The Complex-Valued Exponential and Euler’s Formula

Updated on 20-Jan-2014

Changing the theme has caused some problems with formula pictures in this post and some others. My mates are onto it soon the picture backgrounds will be changed to suitable. Sorry for inconvenience.

WARNING: This derivation is not for the faint of heart. Also, remember to never drink and derive.

Okay lets get into a situation. Suppose I asked you to calculate e(1.28403). “No problem!” you say, whipping out the scientific calculator conveniently stashed away in your front pocket. After a series of rapid-fire number crunching, you get the answer: 3.61116343….
Now what if I asked you to calculate e(1.28403*i)? “But wait!” you cry, waving your calculator in the air. “My calculator doesn’t do calculations with complex numbers! Besides, what does raising a number to i even mean?!

Well, let’s see if we can’t figure this out from various known principles. Anybody who has completed calculus will know that ex can be written as a Taylor series:

Evaluating ex is equivalent to plugging x into this infinitely long polynomial, so let’s try plugging in a complex number in——say, ix, to keep it somewhat simple.

Let’s think back to the properties of imaginary numbers to simplify this equation. By definition, i2 = -1. i3 is just i2 * i, or -i. And i4 = (i2)2 = (-1)2 = 1. i5 = i4 * i, but since i4= 1, i5  is just i. At this point, the cycle repeats itself once again. In general:

Using this, we can simplify our expression for e(ix) from above:

Now we’re getting somewhere! Because i is defined as a square root, i has now dropped out from every even-powered term. What happens if we collect all of our even-powered terms together and factor out an i from all of the odd-powered terms?

“Those power series look awfully familiar!” you exclaim, calculator nearly flying out of your hands as you fling your arms up in excitement. “That’s just the Taylor expansion for cosine and sine!” (And that very fact justifies rearranging this infinite series; even though the two series are alternating, since both halves are absolutely convergent, our rearrangement is allowed.)

The implications of this are astounding. We’ve managed to relate the exponential function with the two core trigonometric functions—two things that at first glance would seem to have no relationship at all—simply by sticking an imaginary unit in.
From here, we can take it one step further (or back, depending on whether you like the general or the specific): let’s evaluate this equation at π.

Your calculator slips out of your hand and clatters on the floor as you bask in this glorious equation.
This is Euler’s identity. The additive identity (0), the multiplicative identity (1), the imaginary unit (i), and two of the most important transcendental numbers (π and e) can be beautifully linked together with three basic mathematical operations (addition, multiplication, and exponentiation).

And in my opinion, this is one of the most beautiful and elegant identities in all of mathematics.

Source: Math Central