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Euler’s Formula

Euler’s Formula

In the discussion above, did you notice that the Taylor / Maclaurin series for sine, cosines, and exponents looked similar? Euler noticed this too, and is credited for discovering how these three functions are related to each other. If you measure angles in radians, then Euler’s formula states \[{e^{jx}} = \cos (x) + j\sin (x)\]

Here are a couple of other ways to write Euler’s formula, which may come in handy. Remember, x must be in radians. \[\begin{array}{l} cos(x) = \frac{1}{2}\left( {{e^{jx}} - {e^{ - jx}}} \right)\\ sin(x) = \frac{1}{{2j}}\left( {{e^{jx}} + {e^{ - jx}}} \right) \end{array}\]

Mathematicians consider Euler’s formula particularly beautiful (they actually use that term) if you set the angle equal to pi radians. Then you get Euler’s identity \[{e^{j\pi }} = - 1\]

What seems remarkable about this identity, is that it ties together concepts that originated from very different problems. The j term came from solving polynomial equations. The pi term is from geometric problems involving circles. The e term comes from looking at the rate that populations tend to grow. At first glance, it is not obvious that any of these terms should be related to each other, but they are.