There is another approach to complex analysis that was initiated already by Lagrange but whose main development is due to Weierstrass. While the main application of Cauchy's thery is to prove that analytic functions have power series expansions, for Weierstrass the power series plays a fundamental rôle. It is possible to define an analytic function as a function that admits a convergent power series expansion. Then one sees that there is uniform convergence in disks and the next step is to prove that such a power series has all orders derivatives.
There is a simple integration-free proof that a power series can be derived term by term inside its radius of convergence, which already gives you that it is infinitely derivable and the power series for the derivatives have the same radius of convergence. This makes Weierstrass approach possible. Elementary functions like exponential and trigonometric functions can be defined by means of corresponding power series and then one shows its usual properties. Consequences of Cauchy's integral formula like Liouville's theorem or Cauchy's inequality can be recovered in this context without integration by means of Parseval's identity (which only involves real integration). And so on...
