[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki23\/archives\/106914#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki23\/archives\/106914","headline":"\u30e4\u30b3\u30d3\u306e\u865a\u6570\u5909\u63db\u5f0f – Wikipedia","name":"\u30e4\u30b3\u30d3\u306e\u865a\u6570\u5909\u63db\u5f0f – Wikipedia","description":"before-content-x4 \u30e4\u30b3\u30d3\u306e\u865a\u6570\u5909\u63db\u5f0f(Jacobi’s imaginary transformation)\u306f\u3001\u6955\u5186\u30c6\u30fc\u30bf\u95a2\u6570\u306b\u95a2\u3059\u308b\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u6052\u7b49\u5f0f\u3067\u3042\u308b[1]\u3002 \u03d13(v\u03c4,\u22121\u03c4)=e\u2212\u03c0i\/4\u03c41\/2e\u03c0iv2\/\u03c4\u03d13(v,\u03c4){displaystyle vartheta _{3}left({frac {v}{tau }},-{frac {1}{tau }}right)=e^{-pi i\/4}tau ^{1\/2}e^{{pi }iv^{2}\/tau }vartheta _{3}left(v,tau right)} after-content-x4 \u03d11(v\u03c4,\u22121\u03c4)=\u2212ie\u2212\u03c0i\/4\u03c41\/2e\u03c0iv2\/\u03c4\u03d11(v,\u03c4){displaystyle vartheta","datePublished":"2022-03-23","dateModified":"2022-03-23","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki23\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki23\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7fd2c10f37fc7f6c433422b5881501da6a12e876","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7fd2c10f37fc7f6c433422b5881501da6a12e876","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/jp\/wiki23\/archives\/106914","about":["Wiki"],"wordCount":14516,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4\u30e4\u30b3\u30d3\u306e\u865a\u6570\u5909\u63db\u5f0f(Jacobi’s imaginary transformation)\u306f\u3001\u6955\u5186\u30c6\u30fc\u30bf\u95a2\u6570\u306b\u95a2\u3059\u308b\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u6052\u7b49\u5f0f\u3067\u3042\u308b[1]\u3002\u03d13(v\u03c4,\u22121\u03c4)=e\u2212\u03c0i\/4\u03c41\/2e\u03c0iv2\/\u03c4\u03d13(v,\u03c4){displaystyle vartheta _{3}left({frac {v}{tau }},-{frac {1}{tau }}right)=e^{-pi i\/4}tau ^{1\/2}e^{{pi }iv^{2}\/tau }vartheta _{3}left(v,tau right)}       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u03d11(v\u03c4,\u22121\u03c4)=\u2212ie\u2212\u03c0i\/4\u03c41\/2e\u03c0iv2\/\u03c4\u03d11(v,\u03c4){displaystyle vartheta _{1}left({frac {v}{tau }},-{frac {1}{tau }}right)=-ie^{-pi i\/4}tau ^{1\/2}e^{{pi }iv^{2}\/tau }vartheta _{1}left(v,tau right)}       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u03d12(v\u03c4,\u22121\u03c4)=e\u2212\u03c0i\/4\u03c41\/2e\u03c0iv2\/\u03c4\u03d14(v,\u03c4){displaystyle vartheta _{2}left({frac {v}{tau }},-{frac {1}{tau }}right)=e^{-pi i\/4}tau ^{1\/2}e^{{pi }iv^{2}\/tau }vartheta _{4}left(v,tau right)}\u03d14(v\u03c4,\u22121\u03c4)=e\u2212\u03c0i\/4\u03c41\/2e\u03c0iv2\/\u03c4\u03d12(v,\u03c4){displaystyle vartheta _{4}left({frac {v}{tau }},-{frac {1}{tau }}right)=e^{-pi i\/4}tau ^{1\/2}e^{{pi }iv^{2}\/tau }vartheta _{2}left(v,tau right)}\u3053\u306e\u6052\u7b49\u5f0f\u306e\u65e5\u672c\u8a9e\u306e\u547c\u79f0\u306f\u5b9a\u307e\u3063\u3066\u304a\u3089\u305a\u3001\u30e4\u30b3\u30d3\u306e\u865a\u6570\u5909\u63db\u5f0f\u3001\u30e4\u30b3\u30d3\u306e\u30e2\u30b8\u30e5\u30e9\u30fc\u5909\u63db\u5f0f\u3001\u3042\u308b\u3044\u306f\u5358\u306b\u30e4\u30b3\u30d3\u5909\u63db\u5f0f\u3068\u3082\u547c\u3070\u308c\u308b\u3002\u30c6\u30fc\u30bf\u95a2\u6570\u306f\u4e8c\u5909\u6570\u306e\u95a2\u6570\u3067\u3042\u308b\u304c\u3001\u7b2c\u4e8c\u5909\u6570\u3092\u7d14\u865a\u6570\u306e\u5b9a\u6570\u3068\u3057\u3066\u7b2c\u4e00\u5909\u6570\u306b\u7740\u76ee\u3059\u308c\u3070\u300c\u865a\u6570\u5909\u63db\u5f0f\u300d\u3068\u3044\u3046\u547c\u79f0\u304c\u7684\u3092\u5c04\u3066\u3001\u7b2c\u4e00\u5909\u6570\u3092\u5b9a\u6570\u3068\u3057\u3066\u7b2c\u4e8c\u5909\u6570\u306b\u7740\u76ee\u3059\u308c\u3070\u300c\u30e2\u30b8\u30e5\u30e9\u30fc\u5909\u63db\u5f0f\u300d\u3068\u3044\u3046\u547c\u79f0\u304c\u7684\u3092\u5c04\u308b\u3002\u516c\u5f0f\u306b\u95a2\u3059\u308b\u6ce8\u610f\u70b9[\u7de8\u96c6]\u03b8i(z,\u03c4){displaystyle theta _{i}(z,tau )}\u306e\u5b9a\u7fa9\u306f\u4e00\u610f\u3067\u306f\u306a\u304f\u3001\u3044\u304f\u3064\u304b\u306e\u6d41\u5100\u304c\u3042\u308a\u6587\u732e\u306b\u3088\u3063\u3066\u7570\u306a\u308b\u306e\u3067\u6ce8\u610f\u304c\u5fc5\u8981\u3067\u3042\u308b[2](\u4e3b\u3068\u3057\u3066\u3001\u03b8ij(z,\u03c4){displaystyle theta _{ij}(z,tau )}\u306e\u5b9a\u7fa9\u306e\u9055\u3044\u304c\u6df7\u4e71\u3092\u751f\u3093\u3067\u3044\u308b)\u3002\u3053\u306e\u8a18\u4e8b\u3067\u306e\u5b9a\u7fa9\u306f\u3001D.Mumford\u306b\u5f93\u3063\u305f[3]\u6b21\u306e\u3088\u3046\u306a\u3082\u306e\u3067\u3042\u308b[2][4]\u3002\u03b80(z,\u03c4):=\u03b801(z,\u03c4):=\u2211n=\u2212\u221e\u221ee\u03c0i\u03c4n2+2\u03c0in(z+12)=1+2\u2211n=1\u221e(\u22121)ne\u03c0i\u03c4n2cos\u20612n\u03c0z,\u03b81(z,\u03c4):=\u2212\u03b811(z,\u03c4):=\u2212\u2211n=\u2212\u221e\u221ee\u03c0i\u03c4(n+12)2+2\u03c0i(n+12)(z+12)=2\u2211n=0\u221e(\u22121)ne\u03c0i\u03c4(n+12)2sin\u2061(2n+1)\u03c0z,\u03b82(z,\u03c4):=\u03b810(z,\u03c4):=\u2211n=\u2212\u221e\u221ee\u03c0i\u03c4(n+12)2+2\u03c0i(n+12)z=2\u2211n=0\u221ee\u03c0i\u03c4(n+12)2cos\u2061(2n+1)\u03c0z,\u03b83(z,\u03c4):=\u03b800(z,\u03c4):=\u2211n=\u2212\u221e\u221ee\u03c0i\u03c4n2+2\u03c0inz=1+2\u2211n=1\u221ee\u03c0i\u03c4n2cos\u20612n\u03c0z.{displaystyle {begin{aligned}theta _{0}(z,tau )&:=theta _{01}(z,tau )\\&:=sum _{n=-infty }^{infty }e^{pi itau n^{2}+2pi inleft(z+{frac {1}{2}}right)}\\&=1+2sum _{n=1}^{infty }(-1)^{n}e^{pi itau n^{2}}cos 2npi z,\\theta _{1}(z,tau )&:=-theta _{11}(z,tau )\\&:=-sum _{n=-infty }^{infty }{e^{pi itau left(n+{frac {1}{2}}right)^{2}+2pi ileft(n+{frac {1}{2}}right)left(z+{frac {1}{2}}right)}}\\&=2sum _{n=0}^{infty }(-1)^{n}e^{pi itau left(n+{frac {1}{2}}right)^{2}}sin(2n+1)pi z,\\theta _{2}(z,tau )&:=theta _{10}(z,tau )\\&:=sum _{n=-infty }^{infty }e^{pi itau left(n+{frac {1}{2}}right)^{2}+2pi ileft(n+{frac {1}{2}}right)z}\\&=2sum _{n=0}^{infty }e^{pi itau left(n+{frac {1}{2}}right)^{2}}cos(2n+1)pi z,\\theta _{3}(z,tau )&:=theta _{00}(z,tau )\\&:=sum _{n=-infty }^{infty }e^{pi itau n^{2}+2pi inz}\\&=1+2sum _{n=1}^{infty }e^{{pi }i{tau }n^{2}}cos 2npi z.end{aligned}}}\u300c\u5ca9\u6ce2\u6570\u5b66\u516c\u5f0f\u96c6\u2162\u300dp.48.\u3067\u306f\u8aa4\u3063\u305f\u5f0f\u304c\u66f8\u304b\u308c\u3066\u3044\u308b\u306e\u3067\u6ce8\u610f\u305b\u3088\u3002\u6955\u5186\u95a2\u6570\u306e\u865a\u6570\u5909\u63db[\u7de8\u96c6]\u30e4\u30b3\u30d3\u306e\u6955\u5186\u95a2\u6570\u306f\u30c6\u30fc\u30bf\u95a2\u6570\u306e\u6bd4\u306b\u3088\u308a\u8868\u3055\u308c\u308b\u3002\u6955\u5186\u95a2\u6570\u306e\u5468\u671f\u3092       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4K,iK\u2032{displaystyle K,iK’}\u3068\u3059\u308b\u3068\u03c4=iK\u2032K{displaystyle tau ={frac {iK’}{K}}}k=(\u03d12(0,\u03c4)\u03d13(0,\u03c4))2{displaystyle k=left({frac {vartheta _{2}(0,tau )}{vartheta _{3}(0,tau )}}right)^{2}}sn\u2061(u,k)=\u03d13(0,\u03c4)\u03d11(u\/2K,\u03c4)\u03d12(0,\u03c4)\u03d14(u\/2K,\u03c4){displaystyle operatorname {sn} (u,k)={frac {vartheta _{3}(0,tau )vartheta _{1}(u\/2K,tau )}{vartheta _{2}(0,tau )vartheta _{4}(u\/2K,tau )}}}cn\u2061(u,k)=\u03d14(0,\u03c4)\u03d12(u\/2K,\u03c4)\u03d12(0,\u03c4)\u03d14(u\/2K,\u03c4){displaystyle operatorname {cn} (u,k)={frac {vartheta _{4}(0,tau )vartheta _{2}(u\/2K,tau )}{vartheta _{2}(0,tau )vartheta _{4}(u\/2K,tau )}}}\u30c6\u30fc\u30bf\u95a2\u6570\u306e\u865a\u6570\u5909\u63db\u5f0f\u306b\u3088\u308a\u03c4\u2032=\u22121\u03c4=iKK\u2032{displaystyle tau ‘=-{frac {1}{tau }}={frac {iK}{K’}}}k\u2032=(\u03d12(0,\u03c4\u2032)\u03d13(0,\u03c4\u2032))2{displaystyle k’=left({frac {vartheta _{2}(0,tau ‘)}{vartheta _{3}(0,tau ‘)}}right)^{2}}sn\u2061(iu,k)=\u03d13(0,\u03c4)\u03d11(iu\/2K,\u03c4)\u03d12(0,\u03c4)\u03d14(iu\/2K,\u03c4)=i\u03d13(0,\u03c4\u2032)\u03d11(u\/2K\u2032,\u03c4\u2032)\u03d14(0,\u03c4\u2032)\u03d12(u\/2K\u2032,\u03c4\u2032)=isn\u2061(u,k\u2032)cn\u2061(u,k\u2032){displaystyle operatorname {sn} (iu,k)={frac {vartheta _{3}(0,tau )vartheta _{1}(iu\/2K,tau )}{vartheta _{2}(0,tau )vartheta _{4}(iu\/2K,tau )}}={frac {ivartheta _{3}(0,tau ‘)vartheta _{1}(u\/2K’,tau ‘)}{vartheta _{4}(0,tau ‘)vartheta _{2}(u\/2K’,tau ‘)}}=i{frac {operatorname {sn} (u,k’)}{operatorname {cn} (u,k’)}}}cn\u2061(iu,k)=\u03d14(0,\u03c4)\u03d12(iu\/2K,\u03c4)\u03d12(0,\u03c4)\u03d14(iu\/2K,\u03c4)=\u03d12(0,\u03c4\u2032)\u03d14(u\/2K\u2032,\u03c4\u2032)\u03d14(0,\u03c4\u2032)\u03d12(u\/2K\u2032,\u03c4\u2032)=1cn\u2061(u,k\u2032){displaystyle operatorname {cn} (iu,k)={frac {vartheta _{4}(0,tau )vartheta _{2}(iu\/2K,tau )}{vartheta _{2}(0,tau )vartheta _{4}(iu\/2K,tau )}}={frac {vartheta _{2}(0,tau ‘)vartheta _{4}(u\/2K’,tau ‘)}{vartheta _{4}(0,tau ‘)vartheta _{2}(u\/2K’,tau ‘)}}={frac {1}{operatorname {cn} (u,k’)}}}\u3068\u306a\u308a\u3001\u6955\u5186\u95a2\u6570\u306e\u865a\u6570\u5909\u6570\u3092\u5f97\u308b\u3002\u03d13(v,\u03c4){displaystyle vartheta _{3}(v,tau )}\u306e\u865a\u6570\u5909\u63db\u5f0f\u306e\u4e21\u8fba\u306e\u6bd4\u3092f(v,\u03c4){displaystyle f(v,tau )}\u3057\u3066\u6052\u7b49\u7684\u306bf(v,\u03c4)=1{displaystyle f(v,tau )=1}\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b\u3002\u30c6\u30fc\u30bf\u95a2\u6570\u306e\u4e8c\u91cd\u5468\u671f\u6027\u306b\u3088\u308af(v,\u03c4)=\u2212i\u03c4e\u03c0iv2\/\u03c4\u03d13(v,\u03c4)\u03d13(v\u03c4,\u22121\u03c4){displaystyle f(v,tau )={frac {{sqrt {-itau }}e^{{pi }iv^{2}\/tau }vartheta _{3}left(v,tau right)}{vartheta _{3}left({frac {v}{tau }},-{frac {1}{tau }}right)}}}f(v+1,\u03c4)=\u2212i\u03c4e\u03c0iv2\/\u03c4+2\u03c0iv\/\u03c4+\u03c0i\/\u03c4\u03d13(v+1,\u03c4)\u03d13(v\u03c4+1\u03c4,\u22121\u03c4)=\u2212i\u03c4e\u03c0iv2\/\u03c4+2\u03c0iv\/\u03c4+\u03c0i\/\u03c4\u03d13(v,\u03c4)e\u03c0i\/\u03c4+2\u03c0iv\/\u03c4\u03d13(v\u03c4,\u22121\u03c4)=f(v,\u03c4){displaystyle f(v+1,tau )={frac {{sqrt {-itau }}e^{{pi }iv^{2}\/tau +2{pi }iv\/tau +{pi }i\/tau }vartheta _{3}left(v+1,tau right)}{vartheta _{3}left({frac {v}{tau }}+{frac {1}{tau }},-{frac {1}{tau }}right)}}={frac {{sqrt {-itau }}e^{{pi }iv^{2}\/tau +2{pi }iv\/tau +{pi }i\/tau }vartheta _{3}left(v,tau right)}{e^{{pi }i\/tau +2{pi }iv\/tau }vartheta _{3}left({frac {v}{tau }},-{frac {1}{tau }}right)}}=f(v,tau )}f(v+\u03c4,\u03c4)=\u2212i\u03c4e\u03c0i(v+\u03c4)2\/\u03c4\u03d13(v+\u03c4,\u03c4)\u03d13(v\u03c4+1,\u22121\u03c4)=\u2212i\u03c4e\u03c0iv2\/\u03c4+2\u03c0iv+\u03c0i\u03c4e\u2212\u03c0i\u03c4\u22122\u03c0iv\u03d13(v,\u03c4)\u03d13(v\u03c4,\u22121\u03c4)=f(v,\u03c4){displaystyle f(v+tau ,tau )={frac {{sqrt {-itau }}e^{{pi }i(v+tau )^{2}\/tau }vartheta _{3}left(v+tau ,tau right)}{vartheta _{3}left({frac {v}{tau }}+1,-{frac {1}{tau }}right)}}={frac {{sqrt {-itau }}e^{{pi }iv^{2}\/tau +2{pi }iv+{pi }itau }e^{-{pi }itau -2{pi }iv}vartheta _{3}left(v,tau right)}{vartheta _{3}left({frac {v}{tau }},-{frac {1}{tau }}right)}}=f(v,tau )}\u3067\u3042\u308b\u304b\u3089\u3001f(v,\u03c4){displaystyle f(v,tau )}\u306fv{displaystyle v}\u306e\u95a2\u6570\u3068\u3057\u3066\u4e8c\u91cd\u5468\u671f\u3092\u6301\u3064\u3002\u307e\u305f\u3001\u30c6\u30fc\u30bf\u95a2\u6570\u306f\u6975\u3092\u6301\u305f\u305a\u3001\u96f6\u70b9\u306f\u03d13(1\u00b12m2+(1\u00b12n)\u03c42,\u03c4)=0{displaystyle vartheta _{3}left({frac {1pm {2m}}{2}}+{frac {(1pm {2n})tau }{2}},tau right)=0}\u03d13(1\u00b12m2\u22121\u00b12n2\u03c4,\u22121\u03c4)=0{displaystyle vartheta _{3}left({frac {1pm {2m}}{2}}-{frac {1pm {2n}}{2tau }},-{frac {1}{tau }}right)=0}\u3067\u3042\u308b\u304b\u3089\u3001f(v,\u03c4){displaystyle f(v,tau )}\u306fv{displaystyle v}\u306e\u95a2\u6570\u3068\u3057\u3066\u8907\u7d20\u5e73\u9762\u5168\u4f53\u3067\u6709\u754c\u3067\u3042\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u30ea\u30a6\u30f4\u30a3\u30eb\u306e\u5b9a\u7406\u306b\u3088\u308av{displaystyle v}\u306b\u306f\u4f9d\u5b58\u3057\u306a\u3044\u3002f(12,\u03c4)=\u2212i\u03c4e\u03c0i\/4\u03c4\u03d13(12,\u03c4)\u03d13(12\u03c4,\u22121\u03c4)=\u2212i\u03c4\u2211n=\u2212\u221e\u221een2\u03c0i\u03c4+n\u03c0i\u2211n=\u2212\u221e\u221een2\u2212\u03c0i\/\u03c4+n\u03c0i\/\u03c4e\u2212\u03c0i\/4\u03c4=\u2212i\u03c4\u2211n=\u2212\u221e\u221e(\u22121)nen2\u03c0i\u03c4\u2211n=\u2212\u221e\u221ee\u2212(2n\u22121)2\u03c0i\/4\u03c4=\u2212i\u03c4\u2211n=\u2212\u221e\u221e(\u22121)nen2\u03c0i\u03c42\u2211n=1\u221ee\u2212(2n\u22121)2\u03c0i\/4\u03c4{displaystyle {begin{aligned}fleft({frac {1}{2}},tau right)&={frac {{sqrt {-itau }}e^{{pi }i\/4tau }vartheta _{3}left({frac {1}{2}},tau right)}{vartheta _{3}left({frac {1}{2tau }},-{frac {1}{tau }}right)}}={frac {{sqrt {-itau }}sum _{n=-infty }^{infty }{e^{n^{2}{pi }itau +n{pi }i}}}{sum _{n=-infty }^{infty }{e^{n^{2}{-pi }i\/tau +n{pi }i\/tau }e^{-{pi }i\/4tau }}}}\\&={frac {{sqrt {-itau }}sum _{n=-infty }^{infty }{(-1)^{n}e^{n^{2}{pi }i{tau }}}}{sum _{n=-infty }^{infty }{e^{-(2n-1)^{2}{pi }i\/4tau }}}}\\&={frac {{sqrt {-itau }}sum _{n=-infty }^{infty }{(-1)^{n}e^{n^{2}{pi }i{tau }}}}{2sum _{n=1}^{infty }{e^{-(2n-1)^{2}{pi }i\/4tau }}}}\\end{aligned}}}f(14,\u03c44)=\u2212i(\u03c4\/4)e\u03c0i\/4\u03c4\u03d13(14,\u03c44)\u03d13(1\u03c4,\u22124\u03c4)=\u2212i\u03c4\u2211n=\u2212\u221e\u221een2\u03c0i\u03c4\/4+n\u03c0i\/22\u2211n=\u2212\u221e\u221ee\u22124n2\u03c0i\/\u03c4+2n\u03c0i\/\u03c4e\u2212\u03c0i\/4\u03c4=\u2212i\u03c4\u2211n=\u2212\u221e\u221einen2\u03c0i\u03c4\/42\u2211n=\u2212\u221e\u221ee\u2212(2n\u22121\/2)2\u03c0i\/\u03c4=\u2212i\u03c4\u2211n=\u2212\u221e\u221einen2\u03c0i\u03c4\/42(\u2211n=1\u221ee\u2212(2n\u22121\/2)2\u03c0i\/\u03c4+\u2211n=1\u221ee\u2212(\u22122n+1\u22121\/2)2\u03c0i\/\u03c4)=\u2212i\u03c4\u2211n=\u2212\u221e\u221einen2\u03c0i\u03c4\/42\u2211n=1\u221ee\u2212(n\u22121\/2)2\u03c0i\/\u03c4{displaystyle {begin{aligned}fleft({frac {1}{4}},{frac {tau }{4}}right)&={frac {{sqrt {-i(tau \/4)}}e^{{pi }i\/4tau }vartheta _{3}left({frac {1}{4}},{frac {tau }{4}}right)}{vartheta _{3}left({frac {1}{tau }},-{frac {4}{tau }}right)}}={frac {{sqrt {-itau }}sum _{n=-infty }^{infty }{e^{n^{2}{pi }itau \/4+n{pi }i\/2}}}{2sum _{n=-infty }^{infty }{e^{{-4n^{2}pi }i\/tau +2n{pi }i\/tau }e^{-{pi }i\/4tau }}}}\\&={frac {{sqrt {-itau }}sum _{n=-infty }^{infty }{i^{n}e^{n^{2}{pi }i{tau }\/4}}}{2sum _{n=-infty }^{infty }{e^{-(2n-1\/2)^{2}{pi }i\/tau }}}}={frac {{sqrt {-itau }}sum _{n=-infty }^{infty }{i^{n}e^{n^{2}{pi }i{tau }\/4}}}{2left(sum _{n=1}^{infty }{e^{-(2n-1\/2)^{2}{pi }i\/tau }}+sum _{n=1}^{infty }{e^{-(-2n+1-1\/2)^{2}{pi }i\/tau }}right)}}\\&={frac {{sqrt {-itau }}sum _{n=-infty }^{infty }{i^{n}e^{n^{2}{pi }i{tau }\/4}}}{2sum _{n=1}^{infty }{e^{-(n-1\/2)^{2}{pi }i\/tau }}}}\\end{aligned}}}\u5206\u5b50\u306en{displaystyle n}\u304c\u5947\u6570\u306e\u9805\u306f\u6b63\u8ca0\u3067\u6253\u3061\u6d88\u3057\u3042\u3046\u304b\u3089\u5076\u6570\u306en{displaystyle n}\u30922n{displaystyle 2n}\u306b\u6539\u3081\u308b\u3002f(14,\u03c44)=\u2212i\u03c4\u2211n=\u2212\u221e\u221ei2ne(2n)2\u03c0i\u03c4\/42\u2211n=1\u221ee\u2212(n\u22121\/2)2\u03c0i\/\u03c4=\u2212i\u03c4\u2211n=\u2212\u221e\u221e(\u22121)nen2\u03c0i\u03c42\u2211n=1\u221ee\u2212(n\u22121\/2)2\u03c0i\/\u03c4=f(12,\u03c4){displaystyle {begin{aligned}fleft({frac {1}{4}},{frac {tau }{4}}right)&={frac {{sqrt {-itau }}sum _{n=-infty }^{infty }{i^{2n}e^{(2n)^{2}{pi }i{tau }\/4}}}{2sum _{n=1}^{infty }{e^{-(n-1\/2)^{2}{pi }i\/tau }}}}={frac {{sqrt {-itau }}sum _{n=-infty }^{infty }{(-1)^{n}e^{n^{2}{pi }i{tau }}}}{2sum _{n=1}^{infty }{e^{-(n-1\/2)^{2}{pi }i\/tau }}}}=fleft({frac {1}{2}},tau right)\\end{aligned}}}\u5148\u306b\u793a\u3057\u305f\u3088\u3046\u306bf(v,\u03c4){displaystyle f(v,tau )}\u306fv{displaystyle v}\u306b\u4f9d\u5b58\u3057\u306a\u3044\u306e\u3067f(v,\u03c4)=f(v,\u03c44)=limn\u2192\u221ef(v,\u03c44n)=lim\u03c4\u2032\u21920f(v,\u03c4\u2032)=f(v,0){displaystyle fleft(v,tau right)=fleft(v,{frac {tau }{4}}right)=lim _{nto infty }fleft(v,{frac {tau }{4^{n}}}right)=lim _{tau ‘to 0}fleft(v,tau ‘right)=f(v,0)}\u3067\u3042\u308a\u3001f(v,\u03c4){displaystyle f(v,tau )}\u306f\u03c4{displaystyle tau }\u306b\u3082\u4f9d\u5b58\u3057\u306a\u3044\u5b9a\u6570\u3067\u3042\u308b\u3002\u305d\u306e\u5024\u306ff(v,\u03c4)=f(0,i)=\u03d13(0,i)\u03d13(0,i)=1{displaystyle f(v,tau )=f(0,i)={frac {vartheta _{3}left(0,iright)}{vartheta _{3}left(0,iright)}}=1}\u3067\u3042\u308b\u3002     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki23\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki23\/archives\/106914#breadcrumbitem","name":"\u30e4\u30b3\u30d3\u306e\u865a\u6570\u5909\u63db\u5f0f – Wikipedia"}}]}]