Fungsi theta asli Jacobi θ 1 u = i πz q = e i πτ e 0.1i π
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{\displaystyle {\begin{aligned}\theta _{1}(u;q)&=2q^{\frac {1}{4}}\sum _{n=0}^{\infty }(-1)^{n}q^{n(n+1)}\sin(2n+1)u\\&=\sum _{n=-\infty }^{\infty }(-1)^{n-{\frac {1}{2}}}q^{\left(n+{\frac {1}{2}}\right)^{2}}e^{(2n+1)iu}\end{aligned}}}
Dalam matematika , Fungsi theta adalah fungsi khusus dari beberapa variabel kompleks . Mereka penting di banyak bidang, termasuk teori varietas abelian dan ruang moduli , dan bentuk kuadrat . Mereka juga telah diterapkan pada teori soliton . Ketika digeneralisasi menjadi aljabar Grassmann , mereka juga muncul di teori medan kuantum .[1]
Bentuk fungsi theta yang paling umum adalah yang terjadi dalam teori fungsi eliptik . Sehubungan dengan salah satu variabel kompleks (secara konvensional disebut z ), fungsi theta memiliki properti yang mengekspresikan perilakunya sehubungan dengan penambahan periode fungsi eliptik terkait, menjadikannya fungsi kuasiperiodik . Dalam teori abstrak ini berasal dari bundel garis kondisi keturunan .
Jacobi theta 1 Jacobi theta 2 Jacobi theta 3 Jacobi theta 4 Ada beberapa fungsi yang terkait erat yang disebut fungsi jacobi theta, dan banyak sistem notasi yang berbeda dan tidak kompatibel untuk fungsi tersebut. Fungsi theta Jacobi (dinamai Carl Gustav Jacob Jacobi ) merupakan fungsi yang ditentukan untuk dua variabel kompleks z dan τ , dimana z dapat berupa bilangan kompleks apa pun dan τ adalah rasio setengah periode , terbatas pada bidang setengah atas , yang berarti ia memiliki bagian imajiner positif. Itu diberikan oleh rumus
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{\displaystyle {\begin{aligned}\vartheta (z;\tau )&=\sum _{n=-\infty }^{\infty }\exp \left(\pi in^{2}\tau +2\pi inz\right)\\&=1+2\sum _{n=1}^{\infty }\left(e^{\pi i\tau }\right)^{n^{2}}\cos(2\pi nz)\\&=\sum _{n=-\infty }^{\infty }q^{n^{2}}\eta ^{n}\end{aligned}}}
dimana q = exp(πiτ )nome η = exp(2πiz )bentuk Jacobi . Pada τ , ini adalah deret Fourier untuk 1-periodik seluruh fungsi dari z . Karenanya, fungsi theta adalah 1-periodik in z :
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{\displaystyle \vartheta (z+1;\tau )=\vartheta (z;\tau ).}
Ternyata juga menjadi τ kuasiperiodik dalam z , dengan
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{\displaystyle \vartheta (z+\tau ;\tau )=\exp[-\pi i(\tau +2z)]\vartheta (z;\tau ).}
Jadi, secara umum,
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{\displaystyle \vartheta (z+a+b\tau ;\tau )=\exp \left(-\pi ib^{2}\tau -2\pi ibz\right)\vartheta (z;\tau )}
untuk semua bilangan bulat a dan b .
Fungsi theta θ 1 q = e i πτ q berubah dengan τ . Fungsi theta θ 1 q = e i πτ q berubah dengan τ . Fungsi theta Jacobi yang didefinisikan di atas terkadang dipertimbangkan bersama dengan tiga fungsi theta tambahan, dalam hal ini ditulis dengan subskrip 0 ganda:
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{\displaystyle \vartheta _{00}(z;\tau )=\vartheta (z;\tau )}
Fungsi bantu (atau setengah periode) ditentukan oleh
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{\displaystyle {\begin{aligned}\vartheta _{01}(z;\tau )&=\vartheta \left(z+{\tfrac {1}{2}};\tau \right)\\[3pt]\vartheta _{10}(z;\tau )&=\exp \left({\tfrac {1}{4}}\pi i\tau +\pi iz\right)\vartheta \left(z+{\tfrac {1}{2}}\tau ;\tau \right)\\[3pt]\vartheta _{11}(z;\tau )&=\exp \left({\tfrac {1}{4}}\pi i\tau +\pi i\left(z+{\tfrac {1}{2}}\right)\right)\vartheta \left(z+{\tfrac {1}{2}}\tau +{\tfrac {1}{2}};\tau \right).\end{aligned}}}
Notasi ini mengikuti Riemann dan Mumford ; Formulasi asli Jacobi adalah dalam istilah nome q = e i πτ τ . Dalam notasi Jacobi θ adalah fungsi tertulis:
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{\displaystyle {\begin{aligned}\theta _{1}(z;q)&=-\vartheta _{11}(z;\tau )\\\theta _{2}(z;q)&=\vartheta _{10}(z;\tau )\\\theta _{3}(z;q)&=\vartheta _{00}(z;\tau )\\\theta _{4}(z;q)&=\vartheta _{01}(z;\tau )\end{aligned}}}
Definisi di atas dari fungsi Jacobi theta sama sekali tidak unik. Lihat Fungsi jacobi theta (variasi notasi) untuk pembahasan lebih lanjut.
Bila kita mengatur z = 0 τ saja, yang ditentukan pada setengah bidang atas (terkadang disebut konstanta teta.) Mak ini dapat digunakan untuk mendefinisikan berbagai bentuk modular , dan untuk mengukur kurva tertentu; khususnya, identitas Jacobi adalah
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{\displaystyle \vartheta _{00}(0;\tau )^{4}=\vartheta _{01}(0;\tau )^{4}+\vartheta _{10}(0;\tau )^{4}}
yang merupakan kurva Fermat dari derajat empat.
Identitas Jacobi menggambarkan bagaimana fungsi theta berubah di bawah kelompok modular , yang dihasilkan oleh τ ↦ τ + 1τ ↦ −1 τ τ dalam eksponen memiliki efek yang sama seperti penjumlahan 1 2 z (n ≡ n 2 mod 2
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{\displaystyle \alpha =(-i\tau )^{\frac {1}{2}}\exp \left({\frac {\pi }{\tau }}iz^{2}\right).}
Kemudian
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{\displaystyle {\begin{aligned}\vartheta _{00}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=\alpha \,\vartheta _{00}(z;\tau )\quad &\vartheta _{01}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=\alpha \,\vartheta _{10}(z;\tau )\\[3pt]\vartheta _{10}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=\alpha \,\vartheta _{01}(z;\tau )\quad &\vartheta _{11}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=-i\alpha \,\vartheta _{11}(z;\tau ).\end{aligned}}}
Alih-alih mengekspresikan fungsi theta dalam istilah z dan τ , kita dapat mengungkapkannya dalam istilah argumen w dan nome q , dimana w = e πiz q = e πiτ
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{\displaystyle {\begin{aligned}\vartheta _{00}(w,q)&=\sum _{n=-\infty }^{\infty }(w^{2})^{n}q^{n^{2}}\quad &\vartheta _{01}(w,q)&=\sum _{n=-\infty }^{\infty }(-1)^{n}(w^{2})^{n}q^{n^{2}}\\[3pt]\vartheta _{10}(w,q)&=\sum _{n=-\infty }^{\infty }(w^{2})^{n+{\frac {1}{2}}}q^{\left(n+{\frac {1}{2}}\right)^{2}}\quad &\vartheta _{11}(w,q)&=i\sum _{n=-\infty }^{\infty }(-1)^{n}(w^{2})^{n+{\frac {1}{2}}}q^{\left(n+{\frac {1}{2}}\right)^{2}}.\end{aligned}}}
Kita melihat bahwa fungsi theta juga bisa didefinisikan dalam istilah w dan q , tanpa referensi langsung ke fungsi eksponensial. Oleh karena itu, rumus-rumus ini dapat digunakan untuk mendefinisikan fungsi Theta di atas bidang lain di mana fungsi eksponensial mungkin tidak dapat didefinisikan di mana-mana, seperti bidang bilangan p -adic .
Fungsi Jacobi theta memiliki wakilan integral berikut:
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{\displaystyle {\begin{aligned}\vartheta _{00}(z;\tau )&=-i\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2uz+\pi u)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{01}(z;\tau )&=-i\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2uz)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{10}(z;\tau )&=-ie^{iz+{\frac {1}{4}}i\pi \tau }\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2uz+\pi u+\pi \tau u)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{11}(z;\tau )&=e^{iz+{\frac {1}{4}}i\pi \tau }\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2uz+\pi \tau u)}{\sin(\pi u)}}\mathrm {d} u.\end{aligned}}}
Lihat Yi (2004).[2] [3]
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{\displaystyle {\begin{aligned}\varphi (e^{-\pi x})&=\vartheta (0;ix)=\theta _{3}(0;e^{-\pi x})=\sum _{n=-\infty }^{\infty }e^{-x\pi n^{2}}\\[8pt]\varphi \left(e^{-\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}\\[8pt]\varphi \left(e^{-2\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt[{4}]{6+4{\sqrt {2}}}}{2}}\\[8pt]\varphi \left(e^{-3\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt[{4}]{27+18{\sqrt {3}}}}{3}}\\[8pt]\varphi \left(e^{-4\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {{\sqrt[{4}]{8}}+2}{4}}\\[8pt]\varphi \left(e^{-5\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt[{4}]{225+100{\sqrt {5}}}}{5}}\\[8pt]\varphi \left(e^{-6\pi }\right)&={\frac {{\sqrt[{3}]{3{\sqrt {2}}+3{\sqrt[{4}]{3}}+2{\sqrt {3}}-{\sqrt[{4}]{27}}+{\sqrt[{4}]{1728}}-4}}\cdot {\sqrt[{8}]{243{\pi }^{2}}}}{6{\sqrt[{6}]{1+{\sqrt {6}}-{\sqrt {2}}-{\sqrt {3}}}}{\Gamma \left({\frac {3}{4}}\right)}}}={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt[{4}]{1}}+{\sqrt[{4}]{3}}+{\sqrt[{4}]{4}}+{\sqrt[{4}]{9}}}}{\sqrt[{8}]{1728}}}\\[8pt]\varphi \left(e^{-7\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\sqrt {{\frac {{\sqrt {13+{\sqrt {7}}}}+{\sqrt {7+3{\sqrt {7}}}}}{14}}\cdot {\sqrt[{8}]{28}}}}={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt[{4}]{7+4{\sqrt {7}}+5{\sqrt[{4}]{28}}+{\sqrt[{4}]{1372}}}}{\sqrt {7}}}\\[8pt]\varphi \left(e^{-8\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {{\sqrt[{8}]{128}}+{\sqrt {2+{\sqrt {2}}}}}{4}}\\[8pt]\varphi \left(e^{-9\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\left(1+\left(1+{\sqrt {3}}\right){\sqrt[{3}]{2-{\sqrt {3}}}}\right)}{3}}\\[8pt]\varphi \left(e^{-10\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {20+{\sqrt {450}}+{\sqrt {500}}+10{\sqrt[{4}]{20}}}}{10}}\\[8pt]\varphi \left(e^{-12\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt[{4}]{1}}+{\sqrt[{4}]{2}}+{\sqrt[{4}]{3}}+{\sqrt[{4}]{4}}+{\sqrt[{4}]{9}}+{\sqrt[{4}]{18}}+{\sqrt[{4}]{24}}}}{2{\sqrt[{8}]{108}}}}\\[8pt]\varphi \left(e^{-16\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\left(4+{\sqrt[{4}]{128}}+{\sqrt[{4}]{1024{\sqrt[{4}]{8}}+1024{\sqrt[{4}]{2}}}}\right)}{16}}\end{aligned}}}
Dua identitas seri berikutnya dibuktikan oleh István Mező :[4]
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{\displaystyle {\begin{aligned}\vartheta _{4}^{2}(q)&=iq^{\frac {1}{4}}\sum _{k=-\infty }^{\infty }q^{2k^{2}-k}\vartheta _{1}\left({\frac {2k-1}{2i}}\ln q,q\right),\\[6pt]\vartheta _{4}^{2}(q)&=\sum _{k=-\infty }^{\infty }q^{2k^{2}}\vartheta _{4}\left({\frac {k\ln q}{i}},q\right).\end{aligned}}}
These relations hold for all 0 < q < 1 . Specializing the values of q , we have the next parameter free sums
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{\displaystyle {\begin{aligned}{\sqrt {\frac {\pi {\sqrt {e^{\pi }}}}{2}}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {3}{4}}\right)}}&=i\sum _{k=-\infty }^{\infty }e^{\pi \left(k-2k^{2}\right)}\vartheta _{1}\left({\frac {i\pi }{2}}(2k-1),e^{-\pi }\right),\\[6pt]{\sqrt {\frac {\pi }{2}}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {3}{4}}\right)}}&=\sum _{k=-\infty }^{\infty }{\frac {\vartheta _{4}\left(ik\pi ,e^{-\pi }\right)}{e^{2\pi k^{2}}}}\end{aligned}}}
Semua angka nol dari fungsi theta Jacobi adalah angka nol sederhana dan diberikan sebagai berikut:
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=
0
⟺
z
=
m
+
n
τ
+
1
2
ϑ
4
(
z
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τ
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=
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z
=
m
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2
{\displaystyle {\begin{aligned}\vartheta (z,\tau )=\vartheta _{3}(z,\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau +{\frac {1}{2}}+{\frac {\tau }{2}}\\[3pt]\vartheta _{1}(z,\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau \\[3pt]\vartheta _{2}(z,\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau +{\frac {1}{2}}\\[3pt]\vartheta _{4}(z,\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau +{\frac {\tau }{2}}\end{aligned}}}
dimana m , n adalah bilangan bulat acak.
Relasi
ϑ
(
0
;
−
1
τ
)
=
(
−
i
τ
)
1
2
ϑ
(
0
;
τ
)
{\displaystyle \vartheta \left(0;-{\frac {1}{\tau }}\right)=(-i\tau )^{\frac {1}{2}}\vartheta (0;\tau )}
digunakan oleh Riemann untuk membuktikan persamaan fungsional untuk fungsi zeta Riemann , dengan menggunakan transformasi Mellin
Γ
(
s
2
)
π
−
s
2
ζ
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s
)
=
1
2
∫
0
∞
(
ϑ
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;
i
t
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−
1
)
t
s
2
d
t
t
{\displaystyle \Gamma \left({\frac {s}{2}}\right)\pi ^{-{\frac {s}{2}}}\zeta (s)={\frac {1}{2}}\int _{0}^{\infty }(\vartheta (0;it)-1)t^{\frac {s}{2}}{\frac {\mathrm {d} t}{t}}}
yang dapat ditampilkan sebagai invarian di bawah substitusi s oleh 1 − s . Integral terkait untuk z ≠ 0Fungsi zeta Hurwitz .
Fungsi theta keempat dan dengan demikian yang lainnya juga terhubung erat ke fungsi gamma-q Jackson melalui relasi[5]
(
Γ
q
2
(
x
)
Γ
q
2
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1
−
x
)
)
−
1
=
q
2
x
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1
−
x
)
(
q
−
2
;
q
−
2
)
∞
3
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q
2
−
1
)
ϑ
4
(
1
2
i
(
1
−
2
x
)
log
q
,
1
q
)
.
{\displaystyle \left(\Gamma _{q^{2}}(x)\Gamma _{q^{2}}(1-x)\right)^{-1}={\frac {q^{2x(1-x)}}{\left(q^{-2};q^{-2}\right)_{\infty }^{3}\left(q^{2}-1\right)}}\vartheta _{4}\left({\frac {1}{2i}}(1-2x)\log q,{\frac {1}{q}}\right).}
Maka η (τ )Dedekind eta function , dan argumen dari fungsi theta sebagai nome q = e πiτ
θ
2
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0
,
q
)
=
ϑ
10
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0
;
τ
)
=
2
η
2
(
2
τ
)
η
(
τ
)
,
θ
3
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0
,
q
)
=
ϑ
00
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0
;
τ
)
=
η
5
(
τ
)
η
2
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1
2
τ
)
η
2
(
2
τ
)
=
η
2
(
1
2
(
τ
+
1
)
)
η
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τ
+
1
)
,
θ
4
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0
,
q
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=
ϑ
01
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0
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τ
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=
η
2
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1
2
τ
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η
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,
{\displaystyle {\begin{aligned}\theta _{2}(0,q)=\vartheta _{10}(0;\tau )&={\frac {2\eta ^{2}(2\tau )}{\eta (\tau )}},\\[3pt]\theta _{3}(0,q)=\vartheta _{00}(0;\tau )&={\frac {\eta ^{5}(\tau )}{\eta ^{2}\left({\frac {1}{2}}\tau \right)\eta ^{2}(2\tau )}}={\frac {\eta ^{2}\left({\frac {1}{2}}(\tau +1)\right)}{\eta (\tau +1)}},\\[3pt]\theta _{4}(0,q)=\vartheta _{01}(0;\tau )&={\frac {\eta ^{2}\left({\frac {1}{2}}\tau \right)}{\eta (\tau )}},\end{aligned}}}
dan,
θ
2
(
0
,
q
)
θ
3
(
0
,
q
)
θ
4
(
0
,
q
)
=
2
η
3
(
τ
)
.
{\displaystyle \theta _{2}(0,q)\,\theta _{3}(0,q)\,\theta _{4}(0,q)=2\eta ^{3}(\tau ).}
Lihat pula Fungsi modular Weber .
Modulus eliptik adalah
k
(
τ
)
=
ϑ
10
(
0
,
τ
)
2
ϑ
00
(
0
,
τ
)
2
{\displaystyle k(\tau )={\frac {\vartheta _{10}(0,\tau )^{2}}{\vartheta _{00}(0,\tau )^{2}}}}
dan modulus eliptik komplementernya adalah
k
′
(
τ
)
=
ϑ
01
(
0
,
τ
)
2
ϑ
00
(
0
,
τ
)
2
{\displaystyle k'(\tau )={\frac {\vartheta _{01}(0,\tau )^{2}}{\vartheta _{00}(0,\tau )^{2}}}}
Fungsi Jacobi theta tidak berubah di bawah aksi subkelompok diskrit dari kelompok Heisenberg . Pembalikan ini disajikan dalam artikel di representasi theta dari kelompok Heisenberg.
^ Tyurin, Andrey N. (30 October 2002). "Quantization, Classical and Quantum Field Theory and Theta-Functions". arΧiv :math/0210466v1 ^ Yi, Jinhee (2004). "Theta-function identities and the explicit formulas for theta-function and their applications". Journal of Mathematical Analysis and Applications . 292 (2): 381–400. doi :10.1016/j.jmaa.2003.12.009 ^ Proper credit for these results goes to Ramanujan. See Ramanujan's lost notebook and a relevant reference at Euler function . The Ramanujan results quoted at Euler function plus a few elementary operations give the results below, so the results below are either in Ramanujan's lost notebook or follow immediately from it.
^ Mező, István (2013), "Duplication formulae involving Jacobi theta functions and Gosper's q -trigonometric functions", Proceedings of the American Mathematical Society , 141 (7): 2401–2410, doi :10.1090/s0002-9939-2013-11576-5 ^ Mező, István (2012). "A q -Raabe formula and an integral of the fourth Jacobi theta function". Journal of Number Theory . 133 (2): 692–704. doi :10.1016/j.jnt.2012.08.025
Abramowitz, Milton ; Stegun, Irene A. (1964). Handbook of Mathematical Functions sec. 16.27ff. ISBN 978-0-486-61272-0 . Akhiezer, Naum Illyich (1990) [1970]. Elements of the Theory of Elliptic Functions . AMS Translations of Mathematical Monographs. 79 . Providence, RI: AMS. ISBN 978-0-8218-4532-5 . Farkas, Hershel M. ; Kra, Irwin (1980). Riemann Surfaces . New York: Springer-Verlag. ch. 6. ISBN 978-0-387-90465-8 . (for treatment of the Riemann theta) Hardy, G. H. ; Wright, E. M. (1959). An Introduction to the Theory of Numbers (edisi ke-4th). Oxford: Clarendon Press. Mumford, David (1983). Tata Lectures on Theta I ISBN 978-3-7643-3109-2 . Pierpont, James (1959). Functions of a Complex Variable . New York: Dover Publications. Rauch, Harry E. ; Farkas, Hershel M. (1974). Theta Functions with Applications to Riemann Surfaces ISBN 978-0-683-07196-2 . Reinhardt, William P.; Walker, Peter L. (2010), "Theta Functions" , dalam Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions ISBN 978-0-521-19225-5 , MR 2723248 Whittaker, E. T. ; Watson, G. N. (1927). A Course in Modern Analysis (edisi ke-4th). Cambridge: Cambridge University Press. ch. 21. (history of Jacobi's θ functions) Harry Rauch with Hershel M. Farkas: Theta functions with applications to Riemann Surfaces, Williams and Wilkins, Baltimore MD 1974, ISBN 0-683-07196-3 .
Templat:PlanetMath attribution
Fungsi polinomial Fungsi aljabar Fungsi dalam Fungsi trigonometri Fungsi berdasarkan Fungsi berdasarkan Fungsi khusus Fungsi lainnya
Perpustakaan nasional Lain-lain
![{\displaystyle \vartheta (z+\tau ;\tau )=\exp[-\pi i(\tau +2z)]\vartheta (z;\tau ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0cb3f2efee89e79116f798a1c8a7078e35fa7f5)
![{\displaystyle {\begin{aligned}\vartheta _{01}(z;\tau )&=\vartheta \left(z+{\tfrac {1}{2}};\tau \right)\\[3pt]\vartheta _{10}(z;\tau )&=\exp \left({\tfrac {1}{4}}\pi i\tau +\pi iz\right)\vartheta \left(z+{\tfrac {1}{2}}\tau ;\tau \right)\\[3pt]\vartheta _{11}(z;\tau )&=\exp \left({\tfrac {1}{4}}\pi i\tau +\pi i\left(z+{\tfrac {1}{2}}\right)\right)\vartheta \left(z+{\tfrac {1}{2}}\tau +{\tfrac {1}{2}};\tau \right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee051ca15813825e13589298c0efd02e3f68e0cf)
![{\displaystyle {\begin{aligned}\vartheta _{00}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=\alpha \,\vartheta _{00}(z;\tau )\quad &\vartheta _{01}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=\alpha \,\vartheta _{10}(z;\tau )\\[3pt]\vartheta _{10}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=\alpha \,\vartheta _{01}(z;\tau )\quad &\vartheta _{11}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=-i\alpha \,\vartheta _{11}(z;\tau ).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e38521d263e968c6643113cb856744c3d8417638)
![{\displaystyle {\begin{aligned}\vartheta _{00}(w,q)&=\sum _{n=-\infty }^{\infty }(w^{2})^{n}q^{n^{2}}\quad &\vartheta _{01}(w,q)&=\sum _{n=-\infty }^{\infty }(-1)^{n}(w^{2})^{n}q^{n^{2}}\\[3pt]\vartheta _{10}(w,q)&=\sum _{n=-\infty }^{\infty }(w^{2})^{n+{\frac {1}{2}}}q^{\left(n+{\frac {1}{2}}\right)^{2}}\quad &\vartheta _{11}(w,q)&=i\sum _{n=-\infty }^{\infty }(-1)^{n}(w^{2})^{n+{\frac {1}{2}}}q^{\left(n+{\frac {1}{2}}\right)^{2}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db65827472877657a7aa66887c63a13ecd71483a)
![{\displaystyle {\begin{aligned}\vartheta _{00}(z;\tau )&=-i\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2uz+\pi u)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{01}(z;\tau )&=-i\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2uz)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{10}(z;\tau )&=-ie^{iz+{\frac {1}{4}}i\pi \tau }\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2uz+\pi u+\pi \tau u)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{11}(z;\tau )&=e^{iz+{\frac {1}{4}}i\pi \tau }\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2uz+\pi \tau u)}{\sin(\pi u)}}\mathrm {d} u.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f19bc02440495b2ff8eda23824fdc3ab905609e)
![{\displaystyle {\begin{aligned}\varphi (e^{-\pi x})&=\vartheta (0;ix)=\theta _{3}(0;e^{-\pi x})=\sum _{n=-\infty }^{\infty }e^{-x\pi n^{2}}\\[8pt]\varphi \left(e^{-\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}\\[8pt]\varphi \left(e^{-2\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt[{4}]{6+4{\sqrt {2}}}}{2}}\\[8pt]\varphi \left(e^{-3\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt[{4}]{27+18{\sqrt {3}}}}{3}}\\[8pt]\varphi \left(e^{-4\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {{\sqrt[{4}]{8}}+2}{4}}\\[8pt]\varphi \left(e^{-5\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt[{4}]{225+100{\sqrt {5}}}}{5}}\\[8pt]\varphi \left(e^{-6\pi }\right)&={\frac {{\sqrt[{3}]{3{\sqrt {2}}+3{\sqrt[{4}]{3}}+2{\sqrt {3}}-{\sqrt[{4}]{27}}+{\sqrt[{4}]{1728}}-4}}\cdot {\sqrt[{8}]{243{\pi }^{2}}}}{6{\sqrt[{6}]{1+{\sqrt {6}}-{\sqrt {2}}-{\sqrt {3}}}}{\Gamma \left({\frac {3}{4}}\right)}}}={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt[{4}]{1}}+{\sqrt[{4}]{3}}+{\sqrt[{4}]{4}}+{\sqrt[{4}]{9}}}}{\sqrt[{8}]{1728}}}\\[8pt]\varphi \left(e^{-7\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\sqrt {{\frac {{\sqrt {13+{\sqrt {7}}}}+{\sqrt {7+3{\sqrt {7}}}}}{14}}\cdot {\sqrt[{8}]{28}}}}={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt[{4}]{7+4{\sqrt {7}}+5{\sqrt[{4}]{28}}+{\sqrt[{4}]{1372}}}}{\sqrt {7}}}\\[8pt]\varphi \left(e^{-8\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {{\sqrt[{8}]{128}}+{\sqrt {2+{\sqrt {2}}}}}{4}}\\[8pt]\varphi \left(e^{-9\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\left(1+\left(1+{\sqrt {3}}\right){\sqrt[{3}]{2-{\sqrt {3}}}}\right)}{3}}\\[8pt]\varphi \left(e^{-10\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {20+{\sqrt {450}}+{\sqrt {500}}+10{\sqrt[{4}]{20}}}}{10}}\\[8pt]\varphi \left(e^{-12\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt[{4}]{1}}+{\sqrt[{4}]{2}}+{\sqrt[{4}]{3}}+{\sqrt[{4}]{4}}+{\sqrt[{4}]{9}}+{\sqrt[{4}]{18}}+{\sqrt[{4}]{24}}}}{2{\sqrt[{8}]{108}}}}\\[8pt]\varphi \left(e^{-16\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\left(4+{\sqrt[{4}]{128}}+{\sqrt[{4}]{1024{\sqrt[{4}]{8}}+1024{\sqrt[{4}]{2}}}}\right)}{16}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43896c679277838602c6386995c5595967b331bc)
![{\displaystyle {\begin{aligned}\vartheta _{4}^{2}(q)&=iq^{\frac {1}{4}}\sum _{k=-\infty }^{\infty }q^{2k^{2}-k}\vartheta _{1}\left({\frac {2k-1}{2i}}\ln q,q\right),\\[6pt]\vartheta _{4}^{2}(q)&=\sum _{k=-\infty }^{\infty }q^{2k^{2}}\vartheta _{4}\left({\frac {k\ln q}{i}},q\right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8c16f39ab9443e220062eaad77d207fd8ff5cb0)
![{\displaystyle {\begin{aligned}{\sqrt {\frac {\pi {\sqrt {e^{\pi }}}}{2}}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {3}{4}}\right)}}&=i\sum _{k=-\infty }^{\infty }e^{\pi \left(k-2k^{2}\right)}\vartheta _{1}\left({\frac {i\pi }{2}}(2k-1),e^{-\pi }\right),\\[6pt]{\sqrt {\frac {\pi }{2}}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {3}{4}}\right)}}&=\sum _{k=-\infty }^{\infty }{\frac {\vartheta _{4}\left(ik\pi ,e^{-\pi }\right)}{e^{2\pi k^{2}}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e539c4d92a7653e109301a20da7afb6e1325a1f)
![{\displaystyle {\begin{aligned}\vartheta (z,\tau )=\vartheta _{3}(z,\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau +{\frac {1}{2}}+{\frac {\tau }{2}}\\[3pt]\vartheta _{1}(z,\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau \\[3pt]\vartheta _{2}(z,\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau +{\frac {1}{2}}\\[3pt]\vartheta _{4}(z,\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau +{\frac {\tau }{2}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0a46570a37094335d8daa1755155d36ad316b6d)
![{\displaystyle {\begin{aligned}\theta _{2}(0,q)=\vartheta _{10}(0;\tau )&={\frac {2\eta ^{2}(2\tau )}{\eta (\tau )}},\\[3pt]\theta _{3}(0,q)=\vartheta _{00}(0;\tau )&={\frac {\eta ^{5}(\tau )}{\eta ^{2}\left({\frac {1}{2}}\tau \right)\eta ^{2}(2\tau )}}={\frac {\eta ^{2}\left({\frac {1}{2}}(\tau +1)\right)}{\eta (\tau +1)}},\\[3pt]\theta _{4}(0,q)=\vartheta _{01}(0;\tau )&={\frac {\eta ^{2}\left({\frac {1}{2}}\tau \right)}{\eta (\tau )}},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51aa288fda023bcc1b7d82366605089d3abd4800)
.
