Euler's formula - proof

By Martin McBride, 2023-09-11
Tags: eulers formula eulers identity radians
Categories: complex numbers imaginary numbers


In this article we will see various ways to prove Euler's formula and Euler's identity.

Euler's formula

Euler's formula is:

Euler's formula

Where θ is measured in radians.

Euler's identity

If we substitute a value of π for θ in we get Euler's formula:

Euler's identity

Since cos π is -1, and sin π is 0, this leads to Euler's identity:

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 where the meaning of the equation 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.

Geometric interpretation of Euler's formula

Here is Euler's formula again:

Euler's formula

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

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:

Euler's formula

Here is that point on an Argand diagram:

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:

Euler's formula

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:

Complex multiplication

This is shown on an Argand diagram here:

Argand diagram

Using the exponential form gives:

Complex multiplication

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:

Complex multiplication

So we have multiplied the moduli and added the angles, which is exactly what we need. This shows that Euler's formula is consistent when we multiply 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 3 different ways here, and then go on to see if 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:

Definition of the exponential function

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.

A second definition of the exponential function relies on a limit:

Definition of the exponential function

Finally, we can define the exponential function via its Maclaurin expansion:

Definition of the exponential function

Proof 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 ? If that means anything, we would surely expect to be able to differentiate it in the normal way. So let's assume:

Proof by differentiation

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

Proof by differentiation

We can now take Euler's formula:

Euler's formula

And rearrange it:

Proof by differentiation

This can be written as:

Proof by differentiation

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 just need to prove that its first derivative is always zero. We will use the product rule to differentiate the LHS:

Proof by differentiation

This gives:

Proof by differentiation

Taking out the common exponential factor gives:

Proof by differentiation

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 by differentiation

Proof by Maclaurin expansion

The Maclaurin expansion of the exponential function is:

Maclaurin expansion of e^x

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

Maclaurin expansion of e^iθ

We can simplify this using the following identities:

Powers of i

Power of i from a repeating sequence of i, -1, -i, 1 ... Applying these identities to the equation above gives:

Maclaurin expansion of e^iθ

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

Maclaurin expansion of e^iθ

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

Maclaurin expansion of sin, 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:

Euler's formula

Summary

We have shown that Euler's formula is consistent with the way that 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 into the imaginary domain, then Euler's function can be proved as a consequence of the definition.

See also



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