# What are the most ugly equations in mathematics? [closed]

I feel like most people focus on only the most simple, elegant equations in math, but are there useful ugly ones? When I say ugly, I mean extremely long and generally useless? I can't exactly find an example, but can any of you can?

P.S. If you can find a useful ugly equation, that would be good too!

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## closed as primarily opinion-based by Andres Caicedo, tetori, Julian Kuelshammer, Dirk, Ayman HouriehOct 26 '13 at 11:39

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

The ones you do not understand. You might find many here dlmf.nist.gov –  Amzoti Oct 26 '13 at 1:11
Being beautiful has a limit; being ugly has no limit. –  Shushan Wen Oct 26 '13 at 1:14
An example of a useless ugly one. –  anon Oct 26 '13 at 1:42
Can we please take this question off hold?? –  Riemann1337 Oct 28 '13 at 1:12
This must be taken off hold. I can't see how it is a opinion based question. –  Sufyan Naeem May 12 at 19:59

The Lagrangian describing all physical processes is pretty ugly :) It is given by:

$$L=-\frac{1}{2}\partial_{\nu}g^{a}_{\mu}\partial_{\nu}g^{a}_{\mu} -g_{s}f^{abc}\partial_{\mu}g^{a}_{\nu}g^{b}_{\mu}g^{c}_{\nu} -\frac{1}{4}g^{2}_{s}f^{abc}f^{ade}g^{b}_{\mu}g^{c}_{\nu}g^{d}_{\mu}g^{e}_{\nu} +\frac{1}{2}ig^{2}_{s}(\bar{q}^{\sigma}_{i}\gamma^{\mu}q^{\sigma}_{j})g^{a}_{\mu} +\bar{G}^{a}\partial^{2}G^{a}+g_{s}f^{abc}\partial_{\mu}\bar{G}^{a}G^{b}g^{c}_{\mu} -\partial_{\nu}W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-M^{2}W^{+}_{\mu}W^{-}_{\mu} -\frac{1}{2}\partial_{\nu}Z^{0}_{\mu}\partial_{\nu}Z^{0}_{\mu}-\frac{1}{2c^{2}_{w}} M^{2}Z^{0}_{\mu}Z^{0}_{\mu} -\frac{1}{2}\partial_{\mu}A_{\nu}\partial_{\mu}A_{\nu} -\frac{1}{2}\partial_{\mu}H\partial_{\mu}H-\frac{1}{2}m^{2}_{h}H^{2} -\partial_{\mu}\phi^{+}\partial_{\mu}\phi^{-}-M^{2}\phi^{+}\phi^{-} -\frac{1}{2}\partial_{\mu}\phi^{0}\partial_{\mu}\phi^{0}-\frac{1}{2c^{2}_{w}}M\phi^{0}\phi^{0} -\beta_{h}[\frac{2M^{2}}{g^{2}}+\frac{2M}{g}H+\frac{1}{2}(H^{2}+\phi^{0}\phi^{0}+2\phi^{+}\phi^{-%%@ })]+\frac{2M^{4}}{g^{2}}\alpha_{h} -igc_{w}[\partial_{\nu}Z^{0}_{\mu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu}) -Z^{0}_{\nu}(W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\mu}\partial_{\nu}W^{+}_{\mu}) +Z^{0}_{\mu}(W^{+}_{\nu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\nu}\partial_{\nu}W^{+}_{\mu})] -igs_{w}[\partial_{\nu}A_{\mu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu}) -A_{\nu}(W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\mu}\partial_{\nu}W^{+}_{\mu}) +A_{\mu}(W^{+}_{\nu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\nu}\partial_{\nu}W^{+}_{\mu})] -\frac{1}{2}g^{2}W^{+}_{\mu}W^{-}_{\mu}W^{+}_{\nu}W^{-}_{\nu}+\frac{1}{2}g^{2} W^{+}_{\mu}W^{-}_{\nu}W^{+}_{\mu}W^{-}_{\nu} +g^2c^{2}_{w}(Z^{0}_{\mu}W^{+}_{\mu}Z^{0}_{\nu}W^{-}_{\nu}-Z^{0}_{\mu}Z^{0}_{\mu}W^{+}_{\nu} W^{-}_{\nu}) +g^2s^{2}_{w}(A_{\mu}W^{+}_{\mu}A_{\nu}W^{-}_{\nu}-A_{\mu}A_{\mu}W^{+}_{\nu} W^{-}_{\nu}) +g^{2}s_{w}c_{w}[A_{\mu}Z^{0}_{\nu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu})-%%@ 2A_{\mu}Z^{0}_{\mu}W^{+}_{\nu}W^{-}_{\nu}] -g\alpha[H^3+H\phi^{0}\phi^{0}+2H\phi^{+}\phi^{-}] -\frac{1}{8}g^{2}\alpha_{h}[H^4+(\phi^{0})^{4}+4(\phi^{+}\phi^{-})^{2}+4(\phi^{0})^{2} \phi^{+}\phi^{-}+4H^{2}\phi^{+}\phi^{-}+2(\phi^{0})^{2}H^{2}] -gMW^{+}_{\mu}W^{-}_{\mu}H-\frac{1}{2}g\frac{M}{c^{2}_{w}}Z^{0}_{\mu}Z^{0}_{\mu}H -\frac{1}{2}ig[W^{+}_{\mu}(\phi^{0}\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}\phi^{0}) -W^{-}_{\mu}(\phi^{0}\partial_{\mu}\phi^{+}-\phi^{+}\partial_{\mu}\phi^{0})] +\frac{1}{2}g[W^{+}_{\mu}(H\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}H) -W^{-}_{\mu}(H\partial_{\mu}\phi^{+}-\phi^{+}\partial_{\mu}H)] +\frac{1}{2}g\frac{1}{c_{w}}(Z^{0}_{\mu}(H\partial_{\mu}\phi^{0}-\phi^{0}\partial_{\mu}H) -ig\frac{s^{2}_{w}}{c_{w}}MZ^{0}_{\mu}(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+}) +igs_{w}MA_{\mu}(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+}) -ig\frac{1-2c^{2}_{w}}{2c_{w}}Z^{0}_{\mu}(\phi^{+}\partial_{\mu}\phi^{-}-\phi^{-%%@ }\partial_{\mu}\phi^{+}) +igs_{w}A_{\mu}(\phi^{+}\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}\phi^{+}) -\frac{1}{4}g^{2}W^{+}_{\mu}W^{-}_{\mu}[H^{2}+(\phi^{0})^{2}+2\phi^{+}\phi^{-}] -\frac{1}{4}g^{2}\frac{1}{c^{2}_{w}}Z^{0}_{\mu}Z^{0}_{\mu}[H^{2}+(\phi^{0})^{2}+2(2s^{2}_{w}-%%@ 1)^{2}\phi^{+}\phi^{-}] -\frac{1}{2}g^{2}\frac{s^{2}_{w}}{c_{w}}Z^{0}_{\mu}\phi^{0}(W^{+}_{\mu}\phi^{-}+W^{-%%@ }_{\mu}\phi^{+}) -\frac{1}{2}ig^{2}\frac{s^{2}_{w}}{c_{w}}Z^{0}_{\mu}H(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+}) +\frac{1}{2}g^{2}s_{w}A_{\mu}\phi^{0}(W^{+}_{\mu}\phi^{-}+W^{-}_{\mu}\phi^{+}) +\frac{1}{2}ig^{2}s_{w}A_{\mu}H(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+}) -g^{2}\frac{s_{w}}{c_{w}}(2c^{2}_{w}-1)Z^{0}_{\mu}A_{\mu}\phi^{+}\phi^{-}-%%@ g^{1}s^{2}_{w}A_{\mu}A_{\mu}\phi^{+}\phi^{-} -\bar{e}^{\lambda}(\gamma\partial+m^{\lambda}_{e})e^{\lambda} -\bar{\nu}^{\lambda}\gamma\partial\nu^{\lambda} -\bar{u}^{\lambda}_{j}(\gamma\partial+m^{\lambda}_{u})u^{\lambda}_{j} -\bar{d}^{\lambda}_{j}(\gamma\partial+m^{\lambda}_{d})d^{\lambda}_{j} +igs_{w}A_{\mu}[-(\bar{e}^{\lambda}\gamma^{\mu} e^{\lambda})+\frac{2}{3}(\bar{u}^{\lambda}_{j}\gamma^{\mu} %%@ u^{\lambda}_{j})-\frac{1}{3}(\bar{d}^{\lambda}_{j}\gamma^{\mu} d^{\lambda}_{j})] +\frac{ig}{4c_{w}}Z^{0}_{\mu} [(\bar{\nu}^{\lambda}\gamma^{\mu}(1+\gamma^{5})\nu^{\lambda})+ (\bar{e}^{\lambda}\gamma^{\mu}(4s^{2}_{w}-1-\gamma^{5})e^{\lambda})+ (\bar{u}^{\lambda}_{j}\gamma^{\mu}(\frac{4}{3}s^{2}_{w}-1-\gamma^{5})u^{\lambda}_{j})+ (\bar{d}^{\lambda}_{j}\gamma^{\mu}(1-\frac{8}{3}s^{2}_{w}-\gamma^{5})d^{\lambda}_{j})] +\frac{ig}{2\sqrt{2}}W^{+}_{\mu}[(\bar{\nu}^{\lambda}\gamma^{\mu}(1+\gamma^{5})e^{\lambda}) +(\bar{u}^{\lambda}_{j}\gamma^{\mu}(1+\gamma^{5})C_{\lambda\kappa}d^{\kappa}_{j})] +\frac{ig}{2\sqrt{2}}W^{-}_{\mu}[(\bar{e}^{\lambda}\gamma^{\mu}(1+\gamma^{5})\nu^{\lambda}) +(\bar{d}^{\kappa}_{j}C^{\dagger}_{\lambda\kappa}\gamma^{\mu}(1+\gamma^{5})u^{\lambda}_{j})] +\frac{ig}{2\sqrt{2}}\frac{m^{\lambda}_{e}}{M} [-\phi^{+}(\bar{\nu}^{\lambda}(1-\gamma^{5})e^{\lambda}) +\phi^{-}(\bar{e}^{\lambda}(1+\gamma^{5})\nu^{\lambda})] -\frac{g}{2}\frac{m^{\lambda}_{e}}{M}[H(\bar{e}^{\lambda}e^{\lambda}) +i\phi^{0}(\bar{e}^{\lambda}\gamma^{5}e^{\lambda})] +\frac{ig}{2M\sqrt{2}}\phi^{+} [-m^{\kappa}_{d}(\bar{u}^{\lambda}_{j}C_{\lambda\kappa}(1-\gamma^{5})d^{\kappa}_{j}) +m^{\lambda}_{u}(\bar{u}^{\lambda}_{j}C_{\lambda\kappa}(1+\gamma^{5})d^{\kappa}_{j}] +\frac{ig}{2M\sqrt{2}}\phi^{-} [m^{\lambda}_{d}(\bar{d}^{\lambda}_{j}C^{\dagger}_{\lambda\kappa}(1+\gamma^{5})u^{\kappa}_{j}) -m^{\kappa}_{u}(\bar{d}^{\lambda}_{j}C^{\dagger}_{\lambda\kappa}(1-\gamma^{5})u^{\kappa}_{j}] -\frac{g}{2}\frac{m^{\lambda}_{u}}{M}H(\bar{u}^{\lambda}_{j}u^{\lambda}_{j}) -\frac{g}{2}\frac{m^{\lambda}_{d}}{M}H(\bar{d}^{\lambda}_{j}d^{\lambda}_{j}) +\frac{ig}{2}\frac{m^{\lambda}_{u}}{M}\phi^{0}(\bar{u}^{\lambda}_{j}\gamma^{5}u^{\lambda}_{j}) -\frac{ig}{2}\frac{m^{\lambda}_{d}}{M}\phi^{0}(\bar{d}^{\lambda}_{j}\gamma^{5}d^{\lambda}_{j}) +\bar{X}^{+}(\partial^{2}-M^{2})X^{+}+\bar{X}^{-}(\partial^{2}-M^{2})X^{-} +\bar{X}^{0}(\partial^{2}-\frac{M^{2}}{c^{2}_{w}})X^{0}+\bar{Y}\partial^{2}Y +igc_{w}W^{+}_{\mu}(\partial_{\mu}\bar{X}^{0}X^{-}-\partial_{\mu}\bar{X}^{+}X^{0}) +igs_{w}W^{+}_{\mu}(\partial_{\mu}\bar{Y}X^{-}-\partial_{\mu}\bar{X}^{+}Y) +igc_{w}W^{-}_{\mu}(\partial_{\mu}\bar{X}^{-}X^{0}-\partial_{\mu}\bar{X}^{0}X^{+}) +igs_{w}W^{-}_{\mu}(\partial_{\mu}\bar{X}^{-}Y-\partial_{\mu}\bar{Y}X^{+}) +igc_{w}Z^{0}_{\mu}(\partial_{\mu}\bar{X}^{+}X^{+}-\partial_{\mu}\bar{X}^{-}X^{-}) +igs_{w}A_{\mu}(\partial_{\mu}\bar{X}^{+}X^{+}-\partial_{\mu}\bar{X}^{-}X^{-}) -\frac{1}{2}gM[\bar{X}^{+}X^{+}H+\bar{X}^{-}X^{-}H+\frac{1}{c^{2}_{w}}\bar{X}^{0}X^{0}H] +\frac{1-2c^{2}_{w}}{2c_{w}}igM[\bar{X}^{+}X^{0}\phi^{+}-\bar{X}^{-}X^{0}\phi^{-}] +\frac{1}{2c_{w}}igM[\bar{X}^{0}X^{-}\phi^{+}-\bar{X}^{0}X^{+}\phi^{-}] +igMs_{w}[\bar{X}^{0}X^{-}\phi^{+}-\bar{X}^{0}X^{+}\phi^{-}] +\frac{1}{2}igM[\bar{X}^{+}X^{+}\phi^{0}-\bar{X}^{-}X^{-}\phi^{0}]$$

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Here I was, wondering why this page was taking forever to load... –  Bruno Joyal Oct 26 '13 at 5:28
Good things come to those who wait :) –  Riemann1337 Oct 26 '13 at 5:29
You just made that equation up! lol jk You win! –  1233dfv Oct 26 '13 at 6:36
Another reason the Standard Model cannot be correct :-). –  copper.hat Oct 26 '13 at 7:09
Apologies to Andres Caicedo, tetori, Julian Kuelshammer, Dirk, Ayman Hourieh, but there is a correct answer to this question, and it's this one. –  Eric Stucky Nov 7 '13 at 13:22

I understand that this is very controversial, but I never liked (certainly very useful) Cardano's formula for the solution of the cubic equation: $$x^3+ax^2+bx+c=0$$ is solved by \begin{align} x=-\frac{a}{3}-\frac{\sqrt[3]{2}\left(-a^2+3b\right)}{3\sqrt[3]{-2a^3+9ab-27c+3\sqrt{3} \sqrt{4b^3-a^2b^2+4a^3c-18abc+27c^2}}}+\\ +\frac{\sqrt[3]{-2a^3+9ab-27c+3\sqrt{3} \sqrt{4b^3-a^2b^2+4a^3c-18abc+27c^2}}}{3\sqrt[3]{2}}. \end{align}

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And this is used to solve the quartic - that solution is even uglier. –  marty cohen Oct 26 '13 at 1:24
You can un-uglify Cardano's formula by expressing it in the form, $$x = \frac{1}{3}\left(-a+y_1^{1/3}+y_2^{1/3}\right)$$ where $y_1,y_2$ are the two roots of its Lagrange resolvent, $$y^2+(2a^3-9ab+27c)y+(a^2-3b)^3=0$$ Note that the constant term is a nice perfect cube . –  Tito Piezas III Oct 26 '13 at 2:05
The repeated plus sign makes it even uglier. –  Karl Kronenfeld Oct 26 '13 at 5:25
Oh, you mean the typo in O.L. post... –  Tito Piezas III Oct 26 '13 at 19:30
Wow! I never even knew that existed. –  Joao Apr 4 '14 at 7:04

I've always found something ugly about complicated formulas for the $n^{th}$ prime number involving multiple summations such as

$$p_n = 1+\sum_{m=1}^{2^n}\left[\sqrt[n]{n}\left(\sum_{x=1}^m\left[ \cos^2 \pi \frac{(x-1)!+1}{x}\right]\right)^{-1/n}\right]$$

where the square brackets denote the floor function. This is from Willans, The Mathematical Gazette, Vol. 48, No. 366, Dec., 1964. There are many other similar-looking ones. Other than for winning bets about whether there is a formula for the $n^{th}$ prime number, there isn't much use for these things.

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Just by looking at the formula one would never guess this'll compute the nth prime. –  Ramchandra Apte Oct 26 '13 at 6:58

Hardy and Ramanujan's formula for the partition function is i think something that should be mentioned
$$p(n)=\frac{1}{2 \sqrt{2}} \sum_{k=1}^v \sqrt{k}\, A_k(n)\, \frac{d}{dn} \exp \left({ \pi\sqrt{\frac23} \frac{\sqrt{n-\frac{1}{24}}}{k} } \right)$$
$$p(n)=\frac{1}{\pi \sqrt{2}} \sum_{k=1}^\infty \sqrt{k}\, A_k(n)\, \frac{d}{dn} \left({ \frac {1} {\sqrt{n-\frac{1}{24}}} \sinh \left[ {\frac{\pi}{k} \sqrt{\frac{2}{3}\left(n-\frac{1}{24}\right)}}\right] }\right)$$
How about the result that any measurable function $f:(X,\mathscr M)\to(\mathbb{R}_+,\mathscr B)$ can be approximated by an increasing sequence of simple functions from below, converging to $f$ pointwise? The formula defining those simple functions is very ugly, involving terms like $2^{2n}$, but the graphs of these functions are very intuitive. Also, this result is useful, as it can be used to prove the celebrated convergence theorems making it possible to interchange integrals and limits.