A Novel Complex Integration Derivation of the Leibniz Series for π

We present a novel derivation of the classical Leibniz series for π using complex integration techniques. Starting from the well-known Gregory-Leibniz series π/4 = Σ((-1)^n)/(2n+1), we develop an alternative derivation pathway through complex integration from 0 to i in the complex plane. Our approach systematically transforms the infinite series into a complex integral, ultimately expressing the result in terms of complex logarithms: π/4 = (1/2i)ln((1+i)/(1-i)) = (1/2i)ln(e^(iπ/2)). This derivation provides a direct connection between the Leibniz series and complex logarithmic functions through contour integration, offering new geometric and analytical insights into this classical result.