Intereting Posts

Is a reversible distribution unique?
Show that $L^1$ is strictly contained in $(L^\infty)^*$
Why does this $u$-substitution zero out my integral?
Weierstrass approximation does not hold on the entire Real Line
Open sets in product topology
Number of point subsets that can be covered by a disk
Continuous functions on a compact set
What is the most rigorous definition of a matrix?
Finding a basis of an infinite-dimensional vector space?
Prove $\int_{-\infty}^\infty\frac{dx}{(1+x^2/a)^n}$ converges
Question on Smith normal form and isomorphism
Expectation of gradient in stochastic gradient descent algorithm
Showing that every connected open set in a locally path connected space is path connected
Finding the value of $\int_0^{\pi/2} \frac{dt}{1+(\tan(x))^{\sqrt{2}}}$
Area of a spherical triangle

(*This summarizes results for cube roots from here and here. The fourth root version is this post*.)

Define $\beta= \tfrac{\Gamma\big(\tfrac56\big)}{\Gamma\big(\tfrac13\big)\sqrt{\pi}}=\frac1{B\big(\tfrac{1}{3},\tfrac{1}{2}\big)}$ with *beta function* $B(a,b)$. Then we have the nice evaluations,

$$\begin{aligned}\frac{3}{5^{5/6}} &=\,_2F_1\big(\tfrac{1}{3},\tfrac{1}{3};\tfrac{5}{6};-4\big)\\

&=\beta\,\int_0^1 \frac{dx}{\sqrt{1-x}\,\sqrt[3]{x^2+4x^3}}\\[1.7mm]

&=\beta\,\int_{-1}^1\frac{dx}{\left(1-x^2\right)^{\small2/3} \sqrt[3]{\color{blue}{9+4\sqrt{5}}\,x}}\\[1.7mm]

&=2^{1/3}\,\beta\,\int_0^\infty\frac{dx}{\sqrt[3]{9+\cosh x}}

\end{aligned}\tag1$$

and,

$$\begin{aligned}\frac{4}{7} &=\,_2F_1\big(\tfrac{1}{3},\tfrac{1}{3};\tfrac{5}{6};-27\big)\\

&=\beta\,\int_0^1 \frac{dx}{\sqrt{1-x}\,\sqrt[3]{x^2+27x^3}}\\[1.7mm]

&=\beta\,\int_{-1}^1\frac{dx}{\left(1-x^2\right)^{\small2/3} \sqrt[3]{\color{blue}{55+12\sqrt{21}}\,x}}\\[1.7mm]

&=2^{1/3}\,\beta\,\int_0^\infty\frac{dx}{\sqrt[3]{55+\cosh x}}

\end{aligned}\tag2$$

Note the powers of *fundamental units*,

$$U_{5}^6 = \big(\tfrac{1+\sqrt{5}}{2}\big)^6=\color{blue}{9+4\sqrt{5}}$$

$$U_{21}^3 = \big(\tfrac{5+\sqrt{21}}{2}\big)^3=\color{blue}{55+12\sqrt{21}}$$

*Those two instances can’t be coincidence.*

- Convergence series with natural logarithm
- Find the principal value of $\int_{-\infty}^{\infty}\frac{1-\cos x}{x^2}\,\mathrm dx$
- Derivative of a vector with respect to a matrix
- Calculating the limit of the derivative of a sum of digamma functions
- Which function ($f$) is continuous nowhere but $|f(x)|$ is continuous everywhere?
- Derivative of a Matrix with respect to a vector

Question:

Is it true this observation can be explained by, let $b=2a+1$, then,

$$\int_0^1 \frac{dx}{\sqrt{1-x}\,\sqrt[3]{x^2+ax^3}}=\int_{-1}^1\frac{dx}{\left(1-x^2\right)^{\small2/3} \sqrt[3]{b+\sqrt{b^2-1}\,x}}=2^{1/3}\int_0^\infty\frac{dx}{\sqrt[3]{b+\cosh x}}$$

Example: We assume it is true and use one of Noam Elkies’ results as,

$$\,_2F_1\big(\tfrac{1}{3},\tfrac{1}{3};\tfrac{5}{6}; -a\big) = \frac{6}{11^{11/12}\, U_{33}^{1/4}}

$$

where $a=\sqrt{11}\,(U_{33})^{3/2}$ with fundamental unit $U_{33}=23+4\sqrt{33}$. Since $b=2a+1=45\,U_{33}$, we then have the nice integral,

$$2^{1/3}\beta\,\int_0^\infty\frac{dx}{\sqrt[3]{45\big(23+4\sqrt{33}\big)+\cosh x}}=\frac{6}{11^{11/12}\,U_{33}^{1/4}}=0.255802\dots$$

where $\beta= \tfrac{\Gamma\big(\tfrac56\big)}{\Gamma\big(\tfrac13\big)\sqrt{\pi}}.\,$ So is it true in general?

- Prove that if $\alpha, \beta, \gamma$ are angles in triangle, then $(tan(\frac{\alpha}{2}))^2+(tan(\frac{\beta}{2}))^2+(tan(\frac{\gamma}{2}))^2\geq1$
- Why is this Definite Integral wrong?
- Is there a fundamental theorem of calculus for improper integrals?
- Are derivatives defined at boundaries?
- Need help with an integral
- How to evaluate $\int_{0}^{+\infty}\exp(-ax^2-\frac b{x^2})\,dx$ for $a,b>0$
- Compute $\sum_{k=1}^{\infty}e^{-\pi k^2}\left(\pi k^2-\frac{1}{4}\right)$
- Knowing which theorem of calculus to use to prove number/nature of solutions
- Is this integral right?
- Understanding the differential $dx$ when doing $u$-substitution

- Global convergence for Newton's method in one dimension
- Fourier transform of $\mathrm{sinc}(4t)$
- Do there exist equations that cannot be solved in $\mathbb{C}$, but can be solved in $\mathbb{H}$?
- Distinguishing equality and isomorphism as relations
- Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \ \mathrm dx$
- Generating functions of partition numbers
- Are the complements of two homeomorphic compact, connected subsets of $\mathbb{R}^2$ homeomorphic?
- Reps of $Lie(G)$ lift to universal cover of $G$. Reps of $G$ descend to highest weight reps of $Lie(G)$?
- Failure of Choice only for sets above a certain rank
- If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
- Simplifying an expression $\frac{x^7+y^7+z^7}{xyz(x^4+y^4+z^4)}$ if we know $x+y+z=0$
- Integral Contest
- Is there a bijection between the reals and naturals?
- Find a formula in terms of k for the entries of Ak, where A is the diagonalizable matrix:
- Prove that the Pontryagin dual of a locally compact abelian group is also a locally compact abelian group.