\pi}{e}$. It is astounding in that we can retrieve $e$ from a formula involving the cosine. It is not surprising if we know the formula $\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$, yet this integral is of a purely real-valued function. It shows how complex analysis actually underlies even the real numbers.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/33819#33819 Answer by basil for What's your favorite equation, formula, identity or inequality? basil 2010-07-29T18:00:40Z 2010-07-29T18:00:40Z <p>d/dx (e^x) =e^x</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/36197#36197 Answer by gspr for What's your favorite equation, formula, identity or inequality? gspr 2010-08-20T15:23:47Z 2010-08-20T15:23:47Z <p>There are many beautiful equations above, so I'll be a bit different and add something nonsensical. Namely</p> <p>$$\langle f\rangle = \frac{\int_\ast f(\phi)e^{\frac{\mathrm{i}}{\hbar}\int_M\mathcal{L}(\phi)}\mathcal{D}\phi}{\int_\ast e^{\frac{\mathrm{i}}{\hbar}\int_M\mathcal{L}(\phi)}\mathcal{D}\phi}.$$</p> <p>Just insert your favourite spacetime manifold $M$ and the classical Lagrangian $\mathcal{L}$ of your choice, and you get to learn the expectation value of any physical observable $f$... as soon as you figure out what the hell $\ast$ and $\mathcal{D}\phi$ are, that is.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/36206#36206 Answer by J. M. for What's your favorite equation, formula, identity or inequality? J. M. 2010-08-20T17:25:43Z 2010-08-20T17:25:43Z <p>With the stuff I've seen in the literature of sequence transformations, I've started to love the formulae for Aitken's Δ² process:</p> <p>$S_n^{\prime}=S_{n+1}-\frac{(\Delta S_n)^2}{\Delta^2 S_n}$</p> <p>and its generalization the Wynn ε algorithm:</p> <p>$\varepsilon_{k+1}^{(n)}=\varepsilon_{k-1}^{(n+1)}+\frac1{\varepsilon_{k}^{(n+1)}-\varepsilon_{k}^{(n)}}$</p> <p>for the latter one especially because it is nicely represented as a lozenge diagram:</p> <p><img src="http://i.imgur.com/vCSQK.png" alt="Wynn epsilon"></p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/36250#36250 Answer by Richard Stanley for What's your favorite equation, formula, identity or inequality? Richard Stanley 2010-08-21T02:39:15Z 2010-08-22T16:14:57Z <p>$$\frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left| \frac{\tan<br> t+\sqrt{7}}{\tan t-\sqrt{7}}\right|\ dt = \sum_{n\geq<br> 1} \left(\frac n7\right)\frac{1}{n^2},$$ 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, <em>Mathematics by Experiment: Plausible Reasoning in the 21st Century</em>, A K Peters, Natick, MA, 2004 (pages 90-91).</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/36255#36255 Answer by Leandro for What's your favorite equation, formula, identity or inequality? Leandro 2010-08-21T04:28:55Z 2010-08-21T04:28:55Z <p>Cauchy integral formula $$f(z)=\frac{1}{2\pi i}\int_{\gamma}\frac{f(w)}{w-z} dw$$</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/36264#36264 Answer by matthias beck for What's your favorite equation, formula, identity or inequality? matthias beck 2010-08-21T06:43:50Z 2010-08-21T06:43:50Z <p>Pick's theorem $A = I + \frac 1 2 B - 1$, where $A$, $I$, and $B$ are the area, number of interior integer points, and number of boundary integer points, respectively, of a polygon with vertices on the integer lattice. Picks identity is fascinating because it computes a continuous quantity completely discretely. (Of course, this is not quite correct, since we have quite a discrete requirement about the vertices of the polygon.) Also, the "1" is not an accident, but the Euler characteristic of the polygon (and so there are various natural extensions of Pick's theorem).</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/36289#36289 Answer by Jay Kangel for What's your favorite equation, formula, identity or inequality? Jay Kangel 2010-08-21T16:05:00Z 2010-08-21T16:05:00Z <p>My favorite equation is</p> <p>$$\frac{16}{64} = \frac{1}{4}.$$</p> <p>What makes this equation interesting is that canceling the $6$'s yields the correct answer. I realized this in, perhaps, third grade. This was the great rebellion of my youth. Sometime later I generalized this to finding solutions to</p> <p>$$\frac{pa +b}{pb + c} = \frac{a}{c}.$$</p> <p>where $p$ is an integer greater than $1$. We require that $a$, $b$, and $c$ are integers between $1$ and $p - 1$, inclusive. Say a solution is trivial if $a = b = c$. Then $p$ is prime if and only if all solutions are trivial. On can also prove that if $p$ is an even integer greater than $2$ then $p - 1$ is prime if and only if every nontrivial solution $(a,b,c)$ has $b = p - 1$.</p> <p>The key to these results is that if $(a, b, c)$ is a nontrivial solution then the greatest common divisor of $c$ and $p$ is greater than $1$ and the greatest common divisor of $b$ and $p - 1$ is also greater than $1$.</p> <p>Two other interesting facts are (i) if $(a, b, c)$ is a nontrivial solution then $2a \leq c &lt; b$ and (2) the number of nontrivial solutions is odd if and only if $p$ is the square of an even integer. To prove the latter item it is useful to note that if $(a, b, c)$ is a nontrivial solution then so is $(b - c, b, b - a)$.</p> <p>For what it is worth I call this demented division.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/36291#36291 Answer by Simon Rose for What's your favorite equation, formula, identity or inequality? Simon Rose 2010-08-21T16:17:32Z 2010-08-21T16:17:32Z <p>One that I just learned recently is $$(1 + q + q^3 + q^6 + q^{10} + q^{15} + \cdots)^4 = \sum_{k=0}^\infty \sigma(2k+1)q^k$$ which states that the number of ways of writing an integer $k$ as a sum of exactly 4 triangular numbers (paying attention to ordering) is equal to the sum of divisors of $2k+1$.</p> <p>If that isn't cool and surprising, I don't know what is.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/36295#36295 Answer by Yujia Qiu for What's your favorite equation, formula, identity or inequality? Yujia Qiu 2010-08-21T16:32:05Z 2010-08-21T16:32:05Z <p>I like Riemann-Roch the most!!!</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/36304#36304 Answer by David Nacin for What's your favorite equation, formula, identity or inequality? David Nacin 2010-08-21T17:50:47Z 2010-08-21T17:50:47Z <p>$\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2} \frac{q-1}{2}}$.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/36305#36305 Answer by Vamsi for What's your favorite equation, formula, identity or inequality? Vamsi 2010-08-21T18:04:40Z 2010-08-21T18:04:40Z <p>$(A-\lambda _1) (A-\lambda _2) \ldots = 0$, the Cayley-Hamilton theorem.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/36308#36308 Answer by Daniel Miller for What's your favorite equation, formula, identity or inequality? Daniel Miller 2010-08-21T18:23:36Z 2010-08-21T18:23:36Z <p>It may be trivial, but I've always found </p> <p>$\sqrt{\pi}=\int_{-\infty}^{\infty}e^{-x^{2}}dx$ </p> <p>to be particularly beautiful.</p> http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality/36334#36334 Answer by Fernando for What's your favorite equation, formula, identity or inequality? Fernando 2010-08-21T22:44:08Z 2010-08-22T01:39:27Z <p>$e=lim_{n\to\infty}\sqrt[p_n]{\prod_{k=1}^np_n}$</p> <p>as seen at <a href="http://gaussianos.com/%25C2%25BFque-tiene-que-ver-el-numero-e-con-los-numeros-primos/%20%22Gaussianos%22" rel="nofollow">Gaussianos</a></p> <p>$(\prod_{k=1}^np_n=p_n$# which is the <a href="http://en.wikipedia.org/wiki/Primorial%20%22primorial%22" rel="nofollow">primorial</a> of the nth prime number $p_n)$</p>