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We derive a precise asymptotic expansion of the complete K\"{a}hler-Einsteinmetric on the punctured Riemann sphere with three or more omitting points. Byusing Schwarzian derivative, we prove that the coefficients of the expansionare polynomials on the two parameters which are uniquely determined by theomitting points. Futhermore, we use the modular form and Schwarzian derivativeto explicitly determine the coefficients in the expansion of the completeK\"{a}hler-Einstein metric for punctured Riemann sphere with $3, 4, 6$ or $12$omitting points.

Hyperbolic Metric, Punctured Riemann Sphere and Modular Functions