# Θ函數

Jacobi theta 1
Jacobi theta 2
Jacobi theta 3
Jacobi theta 4

Θ函數最常見於椭圓函數理論。相對於其「z」 變量，Θ函數是拟周期函数（quasiperiodic function），具有「擬周期性」。在一般下降理論（descent theory）中，此來自線叢條件。

## 雅可比Θ函數

$\vartheta (z; \tau) = \sum_{n=-\infty}^\infty \ e^{(\pi i n^2 \tau +2 \pi i n z ) }$

$\vartheta( z+1; \tau) = \vartheta (z; \tau)$

$\vartheta(z+a+b\tau;\tau) = \ e^{(-\pi i b^2 \tau -2 \pi i b z)}\vartheta(z;\tau)$

## 輔助函數

$\vartheta_{01} (z;\tau) = \vartheta(z+\frac{1}{2};\tau)$
$\vartheta_{10}(z;\tau) = e^{\frac{\pi {\mathrm{i}} \tau}{4} + \pi {\mathrm{i}} z}\vartheta(z+\frac{\tau}{2};\tau)$
$\vartheta_{11}(z;\tau) = e^{\frac{\pi {\mathrm{i}} \tau}{4} + \pi {\mathrm{i}} (z+\frac{1}{2})}\vartheta(z+\frac{\tau+1}{2};\tau).$

$\vartheta(0;\tau)^4 = \vartheta_{01}(0;\tau)^4 + \vartheta_{10}(0;\tau)^4$,

## 雅可比恆等式

$\alpha = (- {\mathrm{i}} \tau)^{\frac{1}{2}} e^{{\pi {\mathrm{i}} z^2}{\tau}}\,$

$\vartheta (\frac{z}{\tau}; -\frac{1}{\tau}) = \alpha \vartheta(z; \tau)$
$\vartheta_{01} (\frac{z}{\tau}; -\frac{1}{\tau}) = \alpha \vartheta_{10}(z; \tau)$
$\vartheta_{10} (\frac{z}{\tau}; -\frac{1}{\tau}) = \alpha \vartheta_{01}(z; \tau)$
$\vartheta_{11} (\frac{z}{\tau}; -\frac{1}{\tau}) = -\alpha \vartheta_{11}(z; \tau)$

## 以nome q表示Θ函數

$\vartheta(w; q) = \sum_{n=-\infty}^\infty w^{2n}q^{n^2}.$

$\vartheta_{01}(w; q) = \sum_{n=-\infty}^\infty (-1)^n w^{2n}q^{n^2},$
$\vartheta_{10}(w; q) = q^{\frac{1}{4}} \sum_{n=-\infty}^\infty w^{2n+1}q^{n^2+n},$
$\vartheta_{11}(w; q) = {\mathrm{i}} q^{\frac{1}{4}} \sum_{n=-\infty}^\infty (-1)^n w^{2n+1}q^{n^2+n}.$

## 乘積表示式

$\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + w^{2}q^{2m-1}\right) \left( 1 + w^{-2}q^{2m-1}\right) = \sum_{n=-\infty}^\infty w^{2n}q^{n^2}.$

$\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp(\pi i \tau n^2) \exp(\pi i z 2n) = \sum_{n=-\infty}^\infty w^{2n}q^{n^2}.$

$\vartheta(z; \tau) = \prod_{m=1}^\infty \left( 1 - \exp(2m \pi i \tau)\right) \left( 1 + \exp((2m-1) \pi i \tau + 2 \pi i z)\right) \left( 1 + \exp((2m-1) \pi i \tau -2 \pi i z)\right)$

$\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + (w^{2}+w^{-2})q^{2m-1}+q^{4m-2}\right),$

$\vartheta(z|q) = \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + 2 \cos(2 \pi z)q^{2m-1}+q^{4m-2}\right)$

$\vartheta_{01}(z|q) = \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 - 2 \cos(2 \pi z)q^{2m-1}+q^{4m-2}\right).$
$\vartheta_{10}(z|q) = 2 q^{1/4}\cos(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + 2 \cos(2 \pi z)q^{2m}+q^{4m}\right).$
$\vartheta_{11}(z|q) = -2 q^{1/4}\sin(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 - 2 \cos(2 \pi z)q^{2m}+q^{4m}\right).$

## 積分表示式

$\vartheta (z; \tau) = -i \int_{i - \infty}^{i + \infty} {e^{i \pi \tau u^2} \cos (2 u z + \pi u) \over \sin (\pi u)} du$
$\vartheta_{01} (z; \tau) = -i \int_{i - \infty}^{i + \infty} {e^{i \pi \tau u^2} \cos (2 u z) \over \sin (\pi u)} du.$
$\vartheta_{10} (z; \tau) = -i e^{iz + i \pi \tau / 4} \int_{i - \infty}^{i + \infty} {e^{i \pi \tau u^2} \cos (2 u z + \pi u + \pi \tau u) \over \sin (\pi u)} du$
$\vartheta_{11} (z; \tau) = e^{iz + i \pi \tau / 4} \int_{i - \infty}^{i + \infty} {e^{i \pi \tau u^2} \cos (2 u z + \pi \tau u) \over \sin (\pi u)} du$

## 與黎曼ζ函數的關係

$\vartheta(0;-\frac{1}{\tau})=(-i\tau)^{\frac{1}{2}} \vartheta(0;\tau)$

$\Gamma\left(\frac{s}{2}\right) \pi^{-\frac{s}{2}} \zeta(s) = \frac{1}{2}\int_0^\infty\left[\vartheta(0;it)-1\right] t^{\frac{s}{2}}\frac{dt}{t}$

## 與维尔斯特拉斯椭圓函數之關係

$\wp(z;\tau) = -(\log \vartheta_{11}(z;\tau))'' + c$

## 與模形式之關係

$\vartheta(0;\tau)=\frac{\eta^2\left(\tau+\frac{1}{2}\right)}{\eta(2\tau+1)}$.

## 解熱方程

$\vartheta (x,it)=1+2\sum_{n=1}^\infty \exp(-\pi n^2 t) \cos(2\pi nx)$

$\frac{\partial}{\partial t} \vartheta(x,it)=\frac{1}{4\pi} \frac{\partial^2}{\partial x^2} \vartheta(x,it)$

t = 0時，Θ函數成為「狄拉克梳状函数」（Dirac comb）

$\lim_{t\rightarrow 0} \vartheta(x,it)=\sum_{n=-\infty}^\infty \delta(x-n)$

## 推廣

F為一n二次型，則有一關連的Θ函數

$\theta_F (z)= \sum_{m\in Z^n} \exp(2\pi izF(m))$

$\theta_F (z) = \sum_{k=0}^\infty R_F(k) \exp(2\pi ikz)$

### 黎曼Θ函數

$\mathbb{H}_n=\{F\in M(n,\mathbb{C}) \; \mathrm{s.t.}\, F=F^T \;\textrm{and}\; \mbox{Im} F >0 \}$

$\theta (z,\tau)=\sum_{m\in Z^n} \exp\left(2\pi i \left(\frac{1}{2} m^T \tau m +m^T z \right)\right)$
$\theta (z,\tau)=\sum_{m\in Z^n} \exp\left(2\pi i \left(\frac{1}{2} m^T \tau m +m^T z \right)\right)$

$\mathbb{C}^n\times \mathbb{H}_n.$的緊緻子集上，黎曼Θ函數絶對一致收歛。

$\theta (z+a+\tau b, \tau) = \exp 2\pi i \left(-b^Tz-\frac{1}{2}b^T\tau b\right) \theta (z,\tau)$

## 参考文献

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0-486-61272-4. (See section 16.27ff.)
• Naum Illyich Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
• Hershel M. Farkas and Irwin Kra, Riemann Surfaces (1980), Springer-Verlag, New York. ISBN 0-387-90465-4 (See Chapter 6 for treatment of the Riemann theta)
• G. H. Hardy and E. M. Wright，An Introduction to the Theory of Numbers, fourth edition (1959) , Oxford University Press
• David Mumford，Tata Lectures on Theta I (1983), Birkhauser, Boston ISBN 3-7643-3109-7
• James Pierpont Functions of a Complex Variable, Dover
• Harry E. Rauch and Hershel M. Farkas, Theta Functions with Applications to Riemann Surfaces, (1974) Williams & Wilkins Co. Baltimore ISBN 0-683-07196-3.