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?
closed as no longer relevant by Robin Chapman, Akhil Mathew, Yemon Choi, Qiaochu Yuan, Pete L. Clark Aug 22 '10 at 9:00
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$\begingroup$ There are numerous significant formula and laws from our mathematical legacy. It should be noticed while we are now enjoying them freely, there are still quite a lot of basic problems that we are not able to solve using the techniques we know. $\endgroup$ – Sunni Mar 4 '10 at 1:20

2$\begingroup$ Voting to close. People are at the repeatingotherpeople'sanswers stage now. $\endgroup$ – Qiaochu Yuan Aug 21 '10 at 18:23

3$\begingroup$ The question has been closed as no longer relevant. It had a long and healthy life, but the large number of answers has become unwieldy. If the question had been asked more recently, it would probably have been closed sooner as being "overly broad". I encourage people who are interested in following up issues raised in the question or the answers with further questions. Please be specific! $\endgroup$ – Pete L. Clark Aug 22 '10 at 9:02

1$\begingroup$ Sadly my favourite $\sum \frac{1}{n^2 +a^2} = \frac{\pi}{a} cth(\pi a)$ wasn't listed $\endgroup$ – Ostap Chervak May 1 '11 at 16:57

$\begingroup$ $$Rf_*R\hbox{Hom}(F,f^!G)\approx R\hbox{Hom}(Rf_!F,G)$$ $\endgroup$ – Steven Landsburg Oct 22 '16 at 22:01
With the stuff I've seen in the literature of sequence transformations, I've started to love the formulae for Aitken's Δ² process:
$S_n^{\prime}=S_{n+1}\frac{(\Delta S_n)^2}{\Delta^2 S_n}$
and its generalization the Wynn ε algorithm:
$\varepsilon_{k+1}^{(n)}=\varepsilon_{k1}^{(n+1)}+\frac1{\varepsilon_{k}^{(n+1)}\varepsilon_{k}^{(n)}}$
for the latter one especially because it is nicely represented as a lozenge diagram:
I'm surprised that nobody said
$e=mc^2$

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13$\begingroup$ I love the "Einstein meets Pythagoras" version: $E=m(a^2+b^2)$ $\endgroup$ – Federico Poloni Aug 20 '10 at 20:46