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We investigate the complex-temperature singularities of the susceptibility ofthe 2D Ising model on a square lattice. From an analysis of low-temperatureseries expansions, we find evidence that as one approaches the point $u=u_s=-1$(where $u=e^{-4K}$) from within the complex extensions of the FM or AFM phases,the susceptibility has a divergent singularity of the form $\chi \simA_s'(1+u)^{-\gamma_s'}$ with exponent $\gamma_s'=3/2$. The critical amplitude$A_s'$ is calculated. Other critical exponents are found to be$\alpha_s'=\alpha_s=0$ and $\beta_s=1/4$, so that the scaling relation$\alpha_s'+2\beta_s+\gamma_s'=2$ is satisfied. However, using exact results for$\beta_s$ on the square, triangular, and honeycomb lattices, we show thatuniversality is violated at this singularity: $\beta_s$ is lattice-dependent.Finally, from an analysis of spin-spin correlation functions, we demonstratethat the correlation length and hence susceptibility are finite as oneapproaches the point $u=-1$ from within the symmetric phase. This is confirmedby an explicit study of high-temperature series expansions.Comment: Latex file, 27 pages of text plus figures appended to file.  ITP-SB-94-37. (further results added to sections 4, 7

Complex-temperature singularities of the susceptibility in the D=2 Ising model. I. Square lattice