Root of quartic equation

The formula on is layered with Δ1, Δ2, Δ, and so on. Then, I spent an hour typing out the root-finding formula without the Δ version. It was really frustrating. The subsequent formulas were typed out by me using LaTeX, so don’t expect to double-click to enlarge them. I also want to apologize for my previous unintentional behavior of misleading people to get likes.

Let's take a close look at the exact root solution of the quartic equation x_{1}, x_{2}, x_{3}, x_{4}

The latex code of the exact root solution of quartic equation

First solution x_{1}

x_{1}=\frac{-b}{4a}-\frac{1}{2}\sqrt{\frac{b^{2}}{4a^{2}}-\frac{2c}{3a}+(\frac{\sqrt[3]{2}(c^{2}-3bd+12ae)}{3a\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}+\frac{\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}{3\sqrt[3]{2}a})} -\frac{1}{2}\sqrt{\frac{b^{2}}{4a^{2}}-\frac{4c}{3a}-(\frac{\sqrt[3]{2}(c^{2}-3bd+12ae)}{3a\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}+\frac{\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}{3\sqrt[3]{2}a}) -\frac{-\frac{b^{3}}{a^{3}}+\frac{4bc}{a^{2}}-\frac{8d}{a}}{4\sqrt{\frac{b^{2}}{4a^{2}}-\frac{2c}{3a}+(\frac{\sqrt[3]{2}(c^{2}-3bd+12ae)}{3a\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}+\frac{\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}{3\sqrt[3]{2}a})}}}

Second solution x_{2}

x_{2}=\frac{-b}{4a}-\frac{1}{2}\sqrt{\frac{b^{2}}{4a^{2}}-\frac{2c}{3a}+(\frac{\sqrt[3]{2}(c^{2}-3bd+12ae)}{3a\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}+\frac{\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}{3\sqrt[3]{2}a})} +\frac{1}{2}\sqrt{\frac{b^{2}}{4a^{2}}-\frac{4c}{3a}-(\frac{\sqrt[3]{2}(c^{2}-3bd+12ae)}{3a\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}+\frac{\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}{3\sqrt[3]{2}a}) -\frac{-\frac{b^{3}}{a^{3}}+\frac{4bc}{a^{2}}-\frac{8d}{a}}{4\sqrt{\frac{b^{2}}{4a^{2}}-\frac{2c}{3a}+(\frac{\sqrt[3]{2}(c^{2}-3bd+12ae)}{3a\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}+\frac{\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}{3\sqrt[3]{2}a})}}}

Third solution x_{3}

x_{3}=\frac{-b}{4a}-\frac{1}{2}\sqrt{\frac{b^{2}}{4a^{2}}-\frac{2c}{3a}+(\frac{\sqrt[3]{2}(c^{2}-3bd+12ae)}{3a\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}+\frac{\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}{3\sqrt[3]{2}a})} -\frac{1}{2}\sqrt{\frac{b^{2}}{4a^{2}}-\frac{4c}{3a}-(\frac{\sqrt[3]{2}(c^{2}-3bd+12ae)}{3a\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}+\frac{\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}{3\sqrt[3]{2}a}) +\frac{-\frac{b^{3}}{a^{3}}+\frac{4bc}{a^{2}}-\frac{8d}{a}}{4\sqrt{\frac{b^{2}}{4a^{2}}-\frac{2c}{3a}+(\frac{\sqrt[3]{2}(c^{2}-3bd+12ae)}{3a\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}+\frac{\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}{3\sqrt[3]{2}a})}}}

Fourth solution x_{4}

x_{4}=\frac{-b}{4a}-\frac{1}{2}\sqrt{\frac{b^{2}}{4a^{2}}-\frac{2c}{3a}+(\frac{\sqrt[3]{2}(c^{2}-3bd+12ae)}{3a\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}+\frac{\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}{3\sqrt[3]{2}a})} +\frac{1}{2}\sqrt{\frac{b^{2}}{4a^{2}}-\frac{4c}{3a}-(\frac{\sqrt[3]{2}(c^{2}-3bd+12ae)}{3a\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}+\frac{\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}{3\sqrt[3]{2}a}) +\frac{-\frac{b^{3}}{a^{3}}+\frac{4bc}{a^{2}}-\frac{8d}{a}}{4\sqrt{\frac{b^{2}}{4a^{2}}-\frac{2c}{3a}+(\frac{\sqrt[3]{2}(c^{2}-3bd+12ae)}{3a\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}+\frac{\sqrt[3]{(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)+\sqrt{-4(c^{2}-3bd+12ae)^{3}+(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}}}}{3\sqrt[3]{2}a})}}}

Related

• https://en.wikipedia.org/wiki/Quartic_equation

Borwein integral is very elegant for smalll value but all of a suddeny becomes very ugly form when it approach some tipping point.

The Borwein integral was first proposed by David Borwein and Jonathan Borwein in 2001 to illustrate that seemingly valid mathematical laws can suddenly fail at some point. This is an integral involving the sinc function. Alright, up to this point, do you think: Oh, what a great pattern! Then proudly began testing the next number. However, the slap in the face came so unexpectedly: rotten, unfinished... it's so shocking. Based on intuition, you may think that the law is like this, but the Borwein integral becomes a strange and long constant term.

Borwein integral takes very elegant mathmatical form for A_{1}, A_{2}, A_{3} all the way up to A_{7}, such as below

${\displaystyle {\begin{aligned}&A_{1}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}\,\mathrm {d} x=\pi /2\\[10pt]&A_{2}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\,\mathrm {d} x=\pi /2\\[10pt]&A_{3}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}{\frac {\sin(x/5)}{x/5}}\,\mathrm {d} x=\pi /2\end{aligned}}}$

However, it may suddenly takes a very long and troublesome form when the number is approaching A_{15}, which becomes a very ugly equation as below:

Latex Code of Borwein integral at A_{8}

${\displaystyle {\begin{aligned}A_{8}&=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\cdots {\frac {\sin(x/13)}{x/13}}{\frac {\sin(x/15)}{x/15}}\,\mathrm {d} x\\[10pt]&={\frac {467807924713440738696537864469}{935615849440640907310521750000}}\pi \\[10pt]&={\frac {\pi }{2}}-{\frac {6879714958723010531}{935615849440640907310521750000}}\pi \\[10pt]&\simeq {\frac {\pi }{2}}-2{,}310057\cdot 10^{-11}\ {\color {red}<}\ {\frac {\pi }{2}}\ .\end{aligned}}}$

{\displaystyle {\begin{aligned}A_{8}&=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\cdots {\frac {\sin(x/13)}{x/13}}{\frac {\sin(x/15)}{x/15}}\,\mathrm {d} x\\[10pt]&={\frac {467807924713440738696537864469}{935615849440640907310521750000}}\pi \\[10pt]&={\frac {\pi }{2}}-{\frac {6879714958723010531}{935615849440640907310521750000}}\pi \\[10pt]&\simeq {\frac {\pi }{2}}-2{,}310057\cdot 10^{-11}\ {\color {red}<}\ {\frac {\pi }{2}}\ .\end{aligned}}}

Related

• https://de.wikipedia.org/wiki/Borwein-Integral

Ramanujan Formula

Reasons of Being Regarded as Ugliest

The lowest scoring formula for ugliness is the Ramanujan formula in many survey results of the "most beautiful" and "ugliest" formulas. It can be found that, like ordinary people, most mathematicians also regard simplicity as an important attribute of the beauty of mathematics and think Ramanujan Craziness Formula are among the most ugliness.

$\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}$

Reasons of Being Regarded as Ugliest

Standard Model Lagrangian Full expanded form may takes a whole page to be displayed correctly. And there is even no well-known latex source code for that.

Do you need to see an ugly equation? There is nothing uglier than the Standard Model lagrangian form. Sorry, this is not a purely mathematical equation, but a physical equation. (Warning: This is intentionally written in coordinate form to make it uglier)

The standard model unifies three of the four fundamental forces: strong force, weak force, and electromagnetic force. Just missing the most difficult attraction to handle. Don't be fooled by the complexity of this formula, it actually only includes five parts.

The first part describes the interaction of gluons, which are medium particles with strong interaction forces, spin 1, and have 8 types.

The second part is very large and is used to describe the interactions between bosons, especially the W and Z bosons, which are medium particles for weak interactions with spin 1 and have three types. There are two types of W bosons, which have unit charges of+1 (W+) and -1 (W −), respectively. W+is the antiparticle of W -. The Z boson (Z0) is electrically neutral and is its own antiparticle.

The third part describes the interaction between elementary particles and weak forces, including the interaction with the Higgs field. Some elementary particles gain mass due to their interaction with the Higgs field.

The fourth part describes the interactions between elementary particles, Higgs particles, and imaginary particles in the Higgs field. Virtual particles are an explanatory concept established in the mathematical calculations of quantum field theory, referring to mathematical terms used to describe subatomic processes such as particle collisions. The fifth part, known as Faddeev Popov ghost particle, is used to eliminate redundancy in weak force interactions. Faddeev Popov ghost particle is an additional field introduced into gauge quantum field theory to maintain consistency in the expression of path integrals, named after Ludwig Faddeev and Victor Popov.

Related

• https://en.wikipedia.org/wiki/Mathematical_formulation_of_the_Standard_Model

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