Ahh, what the hell; it merits explanation, LOL.

A few centuries ago, John Bernoulli stumbled upon a mysterious relation between an imaginary number and a complex logarithm after dispensing his partial fraction decomposition and subsequent integration. There’s no historical record of such, but chances are he may have wanted to determine some instantaneous quantity such as a displacement of a spring. I could never commit the exact timeframe to memory, but this discovery may be concurrent, in time, to Joules discovery of energy, adding to the list of possible instantaneous quantities Bernoulli may have been hunting. Subsequent mathematicians took the rectangular form from the complex logarithm’s argument and transformed it to a polar form. However, the relation wasn’t meaningful and no practical solution to the hypothetical application could be arrived at. Further exacerbating the relation is that a complex logarithm is not unique; it has infinitely many solutions as the function is coterminal like the sinusoids are. That is, its projection on the complex plane is the unit circle; take a ratio at an arbitrary angle, advance 360 degrees or 6.28 radians, you then end up with the same ratio. In complex space, a complex logarithm forms a gradually nondecreasing, monotonic helix.

Euler, intuitively hypothesized that perhaps the inverse relation would contrive a path to a meaningful set of solutions inasmuch as its power series is globally convergent. But being the formal man he was, he took the MacLaurin series of this inverse (2.713 or natural basee) and coupled it with a special case of De Moivre’s Theorem to generate the powers of the imaginary unit. From basic complex number definitions and limits, it can be shown that the cosine and sine argument is 0.5*3.14*nin this case wherenis zero and the set of all positive integers, Z+. Subsequent expansion and associativity of the even and odd exponentials and factorials lead to what is known today as Euler’s Formula. The sine series reflects the fact that the function is odd whereas the cosine series reflects the fact that it’s an even function.

Armed with his formula, he tackled a second order differential equation that has complex conjugate solutions in the form of the aforementioned natural exponential function. Again, they aren’t meaningful because they’re complex. Euler realized he could transform the complex conjugates to real conjugates via his formula bearing in mind that each real solution constitutes half of the full solution and that each linear combination formed by the system of Euler equations (using one negative argument and one positive argument for the cosine and sine) constitutes a new solution because of the preservation of the conjugates. One solution will be intrinsically real while the other will be intrinsically imaginary, but when the math is done they both turn out real in the end based on the properties of the imaginary unit. The fact that the sines have amplitudes is a consequence of the Wronskian determinant of the solutions. For the solutions to be linearly independent, the Wronskian determinant must be nonzero. By inspection, the determinant of solutions forms the sine Pythagorean identity, which is, indeed, nonzero.

One interesting thing to note is he first applied this to a table leg he treated as a slender column. The deflection at buckling is a sinusoid and he proceeded to derive yet another formula … Euler’s Buckling Load Formula for columns. The applications of all of the aforementioned development are abundant: vibrations, wave mechanics, alternating current, probabilistic quantum mechanics … United States speed patterns in traffic, LMAO. I remember a Project Management course I took my senior year. Just about everybody fell asleep in the class whereas I would keep myself awake by deriving theorems like Euler’s Formula, 1st and 2nd FTC, the dif MVT, the integral MVT, trig identities, etc. LOL, kept me awake.