Tagged Questions

For questions about $\theta$ functions (special functions of several complex variables).

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Family of definite integrals involving Dedekind eta function of a complex argument, $\int_0^{\infty} \eta^k(ix)dx$

The Dedekind eta function is denoted by $\eta(\tau)$, and is defined on the upper half-plane ($\Im \tau >0$). Put $\tau = i x$ where $x$ is a positive real number. The function has the following ...
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Curious integrals for Jacobi Theta Functions $\int_0^1 \vartheta_n(0,q)dq$

There are various identities for the Jacobi Theta Functions $\vartheta_n(z,q)$ on the MathWorld page and on the Wikipedia page. But I found no integral identities for these functions. Meanwhile, ...
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Theta series and Jacobi theta functions

I have some difficulties with expressing the following series $\sum\limits_{b=-\infty}^{+\infty}\sum\limits_{a=-\infty}^{+\infty}q^{1 + 3 a + 3 a^2 - 3 b - 3 a b + 3 b^2}$ using standart theta ...
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Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
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Reciprocals of theta functions

I've spent the last few months with partial fraction expansions, and thought to create a function with simple poles over a lattice of zeros, like that of any of the Jacobi theta functions... but I can'...
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Why do the Jacobi theta functions have a natural boundary?

The Jacobi theta functions, like $$\theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz) ,$$ look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the ...
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Expression for Jacobi theta function derivative

Please explain how to get $$\frac{\vartheta'_{1}(z, q)}{\vartheta_{1}(z, q)} = \cot z + 4\sum_{n = 1}^{\infty}\frac{q^{2n}}{1 - q^{2n}}\sin 2nz$$ where $\vartheta_{1}(z, q)$ is one of the Jacobi theta ...
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Coefficients of a power series, an algebraic identity

Given the following 4 functions: \begin{align} \theta_1(0, \tau) &= 2(q^{1/4} + q^{9/4} + q^{25/4} + \ddots), \\ \theta_2(0, \tau) &= 1-2q+2q^4-2q^9+ \ddots), \\ \theta_3(0, \tau) &= 1+2q+...
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How fast does $\Sigma_{n=1}^{\infty} (1- \exp(-x))^{n^2}$ grow?

Let $x$ be a positive real. Define $\theta(x) = \Sigma_{n=1}^{\infty} (1- \exp(-x))^{n^2}$. How fast does $\theta(x)$ grow ? In other words what is a good asymptotic for it when $x$ is large ? Maybe ...
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Is $\sum_n\exp(ian+ibn^2+icn^3)$ known in terms of anything else?

For arbitrary $a,b,c$, does the series $$F(a,b,c)=\sum_{n=-\infty}^\infty\exp\left(ian+ibn^2+icn^3\right),$$ i.e. an evenly-weighed series of exponentials of cubic polynomials, converge to anything ...
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Summation of an infinite q series

When calculating a Partition function, I encounter the following summation $$\sum_{n=0}^{\infty} x^n q^{n^2}.$$ I know that the sum$\sum_{n=-\infty}^{\infty} x^n q^{n^2}$ is a Theta function , but I ...
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What is a mock theta function?

We define a mock theta function as follows: A mock theta function is a function defined by a $q$-series convergent when $|q|<1$ for which we can calculate asymptotic formulae when $q$ tends ...
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Jacobi Theta Functions?

For the Jacobi theta function $\vartheta_3(z|\tau)$ there exists an equality (by Whittaker & Watson) \vartheta_3(z|\tau) = \sum_{n=-\infty}^{\infty} e^{n^2 \pi i \tau + 2 n i z} =...
$$\pi^{-s/2}\zeta(s)\Gamma(s/2)=\pi^{-(1-s)/2}\zeta(1-s)\Gamma((1-s)/2)$$ (That's the equation that want to prove) Hello guys, so I'm trying to prove the functional equation of Riemann Zeta, ...