32 votes
Accepted
What is this modified arithmetico-geometric mean function?
The function $A(x,y)$ has the closed-form expression $$A(x,y)=\frac{\sqrt{y^2-x^2}}{\arccos(x/y)}.$$ This modification of the AGM was introduced by Gauss in 1800, in an unpublished letter. It was ...
- 201k
20 votes
Determination of special values of Eisenstein series
It is well known that $$G_{4k}(i)=\left(4\int_{0}^1\frac{1}{\sqrt{1-x^4}}dx\right)^{4k}\frac{H_{4k}}{(4k)!}, $$ where $H_{4k}$ are called Hurwitz numbers. $H_4=\frac{1}{10}$ and $H_{8}=\frac{3}{10}$ ...
- 3,269
11 votes
What is this modified arithmetico-geometric mean function?
Neat result, but if you wanted to recover the "original" AGM you could have rendered: $x_{n+1}=\frac12(x_n+y_n) [\iff x_n=2x_{n+1}-y_n]$ $y_{n+1}=\sqrt{(2x_{n+1}-y_n)y_n}$ Note that since ...
- 3,028
10 votes
Accepted
What is the surface area of the finite part of the Cayley nodal cubic surface?
Here is the start of an answer - I am afraid that I don't want to spend any more of my time thinking about this problem, but it gets the answer in terms of an integral that looks very computable. ...
- 1,316
10 votes
Asymptotic for Ramanujan's $\tau$-function
No, $\tau(n)$ fluctuates wildly, and it cannot be described in simpler terms. It is "irreducible arithmetic data", and we just love that. Same for its absolute value.
- 114k
10 votes
Accepted
Asymptotic for Ramanujan's $\tau$-function
While an asymptotic for $|\tau(n)|$ does not exist, there are many results that help us to nail down the order of $|\tau(n)|$. First, let us write $$\tau(n)=n^{\frac{11}{2}}f(n).$$ Also, let $d(n)$ ...
- 5,164
9 votes
Accepted
How to work out this elliptic function?
This is a divergent series. But if one applies summation in the sense of Eisenstein, $$\lim_{N\to\infty}\sum_{n=-N}^N\left(\lim_{M\to\infty}\sum_{m=-M}^M\right)$$ then the sum is doubly periodic. ...
- 96.7k
8 votes
Accepted
Can the theory of elliptic functions developed from purely geometric considerations?
One possible source is the book Lawden, D. F. Elliptic functions and applications, New York, NY etc.: Springer-Verlag, 1989, doi:10.1007/978-1-4757-3980-0, (esp. Ch. 4 Geometrical Applications). In ...
- 13.5k
8 votes
Accepted
Conflicting notation for periods of elliptic functions
Probably the reason to denote by $\omega_1$ and $\omega_2$ half periods is that the period of main trigonometric functions $\sin$ and $\cos$ is $2\pi$ but not $\pi$. This notation is used in classical ...
- 13.5k
7 votes
Closed form roots for polynomial $x^9 + ax^6 + bx^5 + cx^3 + d = 0$
There is no way to transform your first polynomial to the special shape of the second polynomial while preserving its Galois group. The Galois group of the second polynomial is solvable, but for ...
- 25.2k
6 votes
Accepted
Bounding an elliptic-type integral
In general one has the following bound on your integral $$ \frac{2}{3}L^{3/2}+ \frac{L^{1/2}}{\sqrt{2}}\ln \left(\frac{1+2K}{2+2(K-L)}\right)\leq \mathrm{integral}\leq \frac{2}{3}L^{3/2}+ L^{1/2} \ln\...
- 4,136
6 votes
Solution of an equation with Jacobi theta function
Have you tried Jacobi's identities for theta-functions? At least for $a=1, b=0$ the identity $$xg(x)=\sqrt{\pi}\,g(\pi/x)$$ implies that the function $xg(x):=xf(x)+x$ is monotone increasing. Edit: ...
6 votes
Accepted
Solution of an equation with Jacobi theta function
Let $\, g(x) := \theta_3(0,\mathrm{e}^{-ax^2 -b}).\,$ Your question about solutions to $\, x + x f(x) = 1 + f(1) \,$ is now about $\, x g(x) = g(1).\,$ Now $\,g(x)\,$ is a bell shaped curve with $\, g(...
- 2,927
6 votes
Accepted
Can a doubly periodic function be locally univalent?
If $f$ is a doubly-periodic meromorphic function on $\Bbb C$ then $f'$ necessarily has zeros – otherwise $1/f'$ would be an entire doubly-periodic function and therefore constant. (More precisely, the ...
- 491
6 votes
Resources on the stationary Schrödinger equation with the soliton potential
The Schrödinger equation with the $\text{sech}^2$ potential (sometimes called the Pöschl–Teller potential) was first studied by Epstein in 1930 [1]. There is an extensive literature on this exactly ...
- 201k
6 votes
Asymptotic for Ramanujan's $\tau$-function
I would like to complement the other answers to mention a couple facts about the erratic nature of the signs of $\tau(n)$. One result is the result of Wilton saying that $$ \sum_{n \leq N} \frac{\tau(...
- 4,711
5 votes
Accepted
Generating function of the product of Legendre polynomials
Using the recursion relation $$ P_{n-1} (x) = x P_n (x) - \frac{x^2 -1}{n} \frac{d}{dx} P_n (x) \ , $$ you can reduce your expression to a sum of the generating function you quote and a combined ...
- 7,069
5 votes
Can the theory of elliptic functions developed from purely geometric considerations?
Some elementary parts of the theory of elliptic functions can indeed be developed in this way. To those books listed by Alexey Ustinov I can add a large treatise by G. Halphen, Traité des fonctions ...
- 96.7k
5 votes
Determination of special values of Eisenstein series
For real Eisenstein series $$ \sum_{(m,n)\ne(0,0)}\frac{1}{|m\tau+n|^s}, $$ the Kronecker limit formula gives the value at $s=1$ in terms of the Dedekind eta function. See https://en.wikipedia.org/...
- 48.4k
5 votes
Accepted
Are these 5 the only eta quotients that parameterize $x^2+y^2 = 1$?
For your question $2$ all that is needed is a quick search of 'eta07.gp' for three terms identities with even rank and with all even exponents. I turned up $t_{12,12,40}$ and the other $5$ members of ...
- 2,927
5 votes
Accepted
What are the modularity properties of Weierstrass sigma function?
The classical Weierstrass sigma function is not exactly $\sigma_L$ but rather, with your notations, $\sigma(z,\tau)=e^{az^2} \sigma_L(2\pi iz)(q)$ for some constant $a$, see Remark 5.3 in Ando, ...
- 21.1k
4 votes
Are these 5 the only eta quotients that parameterize $x^2+y^2 = 1$?
Courtesy of Somos' answer, we can find more eta quotient parameterizations to $x^2+y^2 = 1$. Define, $$a(\tau) = \frac{\eta^3(\tau)\,\eta(6\tau)}{2\,\eta^3(2\tau)\,\eta(3\tau)},\quad\quad b(\tau) = \...
- 13k
4 votes
Special values of the modular J invariant
The value of $\eta(i\sqrt{6})$ and $\eta(i\sqrt{3/2})$ involves the use of gamma function values on a 24 basis, so we have: $$\eta(i\sqrt{6})=\frac{1}{2^{3/2}3^{1/4}}\big(\sqrt{2}-1\big)^{1/12}\frac{\...
- 141
4 votes
Accepted
Infinite sum and product associated with the Weierstrass elliptic function
From standard definitions of $e_1,e_2,e_3$ and $\theta_2$ in reference works such as Abramowitz and Stegun, (e.g., $\theta_2(0,x)=2x^{1/4}\sum_{n>0} x^{n(n-1)}$), it turns out that the following ...
- 2,927
4 votes
Expression for infinite product
As the other answer pointed out, with the typo fixed, the equation is $$\frac{4}{R}\prod_{n=1}^{\infty} \left(\frac{1+R^{-4n}}{1+R^{-4n+2}}\right)^4= \frac{1}{R}\left(2+2 \sum_{n=1}^ {\infty} \frac{1}...
- 2,927
4 votes
Arguments where the $j$-invariant has non-trivially real values — and a conjectured duality
Recall that the $j$-invariant identifies $\Bbb{H}/{\rm{PSL}}_2(\Bbb{Z})$ with $\Bbb{C}$; and a fundamental domain for the action of ${\rm{PSL}}_2(\Bbb{Z})$ on $\Bbb{H}$ is $D:=\left\{z\in\Bbb{H}\,\big|...
- 3,659
4 votes
Accepted
Factorisation of division polynomial
The most general identity for division polynomials is $$ \psi_{a+b}\psi_{a-b}\psi_{c+d}\psi_{c-d}=\psi_{a+c}\psi_{a-c}\psi_{b+d}\psi_{b-d}-\psi_{b+c}\psi_{b-c}\psi_{a+d}\psi_{a-d},$$ where all $a,b,c,...
- 13.5k
4 votes
Accepted
Are $j(\tau)$ and $\wp \left(\frac{\omega_1}{n},\omega_1,\omega_2\right)$ always expressible by radicals over $\mathbb{Q}$?
[Edit: removed incorrect statement and added a better reference] [Edit: another correction and expansion] Note that if you rescale the lattice by a constant $\wp\left(\frac{\omega_1}n,\omega_1,\...
- 863
3 votes
Determination of special values of Eisenstein series
$$\wp(z) = \frac1{z^2}+\sum_{(n,m)\ne (0,0)} \frac1{(z+ni+m)^2}-\frac1{(ni+m)^2}$$ is the unique even $\Bbb{Z}+i\Bbb{Z}$ periodic meromorphic function with only one double pole at $0$ where $\wp(z) = ...
- 3,434
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