Euler's formula: proof and geometric interpretation

In this article, we will cover Euler's formula and the famous Euler's identity. We will also explore what it means to raise a number to an imaginary power and prove Euler's formula in two different ways: by differentiation and by the Maclaurin series expansion.

Euler's formula

Euler's formula is:

Where θ is measured in radians.

Euler's identity

If we set θ equal to π, we get this:

Since cos π is -1, and sin π is 0, we can simplify this expression. The result is called Euler's identity:

If we prove Euler's formula, this will also prove Euler's identity.

What do we mean by proof?

Quite often in mathematics, we might have an equation whose meaning is extremely clear, and we simply need to prove that it is true. This applies, for example, to Pythagoras' theorem - we know what a triangle is, and we know how to square and add numbers, so all we need to do is prove that the formula is correct.

With Euler's formula, things aren't quite so clear-cut. We are raising e to the power of an imaginary number. To take the naive definition of powers, we are multiplying e by itself an imaginary number of times. What does that even mean?

An alternative way to look at this is to say that we have decided to define Euler's formula to be true. The task now is to decide if that definition is a good one. Is it sensible, consistent, and useful to say that, by definition, Euler's formula tells us what it means to raise a number to an imaginary power.

Our approach will be to look at the formula from several angles. Firstly, we will consider the geometric interpretation of the formula and how it relates to what we know about the modulus-argument form and multiplication of complex numbers. Secondly, we will look at several definitions of the exponential function for real numbers, and ask if Euler's formula appears to be correct when we extend those definitions into the imaginary domain.

We will see that it makes a lot of sense to accept Euler's formula as the definition of what it means to raise a number to an imaginary power.

The process of extending a function defined on the real numbers to the complex numbers in a way that preserves its key properties is called analytic continuation. Euler's formula is the unique analytic continuation of the real exponential function to imaginary arguments.

Geometric interpretation of Euler's formula

Here is Euler's formula again:

The right-hand side has a simple geometric interpretation. Here it is represented on an Argand diagram:

The complex number with real part cos θ and imaginary part sin θ is a distance 1 from the origin and makes an angle θ with the x-axis. In other words, it has modulus 1 and argument θ.

If we multiply both sides of Euler's formula by r, where r >= 0, we get the following:

Here is that point on an Argand diagram:

The complex number with real part r cos θ and imaginary part r sin θ is a general complex number with modulus r and argument θ. So, based on the previous formula, the expression:

Also represents a general complex number with modulus r and argument θ.

Multiplication

When we multiply two complex numbers in modulus-argument form, we multiply the 2 moduli, and add the 2 angles:

This is shown on an Argand diagram here:

Using the exponential form gives:

When we multiply 2 exponential terms that have real exponents, we add the exponents. If we assume that imaginary exponents work in the same way, this expression becomes:

So we have multiplied the moduli and added the angles, which is exactly what we need. This shows that Euler's formula holds when multiplying two complex numbers. That is a good start, although it doesn't really explain why the exponent has a factor of i.

If you look at the modulus-argument article linked previously, you will see that Euler's formula is also consistent when raising a complex number to a real power, or finding a real root of a complex number.

Definitions of the exponential function

There are several ways to define the exponential function. We will look at two different ways here, then see whether those definitions still make sense in terms of Euler's formula.

The first definition of the exponential function is that it is the only function f that satisfies these conditions:

This follows since the exponential function is the only non-trivial function that is its own derivative. Of course, if we multiply the exponential function by any real number n, the resulting function is also its own derivative. By specifying that the value of the function must be 1 when x is 0, we eliminate any n other than 1.

The second definition the exponential function uses its Maclaurin expansion:

Proof of Euler's formula by differentiation

We saw previously that a defining characteristic of the exponential function is that it is its own derivative.

What about the derivative of e to the power of iθ? If that means anything, we would surely expect to be able to differentiate it in the normal way. So let's assume:

Notice the extra factor of i since the exponent is iθ (due to the chain rule). Just for good measure, as we will be needing this soon, here is the negative case:

We can now take Euler's formula:

And rearrange it:

This can be written as:

It is easy to verify that when θ is 0, this equation is true (e to the power 0 is 1, and cos 0 is 1).

To prove that this expression is always equal to 1, we need to prove that its first derivative is always zero. We will use the product rule to differentiate the LHS:

This gives:

Taking out the common exponential factor gives:

The sine and cosine terms sum to 0 (remembering that i squared is -1), so the whole expression is zero. This proves that Euler's formula leads to the result:

Proof of Euler's formula by Maclaurin expansion

The Maclaurin expansion of the exponential function is:

We will substitute a value of iθ for x to extend this definition to the imaginary domain:

We can simplify this using the following identities:

The powers of i cycle with period 4. The first 4 values are i, -1, -i, 1, and then the pattern repeats. Applying these identities simplifies the series considerably:

We can separate the real and imaginary terms in this equation:

Let's compare this with the standard Maclaurin expansions for sin and cos:

The cos series is identical to the real series in the previous exponential function, and the sin series is identical to the imaginary part. So this proves once again that:

Applications of Euler's formula

Euler's formula has various applications, including:

• Oscillations and waves: e^(iωt) is ubiquitous in physics and engineering for representing sinusoidal signals.
• Roots of unity: e^(2πik/n) gives the n-th roots of unity, fundamental to number theory and the discrete Fourier transform.
• Complex logarithms and powers: Euler's formula is the foundation for defining the complex logarithm and complex exponential functions for complex numbers generally.

Related forms

It is possible to express the sine and cosine functions in terms of Euler's formula. Here is Euler's formula:

Here is a related formula obtained by taking the negative of the angle:

We can subtract these two formulas to eliminate the cos term, leaving us with an expression of sin as a function of the two exponentials:

Similarly, if we add these two formulas, we eliminate the sin term, leaving us with an expression of cos as a function of the two exponentials:

It is interesting to compare these two forms with the standard definitions of the hyperbolic functions, sinh and cosh.

Summary

We have shown that Euler's formula is consistent with how complex-number multiplication, powers, and roots work. We have also shown that, for 2 commonly used definitions of the exponential function, if we assume that the definitions can be extended to the imaginary domain, then Euler's formula follows as a consequence of the definitions.

Related articles

• Imaginary and complex numbers
• Complex number arithmetic
• Argand diagrams
• Why does complex number multiplication cause rotation?
• Modulus-argument form of complex numbers
• De Moivre's theorem
• i to the power i
• Complex powers and roots of complex numbers
• Semiprocal numbers - z to the power i
• Complex polynomials
• Complex number trigonometry functions