Euler's Formula

Euler’s formula takes in angle an input and returns a complex number that represents a point on the unit circle in the complex plane that corresponds to the angle. For example, given the angle of radians, Euler’s formula returns the complex number which is the right-most point on the unit circle in the complex plane.

Note, the notation is shorthand for the exponential function. Shown below is the formula written explicityly with the exponential function.

When using a computational medium that supports complex numbers this is useful to know. Conceptually, the definition of the exponential function can be used to verify the formula as dicussed in the explanation below.

The connection between the exponential function and the trigonometric functions sine and cosine is suprising and gives this formula notoriety. However, as mentioned above is shorthand for the exponential function.

The example below derives Euler’s formula starting with the power series definition of the exponential function^[1].

1. Start with the power series definition of the exponential function.

2. Subsitute the complex input into the function as input.

3. Expand the expressions in the numerators.

4. Notice the places where the complex constant appears. Everywhere the complex constant is raised to a power greater than one, such as , and we can substitute into the expression.

5. Simplify the expressions which flips some of the signs. The expressions that still contain the constant are highlighted blue.

6. Group the expressions containing and those that do not and then factor out the complex constant.

7. Observe that the two expresions represent the power series definitions of sine and cosine^[2][3].

   Multiply both sides of the power series of sine by the complex constant.

8. Substitute the definitions into the expression.

   This gives us Euler’s formula which we can write in its shorthand form shown below.

Complex Numbers

Complex numbers are an extension of the real number system with useful properties that model two dimensional space and trigonometry.

Exponential Function

The exponential function models exponential growth. The output of the function at any given point is equal to the rate of change at that point.

Sine Function

The sine function returns the sine of a number provided in radians. In geometric terms, the function returns the vertical component of the point formed by the angle on the unit circle.

Cosine Function

The cosine function returns the cosine of an angle provided in radians. In geometric terms, the function returns the ratio of the right-triangle's adjacent side over its hypotenuse.

1. Derive Definition of Exponential Function (Taylor Series)
   Wumbo (internal)
   https://wumbo.net/example/derive-exponential-function-taylor-series/
2. Derive Sine Function (Taylor Series)
   Wumbo (internal)
   https://wumbo.net/example/derive-sine-taylor-series/
3. Derive Cosine Function (Taylor Series)
   Wumbo (internal)
   https://wumbo.net/example/derive-cosine-taylor-series/