# 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

${\displaystyle \vartheta (z;\tau )=\sum _{n=-\infty }^{\infty }\ e^{(\pi in^{2}\tau +2\pi inz)}}$

${\displaystyle \vartheta (z+1;\tau )=\vartheta (z;\tau )}$

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

## 輔助函數

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

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

## 雅可比恆等式

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

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

## 以nome q表示Θ函數

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

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

## 乘積表示式

${\displaystyle \prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+w^{2}q^{2m-1}\right)\left(1+w^{-2}q^{2m-1}\right)=\sum _{n=-\infty }^{\infty }w^{2n}q^{n^{2}}.}$

${\displaystyle \vartheta (z;\tau )=\sum _{n=-\infty }^{\infty }\exp(\pi i\tau n^{2})\exp(\pi iz2n)=\sum _{n=-\infty }^{\infty }w^{2n}q^{n^{2}}.}$

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

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

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

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

## 積分表示式

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

## 與黎曼ζ函數的關係

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

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

## 與维尔斯特拉斯椭圓函數之關係

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

## 與模形式之關係

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

## 解熱方程

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

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

t = 0時，Θ函數成為「狄拉克梳状函数」（Dirac comb）

${\displaystyle \lim _{t\rightarrow 0}\vartheta (x,it)=\sum _{n=-\infty }^{\infty }\delta (x-n)}$

## 推廣

F為一n二次型，則有一關連的Θ函數

${\displaystyle \theta _{F}(z)=\sum _{m\in Z^{n}}\exp(2\pi izF(m))}$

${\displaystyle \theta _{F}(z)=\sum _{k=0}^{\infty }R_{F}(k)\exp(2\pi ikz)}$

### 黎曼Θ函數

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

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

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

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

## 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.