Notes on the Euler formula - Eli Bendersky's website

A First Look at Euler’s Formula

Euler’s formula states that for any real x:

Despite its central role across mathematics and engineering, the expression is initially puzzling. What does a complex exponent mean, and what ties it to trigonometry?

Two Ways to Represent Complex Numbers

Complex numbers have two canonical representations:

• Cartesian: a sum of a real and an imaginary part.
• Polar: a magnitude and an angle.

Conversion between the two forms uses standard trigonometric relations. Given the Cartesian form, you can compute the magnitude and angle. Given the polar form, you can recover the real and imaginary components.

Why Exponentials Appear

Consider multiplying two complex numbers in polar form. Their magnitudes multiply, but their angles add. This follows directly from basic trigonometric identities.

Now suppose a hypothetical function f(x) represents a complex number such that multiplication of two such numbers corresponds to adding their arguments. What familiar real-valued function behaves this way? An exponential. This is not a proof, but it hints that complex numbers and exponentials are deeply connected.

Another clue comes from differentiation. Treating the polar-form expression as a function of the angle and taking its derivative yields a result that factors out i — exactly the derivative you would expect from an exponential function.

Proof via Power Series

The canonical proof of Euler’s formula relies on Maclaurin series expansions for the exponential, sine, and cosine functions.

The Maclaurin series for a function is a polynomial approximation around zero:

Substituting ix for x in the exponential series and applying the cyclic powers of i gives a series that can be regrouped into real and imaginary parts. The real part matches the Maclaurin series for cosine; the imaginary part matches the series for sine. This directly proves the formula.

Proof via Derivatives

An elegant alternative defines a function as the ratio of the complex exponential to its trigonometric counterpart (cosine plus i times sine).

Differentiating this ratio shows it is constant — its derivative is zero. Evaluating it at x = 0 gives a value of 1, so the numerator and denominator are equal for all x. This yields Euler’s formula without relying on series expansions.

Visualizing the Complex Exponential

Plotting e^(ix) is challenging in general because both the input and output are two-dimensional, requiring a four-dimensional plot. However, restricting the input to real x reduces the problem to three dimensions:

• Input: the real angle x
• Outputs: the real and imaginary parts of the result

Looking at the two output components separately as functions of x confirms the expected behavior: the real part traces a cosine, and the imaginary part traces a sine.

Projecting the three-dimensional curve onto the plane of the output values shows the path traced out by the complex exponential:

The result is the unit circle. As a point moves counter-clockwise around this circle, its horizontal coordinate equals the cosine of its angle, and its vertical coordinate equals the sine. This beautifully links trigonometric functions to circular geometry.

Euler’s Identity

A famous special case binds together five fundamental constants of mathematics:

This follows immediately from the general formula by substituting x = π, since cosine of π is −1 and sine of π is 0.

De Moivre’s Formula

Raising the complex exponential to an integer power n and then applying Euler’s formula to both sides gives De Moivre’s formula:

This identity is a practical tool for calculations involving complex numbers and serves as a rich source of trigonometric identities.

An extended version handles fractional powers and is used to find roots of complex numbers. Caution is required here: the complex root function, like its real counterpart, is multi-valued. It maps a single input to several possible outputs.

The fractional-power form states that for integer k, the expression:

is valid because raising it back to the n-th power returns the original number for any such k, thanks to the periodicity of sine and cosine.

[1]	In other words, multiplying by a complex number combines a scaling and rotation operations. Multiplying any z by scales (multiplies) z's magnitude by r and rotates it (counter-clockwise) by angle .