Its first published proof used the calculus of variations, although a letter to Bessel shows that Gauss was aware of the result 14 years earlier. In an 1883 article, a later mathematician offered a proof that removed the unstated hypothesis that the 
Its first published proof used the calculus of variations, although a letter to Bessel shows that Gauss was aware of the result 14 years earlier. In an 1883 article, a later mathematician offered a proof that removed the unstated hypothesis that the function's derivative must be continuous; however, the existence and continuity of all derivatives follow from an important corollary. FTP, name this fundamental result of complex analysis, which states that the integral of an analytic function around a closed loop is 0.
Its first published proof used the calculus of variations, although a letter to Bessel shows that Gauss was aware of the result 14 years earlier.
Cauchy-Goursat Theorem or Cauchy's Integral Theorem (do not accept Cauchy's Integral Formula)