# 자코비 세타함수

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## 개요

• $$q=e^{2\pi i \tau}$$, $$x=e^{\pi i \tau}$$라 두자
• 세타함수의 정의 (spectral decomposition of heat kernel)

$\theta(\tau)=\theta_3(\tau)=\sum_{n=-\infty}^\infty q^{n^2/2}=\sum_{n=-\infty}^\infty x^{n^2}= \sum_{n=-\infty}^\infty e^{\pi i n^2\tau}$ $\theta_{2}(\tau)= \sum_{n=-\infty}^\infty q^{(n+\frac{1}{2})^2/2}$ $\theta_{4}(\tau)= \sum_{n=-\infty}^\infty (-1)^n q^{n^2/2}$

## 많이 사용되는 또다른 정의

• 전통적인 세타함수$\theta(\tau)=\theta_3(\tau)=\sum_{n=-\infty}^\infty q^{n^2/2}= \sum_{n=-\infty}^\infty e^{\pi i n^2\tau}$
• 현대의 수학문헌에서는 다음과 같은 함수도 같은 이름으로 자주 사용됨$\Theta(\tau)=\sum_{n=-\infty}^\infty q^{n^2}= \sum_{n=-\infty}^\infty e^{2\pi i n^2\tau}\,\quad (q=e^{2\pi i \tau})$
• $$\Theta(\tau)$$ 는 $$\Gamma_0(4)$$에 대한 모듈라 형식이 됨$\Gamma_0(4) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbf{Z}) : \begin{pmatrix} a & b \\ c & d \end{pmatrix} \equiv \begin{pmatrix} {*} & {*} \\ 0 & {*} \end{pmatrix} \pmod{4} \right\}$

## 여러가지 공식들

• $$\theta_2^4(q)+\theta_4^4(q)=\theta_3^4(q)$$
• $$\theta_3^2(q^2)+\theta_2^2(q^2)=\theta_3^2(q)$$
• $$\theta_3^2(q^2)-\theta_2^2(q^2)=\theta_3^2(q)$$

## 세타함수의 모듈라 성질

정리

세타함수는 다음의 변환 성질을 만족한다 $\theta(-\frac{1}{\tau})=\sqrt{\frac{\tau}{i}} \theta({\tau})=\sqrt{-i\tau}\theta({\tau})$ 여기서 $$-\frac{\pi}{4}<\arg \sqrt{-i\tau}<\frac{\pi}{4}$$ 이 되도록 선택

증명

포아송의 덧셈 공식을 사용한다. $\sum_{n\in \mathbb Z}f(n)=\sum_{n\in \mathbb Z}\hat{f}(n)$

$$f(x)=e^{\pi i x^2\tau}$$의 푸리에 변환은 다음과 같이 주어진다. $\hat{f}(\xi)=\sqrt{\frac{i}{\tau}}e^{-\pi i\frac{\xi^2}{\tau}}$ 따라서 $\theta(\tau)= \sum_{n\in \mathbb Z} \exp(\pi i n^2\tau)=\sum_{n\in \mathbb Z}f(n)=\sum_{n\in \mathbb Z}\hat{f}(n)=\sqrt{\frac{i}{\tau}}\sum_{n\in \mathbb Z}e^{-\pi i n^2 \frac{1}{\tau}}=\sqrt{\frac{i}{\tau}}\theta(-\frac{1}{\tau})$ ■

• $$\tau=iy, y>0$$ 으로 쓰면, 다음과 같이 표현된다 $\theta(\frac{i}{y})=\sqrt{y} \theta({iy})$
• $$\Gamma(2)$$에 대한 모듈라 형식이 됨$\Gamma(2) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbf{Z}) : \begin{pmatrix} a & b \\ c & d \end{pmatrix} \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \pmod{2} \right\}$

### 더 일반적인 모듈라 변환

더 일반적으로, $$ad-bc=1$$, $$ab\equiv 0\pmod 2$$, $$cd\equiv 0\pmod 2$$, $$c>0$$인 정수 a,b,c,d에 대하여 다음이 성립한다 $\theta \left( \frac {a\tau+b} {c\tau+d}\right) =\epsilon(c,d) \sqrt{-i\left(c\tau+d\right)}\theta(\tau) \label{mod}$ 여기서 $$-\frac{\pi}{4}<\arg \sqrt{-i(c\tau+d)}<\frac{\pi}{4}$$ 이 되도록 선택하며 ($$\Re\left(-i(c\tau+d)\right) >0$$이다), $\epsilon(c,d)=\frac{\sqrt{c}}{S(-d,c)}$ 이고 $$S(p,q)=\sum_{r=0}^{q-1} e^{\pi i pr^2/q}$$는 가우스 합.

## cusp에서의 행동과 가우스합

### 0 근방에서의 행동

• $$y>0$$가 매우 작을 때,

$\theta(iy)\sim \frac{1}{\sqrt{y}}$ (증명) $\theta(\frac{i}{y})=\sqrt{y} \theta({iy})$ ■

### 일반적인 유리수(cusp)에서의 행동

• $$pq$$가 짝수인 자연수 p,q에 대하여 $$y>0$$가 매우 작을 때,

$\theta(\frac{p}{q}+iy)\sim \frac{1}{q}S(p,q)\frac{1}{\sqrt{y}}$ 여기서 $$S(p,q)=\sum_{r=0}^{q-1} e^{\pi i pr^2/q}$$는 가우스 합. 다음과 같이 쓸 수 있다 $\lim_{\epsilon \to 0}\sqrt{\epsilon} \theta(\frac{p}{q}+i\epsilon)=\frac{1}{q}S(p,q)$$\lim_{\epsilon \to 0}\sqrt{\epsilon} \theta(-\frac{p}{q}+i\epsilon)=\frac{1}{q}\overline{S(p,q)}$

• 이 정리에 세타함수의 모듈라 성질을 적용하면, 가우스합의 상호법칙을 얻는다

$\sqrt{q}\overline{S(q,p)}=e^{-\pi i/4}\sqrt{p}S(p,q)$

### 증명1

$$\tau =\frac{p}{q}+i y$$와 다음의 행렬 $\left( \begin{array}{cc} a & b \\ q & -p \\ \end{array} \right)$ 에 모듈라 성질 \ref{mod}를 적용하면, 다음을 얻는다 $\theta \left(\frac{a}{q}+\frac{i}{q^2 y}\right)= \frac{\sqrt{q}}{S(p,q)} \sqrt{q y}\theta \left(\frac{p}{q}+i y\right)=\frac{q \sqrt{y}}{S(p,q)}\theta \left(\frac{p}{q}+i y\right)$ ■

### 증명2

다음을 생각하자 $\theta(\frac{p}{q}+i\epsilon)=\sum_{n=-\infty}^\infty e^{\pi i n^2(\frac{p}{q}+i\epsilon)}= \left(\sum_{r=0}^{q-1}e^{\pi i p r^2/q}\right)\left(\sum_{l=-\infty}^\infty e^{-\pi \epsilon (ql+r)^2}\right)$ 여기서 $$n=ql+r$$로 두었음.

따라서, $\sqrt{\epsilon} \theta(\frac{p}{q}+i\epsilon)=\left(\frac{1}{q} \sum_{r=0}^{q-1}e^{i\pi p r^2/q}\right)\left(\sum_{l=-\infty}^\infty e^{-\pi \epsilon (ql+r)^2} (\sqrt{\epsilon}q)\right).$

여기서 $$\Delta{x}=\sqrt{\epsilon}q$$로 두면, $\sum_{l=-\infty}^\infty e^{-\pi \epsilon (ql+r)^2} ( \sqrt{\epsilon}q)=\sum_{x\in\sqrt{\epsilon}(q\mathbb{Z}+r)}e^{-\pi x^2}\Delta x.\label{thg}$ \ref{thg}에서 $$\epsilon \to 0$$을 취하면 리만합은 1차원 가우시안 적분 으로 수렴하게 된다. 따라서 $\lim_{\epsilon \to 0}\sqrt{\epsilon} \theta(\frac{p}{q}+i\epsilon)=\left(\frac{1}{q} \sum_{r=0}^{q-1}e^{i\pi p r^2/q}\right)(\int_{-\infty}^\infty e^{-\pi x^2}\,dx)=\frac{1}{q}S(p,q)$ ■

## 데데킨트 에타함수와의 관계

$$\theta(\tau)=\frac{\eta(\tau)^5}{\eta(2\tau)^2\eta(\frac{\tau}{2})^2}$$

삼중곱 공식을 이용

$$\theta(\tau)=\sum_{n=-\infty}^\infty q^{n^2/2}=\sum_{n=-\infty}^\infty x^{n^2}=\prod_{m=1}^\infty \left( 1 - x^{2m}\right) \left( 1 + x^{2m-1}\right) \left( 1 + x^{2m-1}\right)$$

$$q=e^{2\pi i \tau}$$, $$x=e^{\pi i \tau}$$

## singular value k와의 관계

$$k=k(\tau)=\frac{\theta_2^2(\tau)}{\theta_3^2(\tau)}$$

$$k'=\sqrt{1-k^2}=\frac{\theta_4^2(\tau)}{\theta_3^2(\tau)}$$

## 세타함수와 AGM iteration

$$\frac{\theta_3^2(q)+\theta_4^2(q)}{2}=\theta_3^2(q^2)$$

$$\sqrt{\theta_3^2(q)\theta_4^2(q)}=\theta_4^2(q^2)$$

따라서 $$a_n=\theta_3^2(q^{2^n}),b_n=\theta_4^2(q^{2^n})$$ 라 하면, $$a_n, b_n$$은 AGM iteration 을 만족하고 $$\lim_{n\to\infty}a_n=1$$이고, $$1=M(\theta_3^2(q),\theta_4^2(q))$$가 된다.

## 제1종타원적분과의 관계

(정리)

주어진 $$0<k<1$$ 에 대하여, $$k=k(q)=\frac{\theta_2^2(q)}{\theta_3^2(q)}$$를 만족시키는 $$q$$가 존재한다. 이 때,

$$M(1,k')=\theta_3^{-2}(q)$$ 와 $$K(k) = \frac{\pi}{2}\theta_3^2(q)$$가 성립한다.

여기서 $$K(k)$$는 제1종타원적분 K (complete elliptic integral of the first kind).

(증명)

$$1=M(\theta_3^2(q),\theta_4^2(q))=\theta_3^{2}(q)M(1,\frac{\theta_4^2(q)}{\theta_3^2(q)})=\theta_3^{2}(q)M(1,k')$$

그러므로, $$M(1,k')=\theta_3^{-2}(q)$$이다.

한편, 란덴변환에 의해 $$K(k)=\frac{\pi}{2M(1,\sqrt{1-k^2})}$$가 성립(산술기하평균함수(AGM)와 파이값의 계산 , 란덴변환(Landen's transformation) 참조)하므로, $$K(k) = \frac{\pi}{2}\theta_3^2(q)$$도 증명된다. (증명끝)

## special values

$$\theta_3(i)=\frac{\sqrt[4]{\pi}}{\Gamma(\frac{3}{4})}=1.08643481121\cdots$$

(증명)

감마함수의 다음 성질을 사용하면$$\Gamma(1-z) \; \Gamma(z) = {\pi \over \sin{(\pi z)}} \,\!$$

$$\Gamma(\frac{1}{4})\Gamma(\frac{3}{4}) = \sqrt{2}{\pi}$$

위에서 증명한 제1종타원적분 K (complete elliptic integral of the first kind)과의 관계로부터

$$K(k(\tau)) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}} = \frac{\pi}{2}\theta_3^2(\tau)$$

$$\frac{\pi}{2}\theta_3^2(i)=K(k_1)=K(\frac{1}{\sqrt{2}})=\frac{1}{4}B(1/4,1/4)=\frac{\Gamma(\frac{1}{4})^2}{4\sqrt{\pi}}$$

$$\theta_3^2(i)=\frac{\Gamma(\frac{1}{4})^2}{2{\pi}^{3/2}}=\frac{\sqrt{\pi}}{\Gamma(\frac{3}{4})^2}$$ ■

$$\theta_3(i)=\sum_{n=-\infty}^\infty e^{-\pi n^2}=1+2\sum_{n=1}^\infty e^{-\pi n^2}=\frac{\sqrt[4]{\pi}}{\Gamma(\frac{3}{4})}$$

## 재미있는 사실

$$f(\tau)=1+2\sum_{n=1}^{\infty}e^{\pi i n \tau}$$

$$f(i)=1+2\sum_{n=1}^{\infty} e^{-n\pi}= \frac{e^{\pi} + 1} {e^{\pi} - 1}$$

$$\theta(\tau)= \sum_{n=-\infty}^\infty e^{\pi i n^2\tau}$$

$$\theta(i)=\sum_{n=-\infty}^\infty e^{-\pi n^2}=1+2\sum_{n=1}^\infty e^{-\pi n^2}=\frac{\sqrt[4]{\pi}}{\Gamma(\frac{3}{4})}$$

$$\sum_{n=0}^{\infty} e^{-\pi n}=\frac{e^{\pi}}{e^{\pi}-1}$$

$$\sum_{n=0}^\infty e^{-\pi n^2}=\frac{\sqrt[4]{\pi}}{2\Gamma(\frac{3}{4})}+\frac{1}{2}$$

$$\sum_{n=0}^\infty e^{-\pi n^3}=?$$

$$\sum_{n=0}^\infty e^{-\pi n^4}=?$$

## 메타데이터

### Spacy 패턴 목록

• [{'LOWER': 'theta'}, {'LEMMA': 'function'}]

## 노트

### 말뭉치

1. See Jacobi theta functions (notational variations) for further discussion.[1]
2. More broadly, this section includes quasi-doubly periodic functions (such as the Jacobi theta functions) and other functions useful in the study of elliptic functions.[2]
3. JacobiTheta(j, z, tau) , rendered as θ j ⁣ ( z , τ ) \theta_{j}\!\left(z , \tau\right) θ j ​ ( z , τ ) , denotes a Jacobi theta function.[3]
4. There are four Jacobi theta functions, identified by the index j ∈ { 1 , 2 , 3 , 4 } j \in \left\{1, 2, 3, 4\right\} j ∈ { 1 , 2 , 3 , 4 } .[3]
5. The values of the Jacobi theta functions at z = 0 z = 0 z = 0 are known as theta constants.[3]
6. ( z , τ ) , represents the order r r r derivative of the Jacobi theta function with respect to the argument z z z .[3]
7. The following table illustrates the quasi-double periodicity of the Jacobi theta functions.[4]
8. Any Jacobi theta function of given arguments can be expressed in terms of any other two Jacobi theta functions with the same arguments.[4]
9. The plots above show the Jacobi theta functions plotted as a function of argument and nome restricted to real values.[4]
10. The Jacobi theta functions satisfy an almost bewilderingly large number of identities involving the four functions, their derivatives, multiples of their arguments, and sums of their arguments.[4]
11. Previously, we proved an addition formula for the Jacobi theta function, which allows us to recover many important classical theta function identi- ties.[5]
12. We rst need to introduce the Jacobi theta functions.[5]
13. Let 1, 2, 3, and 4 be the Jacobi theta functions.[5]
14. This identity includes many well-known addition formulas for the Jacobi theta functions.[5]
15. We use Jacobi theta functions to construct examples of Jacobi forms over number fields.[6]
16. We determine the behavior under modular transformations by regarding certain coefficients of the Jacobi theta functions as specializations of symplectic theta functions.[6]
17. In addition, we show how sums of those Jacobi theta functions appear as a single coefficient of a symplectic theta function.[6]
18. For those parameters for which this equation reduces to the heat equation, Θ(x,t) reduces to the third Jacobi Theta function.[7]
19. They generalize modular forms, have an associated weight and index (which is a positive half-integer), and include the classical Jacobi theta function.[8]
20. We rst recall some properties of the Jacobi theta function.[8]
21. Previously, we proved an addition formula for the Jacobi theta function, which allows us to recover many important classical theta function identities.[9]
22. This section is about a more general theta function, called the Jacobi theta function.[10]
23. The Jacobi theta function has the following properties: Two-fold periodicity in z (up to a phase, for xed ).[10]