Euler's Formula

Euler's formula, the most remarkable formula in mathematics according to Richard Feynman, states that for any real number \(x\),

\[ e^{ix} = \cos x + i \sin x \]

To appreciate Euler's formula and it's application to geometric algebra, and ultimately relativity, we need to understand why the imaginary exponent of this special number \(e\) results in the trigonometric functions.

If you already understand how to derive Euler's formula, you may want to skip over to Euler's formula and Geometric Algebra.

Finding a function whose derivative is the function itself

Euler's formula comes about by first considering a function of \( x \), for which the derivative of the function for any value of \(x\) has the same value as function itself. That is:

\[ \frac{d}{dx}f(x) = f(x) \]

If we start with a guess of \( f(x) = 1 + x \) then we see it is not quite right as

\[ \frac{d}{dx}(1 + x) = 1 \ne (1 + x) \]

but we can improve it by adding another term

\[ f(x) = 1 + x + \frac{x^2}{2} \]

which is a little closer in that

\[ \frac{d}{dx}(1 + x + \frac{x^2}{2}) = 1 + x \ne f(x) \]

but still not the same. So we improve it again with

\[ f(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{2\times 3}\]

and so on, until we end up with an infinite sum of terms:

\[ f(x) = \frac{x^0}{0!} + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + ... = \sum_{k=0}^{\infty} \frac{x^k}{k!} \]

We now have a function satisfying \( \frac{d}{dx} f(x) = f(x) \) and can check our solution with:

\[ \begin{aligned} \frac{d}{dx} f(x) &= \frac{d}{dx} \sum_{k=0}^{\infty} \frac{x^k}{k!} \\ &= \sum_{k=0}^{\infty} \frac{kx^{k-1}}{k!} \\ &= \sum_{k=1}^{\infty} \frac{x^{k-1}}{(k - 1)!} \\ &= \sum_{k=0}^{\infty} \frac{x^k}{k!} \\ &= f(x) \end{aligned} \]

Great, but where does \(e\) fit in?

Understanding what an exponent is

We learn in high school that \( 3^2 \times 3 = 3^3 \) and \( 2^2 \times 2 \times 2 = 2^4 \) and later that \( x^2 \times x^2 = x^4 \) etc., but the general definition of exponentiation is that when multiplying two numbers with the same base (\( 3^2 \times 3^1 \)), the powers (or exponents) are added (\( 3^{2+1} = 3^3 \)):

\[ b^m \times b^n = b^{m + n} \]

The definition of e

If we take our function which satisfies the property \(\frac{d}{dx}f(x) = f(x)\) we can show that it behaves just like an exponent should, in that:

\[ f(m) \times f(n) = f(m+n) \]

To show this requires quite a bit of re-arranging:

\[ \begin{aligned} f(m) \times f(n) &= \sum_{k=0}^{\infty} \frac{m^k}{k!} \times \sum_{k=0}^{\infty} \frac{n^k}{k!} \\ &= (\frac{m^0}{0!} + \frac{m^1}{1!} + \frac{m^2}{2!} + \frac{m^3}{3!} + ...) \times (\frac{n^0}{0!} + \frac{n^1}{1!} + \frac{n^2}{2!} + \frac{n^3}{3!} + ...) \\ &= 1 + (\frac{m^1}{1!} + \frac{n^1}{1!}) + (\frac{m^2}{2!} + \frac{m^1 n^1}{1! 1!} + \frac{n^2}{2!}) + (\frac{m^3}{3!} + \frac{m^2 n^1}{2! 1!} + \frac{m^1 n^2}{1! 2!} + \frac{n^3}{3!}) ... \\ &= 1 + (\frac{m^1}{1!} + \frac{n^1}{1!}) + (\frac{m^2}{2!} + \frac{2}{2}\frac{m^1 n^1}{1! 1!} + \frac{n^2}{2!}) + (\frac{m^3}{3!} + \frac{3}{3}\frac{m^2 n^1}{2! 1!} + \frac{3}{3}\frac{m^1 n^2}{1! 2!} + \frac{n^3}{3!}) ... \\ &= \sum_{k=0}^{\infty} \frac{(m + n)^k}{k!} \\ &= f(m + n) \end{aligned} \]

Also similar to an exponent, we can see that for our function, \( f(0) = 1\):

\[ \begin{aligned} f(0) &= \sum_{k=0}^{\infty} \frac{0^k}{k!} \\ &= \frac{0^0}{0!} + \frac{0^1}{1!} + \frac{0^2}{2!} + ... \\ &= 1 + 0 + 0 + ... \\ &= 1 \end{aligned} \]

Given that our function \(f(x)\) behaves just like an exponent should behave, we can define our function simply as a power of some specific number \(e\):

\[ f(x) = e^x \]

and work out what this number \(e\) is, with

\[ e^1 = f(1) = \sum_{k=0}^{\infty} \frac{1}{k!} \approx 2.71828... \]

Now any value of our function can be evaluated simply as a power of \(e\):

\[ f(3) = e^3 = 2.71828... ^3 = 20.08534...\]

Derivatives of exponentials

What if we need to work with other exponents of \(e\), such as \(e^{3x}\)? How do we work out the derivative - how fast \(e^{3x}\) is changing as \(x\) changes? Your memory may (or may not) tell you that:

\[ \frac{d}{dx} e^{3x} = 3e^{3x} \]

or more generally, if \(g\) is another function of \(x\), that

\[ \frac{d}{dx} e^{g} = \frac{dg}{dx}e^{g} \]

This is all we need for our tooling here, though we can also see why this is the case given the chain rule for derivatives which says that:

\[ \frac{d}{dx}f(g) = \frac{dg}{dx} \times \frac{d}{d g}f(g) \]

together with the special property of our function \(f(x) = e^x\) that the derivative at any point is equal to the function itself, \(\frac{d}{dx}f(x) = f(x)\).

Euler's formula

Now we have enough background to appreciate the beauty of Euler's formula. Euler, like Roger Cotes before him, noticed that if he evaluated this exponent function with a special type of value, an "imaginary" value whose square is negative, the result is a combination of the trigonometric functions \(\cos\) and \(sin\). That is, if we create an imaginary unit \(i\) with the property that \(i^2 = -1\), and evaluate \(e^{ix}\):

\[ \begin{aligned} e^{ix} &= \sum_{k=0}^{\infty} \frac{(ix)^k}{k!} \\ &= \frac{(ix)^0}{0!} + \frac{(ix)^1}{1!} + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} ... \\ &= (1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} ...) + i(\frac{x^1}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} ...) \\ &= \sum_{k=0}^{\infty}(-1)^{k}\frac{x^{2k}}{2k!} + i \sum_{k=0}^{\infty}(-1)^k\frac{x^{2k+1}}{(2k+1)!} \\ &= \cos(x) + i\sin(x) \end{aligned} \]

That is, the two separate summations are the actual definitions of \(\cos(x)\) and \(\sin(x)\).

Notice that, with our introduction to Geometric Algebra, we did not need to invent an imaginary unit \(i\) here, as we have already seen that the product of basis vectors \(\mathbf{xy}\) has the same property that \((\mathbf{xy})^2 = -1\), but we will pick up this thread in Euler's formula and Geometric Algebra next.

Derivatives of cos and sin

It's worth noting that we can calculate the derivative of \(e^{ix}\) and as a result work out the standard derivations of \(\cos\) and \(sin\) that we may have memorized in our schooling:

\[ \begin{aligned} \frac{d}{dx}e^{ix} &= ie^{ix} \\ &= i\cos(x) + i^2\sin(x) \\ &= -\sin(x) + i\cos(x) \end{aligned} \]

and so comparing the real parts of \(e^{ix}\) and \(\frac{d}{dx}e^{ix}\) show that

\[ \frac{d}{dx}\cos(x) = -\sin(x)\]

while comparing the imaginary parts show that

\[ \frac{d}{dx}\sin(x) = \cos(x) \]