An Accidental Hint at Euler's Equation

I had this thought at lunch today. It's not a sophisticated treatment of the subject, but I found it amusing.

We know that when you differentiate a sinusoid twice, you get back a scaled and negated version of the original. Here the scaling factor is unity.

$y = sin(t) \\ \ddot{y} = -sin(t) = -y$

We also know that double-differentiating an exponential gets us back a scaled (but not negated) version of the original. Here again, the scaling factor is unity.

$y = e^t \\ \ddot{y} = e^t = y$

These two behaviors are tantalizingly similar. So how might we get the exponential to behave like the sinusoid with respect to double-differentiation? Maybe we could throw in the square root of negative one.

$y = e^{j t} \\ \ddot{y} = j^2 e^{j t} = -e^{j t} = -y$

Making the exponent complex makes the exponential behave like a sinusoid with respect to double-differentiation. Hence, we've stumbled onto something like Euler's equation (shown below for reference).

$e^{j t } = cos \, t + j \, sin \, t$

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An Accidental Hint at Euler's Equation

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