Introduction

e^(iθ) = (cosθ + isinθ)

Where: θ=wt

e^(iwt) = [cos(wt) + isin(wt)]

Euler's formula traces a circle in the complex number plane as the value of t increases over time . So it applies to oscillating motion which has an angular frequency(w) that traces a circle, as well as rotating and circular motion. The total expression is the exponential function which has a real component on the real horizontal axis, and an imaginary component on the imaginary(i) vertical axis. The Taylor series for this exponential function reveals the fact that the powers of i, are the reason this exponential function traces a perfect circle in the complex number plane.

Where:

[i⁰=1], [i¹=i], [i²=-1], [i³=-i]

The negative coefficients keep the function from experiencing exponential growth; and the coefficient sequence [1, i, -1, -i] repeats every four terms and corresponds to a 90° vector rotation in the complex number plane for each term. Which completes a full circle of 360°.

The fact that an exponential function can describe a circular geometry is remarkable. But also the fact that the derivative of e^(ax) with respect to x is ae^(ax) makes the function easy to work with. So Euler's formula and the complex number plane are powerful mathematical tools.





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