Review Formula Euler: Pembuktian dan Aplikasinya

Abstract

Euler’s formula  provides the fundamental relation between trigonometric functions and complex exponential functions. This formula, which is called “the most remarkable formula in mathematics”, can be proven using various approaches other than the “original” proof via Maclaurin series expansion. In this paper, six ways of proving Euler’s formula are given using the methods of series expansion, differentiation, differential equation, integration and trigonometric identities, complex number representation, and uniqueness solutions of differential equations to provide various perspectives regarding the relationship of this formula to topics in the field of mathematics. Several applications in the field of mathematics and electrical engineering are discussed. In the field of trigonometry, Euler’s formula can be used to prove the identity of double angles,  and , the identity of the sum of angles,  dan , as well as the values of complex trigonometry. By applying Euler’s formula, a practical formula is obtained for calculating complex powers of real numbers and a one-liner proof of the de Moivre’s formula. The impedance of a capacitor can also be calculated using the Euler formula. In the analysis of RC and RLC circuits, Euler’s formula can be used to produce an equation of the relationship between input voltage and capacitor voltage, namely that the capacitor voltage is the result of multiplying the input voltage by a complex number.