Euler's Formula

SINE is a self-loving curve. Just as to caress oneself is to live the changes in one's skin, to be the nerves that register motion, the sine curve reproduces itself in derivative. The cosine sounds and feels like a complement to the sine, but it's really just the same sinewy sinusoid shifted along the straight stretch a quarter crank: Sines are self-pleasuring beasts. It takes them four tries to find the straight and narrow: like a baseball diamond, they have to turn the corner four times before they find their way home.

Reciproals are quicker lovers. Up, down, see you 'round. Two shifts will get you back where you started, but a reciprocal is really a power of minus one, and Euler's staunch constant is the bearer of powers. Derivatives can't hurt this old fellow; they only extrude his juice. If -1 is his juice, two steps is his programme.

So how can we introduce Euler to the sinusoid? They're playboys in different worlds, one always dancing in 2:2 time, one in 4:4. The answer is: slow down the music! Not so fast, Mr. E! What's an exponent that takes four steps to self-replicate? Why, we've got to chop that reciprocal geometrically in half, and that means taking the square root of -1. i, carumba!

So it's the apple of e's i we seek. Slip that ghost of an exponent into that faithful e to the x and watch it churn. e to the x. e to the minus x. e to the minus ix. Now it's just e again. All the while trailing clear coeffiecents like a poorly conceived hybrid.

But maybe not so unsustainable, because i's coalesce into minus signs and minus signs into plus signs and plus signs, as we all know, are reabsorbed by the environment. So it goes. This engine now chugs in time with the sine and his sister, and it's time to join them. Here goes, simplicity is a virtue. e^ix = cos x + i sin x. Simple enough? No frills, no worries? Good. Now stuff that sucker with pi and you're ready for a mesmerizing Thanksgiving.