Complex numbers Expressed in Exponential Form | Proof скачать в хорошем качестве

Complex numbers Expressed in Exponential Form | Proof

In this video, I explore complex numbers in exponential form and prove the fundamental equation: z = re^(iθ). I begin with a recap of converting a complex number from rectangular form (z = x + iy) to polar (modulus-argument) form: z = r(cos θ + i sin θ). Then, I demonstrate how the Maclaurin series for cos θ, sin θ, and e^x can be used to prove that a complex number can be expressed in exponential form. By substituting x = iθ into the e^x series and carefully grouping the real and imaginary terms, I reveal the connection between these series and derive Euler's formula: e^(iθ) = cos θ + i sin θ. Multiplying both sides by r leads us to the exponential form. Finally, I walk through a practical example, showing how to express a complex number in exponential form by calculating its modulus and argument.