The most astonishing formula of mathematics

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Taylor polynomials of the sine function

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What you see below is one of the most beautiful formulas in mathematics.

Euler's identity

A single equation, establishing a relation between ee, π\pi, the imaginary number ii, and 11. It is mind-blowing.

This is what's behind the sorcery.

First, let's go back to square one: differentiation.

The derivative of a function at a given point describes the slope of its tangent plane.

Slope of the tangent plane

By definition, the derivative is the limit of difference quotients: slopes of line segments that get closer and closer to the tangent.

These slopes are called "difference quotients".

Difference quotients

Here is the precise definition of the derivative in its entirety.

Definition of the derivative

Take a mental note, as this is going to be essential. Due to what limits are, we can write the derivative as the difference quotient plus a small error term.

Definition of the derivative

This is a minor but essential change in our viewpoint. By rearranging the terms, we see that a differentiable function equals a linear part + error.

Differentiation as linear approximation

In essence, differentiation is the same as a linear approximation. I am not going to lie: this is mindblowing.

If you don't believe me, check out this plot.

Differentiation as linear approximation

Around x0x_0, the linear function given by the derivative is pretty close to our function. In fact, this is the best possible local linear approximation.

If the linear approximation is not good enough, can we do better?

Sure. For instance, the first and second derivatives give the best local quadratic approximation.

Second order derivatives as best quadratic approximation

You guessed right. The first nn derivatives determine the best local approximation by an nn-th degree polynomial.

This is the nn-th Taylor polynomial around x0x_0.

Taylor polynomials

For instance, this is what the Taylor polynomials around zero look like for sin(x)\sin(x).

Taylor polynomials for the sine function

As you can see, higher-order Taylor polynomials are almost indistinguishable from sin(x)\sin(x).

This is not an accident.

Hold on to your seats: if you think big and let n to infinity, the resulting Taylor expansion can often yield the original function!

(These are called analytic functions, but the terminology is not important for us.)

Taylor series

For example, this holds for our favorite trigonometric functions, sine and cosine.

(You can verify this by hand using the derivatives of sine and cosine.)

Taylor expansion of the trigonometric functions

The Taylor expansion of the amazing exponential function also yields itself.

Taylor expansion of the exponential function

This is one of the most important formulas in mathematics. Why?

Because we can use this to extend the exponential function onto the complex plane! (I don't want to scare you, but you can even plug in matrices. We'll stick to complex numbers, though.)

Complex exponential function

Now, by letting z=ixz = i x, we stumble upon something staggering. The complex exponential function is a linear combination of trigonometric functions!

The first time I learned about this, my head exploded.

Euler's formula

This is Euler's formula in its full might.

Euler's formula

By plugging in ππ, we get the famous Euler's identity. This result wins almost all "what's the most beautiful formula of mathematics" contests.

Euler's identity

Besides its staggering beauty, there is much more to it.

Euler's formula connects the polar form of complex numbers with the exponential function.

This fundamental identity underlies the entire field of science and engineering.

Polar form of complex numbers

This how the polar form is visualized on the complex plane.

Polar form of complex numbers

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