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附录
这里对双正弦函数和与之相关的函数的性质做一个简单的介绍.双正弦函数的理论可以追溯到Barnes的论文.与量子二重对数函数之间紧密的联系是由Faddeev和Kashaev通过与量子反向散射方法给出的.双正弦函数的基本的性质已经在文献[13-15]中列了出来.
为了简单起见,引入符号ω=(ω1,ω2),其中要求ω1,ω2>0.双正弦函数S2(x|ω)可以通过一个环路积分来定义 S2(x|ω)=exp∫Csinh(x-ω1+ω22)t2sinhω1 t2sinhω2 t2ln(-t)dt2πit(A1)
定义成立的范围为 0<Re x<ω1+ω2(A2) 在这个环路积分中,积分路径是从∞+0i到0,然后绕过原点之后再到∞-0i.与此同时还有一个与(A1)等价的积分表示由如下式子给出: S2(x|ω)=exp(πi2B2,2(x|ω→))·exp∫R+0iexp(xt)(exp(ω1 t)-1)(exp(ω2 t)-1)dtt(A3)
式中, B2,2(x|ω)=x2ω1ω2-ω1+ω2ω1ω2 x+ω21+ω22+3ω1ω26ω1ω2(A4) 从式(A4)中, 可以注意到有这样的性质:B2,2(ω1+ω2-x|ω)=B2,2(x|ω). 同样的,还可以从式(A1)中得到另一个等价的积分表示: S2(x|ω)=exp∫∞0sinh(z-ω1+ω22)t2sinhω1 t2sinhω2 t2 -2x-ω1-ω2ω1ω2 tdtt(A5) 上述的两种积分表示在数值计算都非常难以计算,只能做解析证明时使用.所以如果希望能够做数值求解,就需要去将双正弦函数展开为一个收敛的数列求和.通过留数定理来计算式(A5),就可以得到双正弦函数的序列展开,适用范围是Im z>0或Im z<0. S2(x|ω)=
expπi2B2,2(x|ω→) + ∑∞n=11nexp(2πinxw1)exp(2πinw1w2)-1+exp(2πinxw1)exp(2πinw2w1)-1, Im(x)>0;
exp-πi2B2,2(x|ω→) + ∑∞n=11nexp(-2πinxw1)exp(-2πinw1w2)-1+exp(-2πinxw1)exp(-2πinw2w1)-1, Im(x)<0 (A6)
通过对积分路径C做一个合适的变换,可以将双正弦函数(A1)的定义域解析延拓至除了ω1ω2∈R-之外ω1,ω2取任意复数.同样的,在Imω1ω2>0区域内也可以得到一个乘积形式的展开: S2(x|ω)=expπi2B2,2(x|ω→)∏∞m=0(1-q2mexp(2πixω2))∏∞m=1(1-q-2mexp(2πixω1))=
exp(-πi2B2,2(x|ω→))∏∞m=0(1-q-2mexp( -2πixω1))∏∞m=1(1-q2mexp( -2πixω2))(A7)
式中, q=exp(πiω1ω2), q=exp(πiω2ω1)(A8) 由theta函数θ1(x|ω1ω2)的模变换法则可以得到式(A6)与(A7)是等价的.
双正弦函数S2(x|ω)的零点和极点是其非常重要的性质,他们的位置分别是 poles: x=n1ω1+n2ω2, n1,n2≥1; zeros: x=n1ω1+n2ω2, n1,n2≤0(A9)
并且在零点附近,函数满足如下的关系: limx→0 S2(x|ω)=2πxω1ω2(A10)
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