Generalized Kramers–Kronig (K–K) type dispersion relations and sum rules are derived in the static limit for the moments of the degenerate four wave mixing susceptibility. The degenerate nonlinear susceptibility is different from a typical use of the conventional K–K dispersion relations, which assume absence of complex poles of a function in the upper half of complex frequency plane, whereas degenerate susceptibility has simultaneous poles in both half planes. In the derivation of the generalized K–K relations the poles and their order are taken into account by utilization of the theorem of residues. The conventional K–K relations can be used to estimate the real and imaginary parts of the second and higher powers of the susceptibility as the effect of the poles is reduced due to a faster convergence of the dispersion relations. The present theory is directly applicable to higher order susceptibilities and can be used in testing of theoretical models describing the degenerate four wave mixing susceptibility in nonlinear optical and terahertz spectroscopy.

在简并四波混合磁化率的矩的静态极限中导出了广义克莱默斯-克罗尼格 (K-K) 型色散关系和求和规则。简并非线性磁化率不同于传统 K-K 色散关系的典型用法，后者假设复频率平面上半部分不存在函数的复极点，而简并磁化率在两个半平面中同时具有极点。在推导广义 K-K 关系时，利用余数定理考虑了极点及其阶次。传统的 K-K 关系可用于估计磁化率的二次幂和更高次幂的实部和虚部，因为极点的影响由于色散关系的收敛速度更快而降低。