The Euler formula

Euler's formula is a beautiful mathematical relationship that links complex exponential functions to trigonometry. It states that for any real number \( x \), \( e^{ix} = \cos x + i \sin x \). Here, \( e \) is the base of natural logarithms, \( i \) represents the imaginary unit, and \( \cos \) and \( \sin \) are the cosine and sine functions. Essentially, this formula shows how complex exponential functions encode rotations and oscillations, providing a powerful way to analyze waves, signals, and many areas in engineering and physics in a unified framework.