Signals are typically represented using sine and cosine functions, yet the notation $e^{ix}$ also plays a central role in digital signal processing (DSP), particularly in the Fourier Series, Fourier Transform, and related concepts. Euler’s Formula bridges these two representations, and Euler’s Identity encapsulates this relationship in an elegant equation that involves the imaginary unit $i$, two irrational numbers $\pi$ and $e$, and the fundamental integers 0 and 1.
Understanding how to prove Euler’s Formula gives insight into the deep connections between exponential and trigonometric functions. It also reveals why each element appears in Euler’s Identity. In DSP, this relationship allows us to work more flexibly with sinusoidal signals by using exponential notation, often simplifying mathematical manipulations by isolating real and imaginary components.
There are two functions defined as:
$$ f_1(x) = e^{ix} $$
$$ f_2(x) = \cos{x} +i\sin{x} $$
The derivatives of the functions are:
$$ f_1'(x) = ie^{ix} = if_1(x) $$
$$ f_2'(x) = -\sin x +i\cos{x} = if_2(x) $$
There is third function defined as:
$$ g(x)=\frac{f_1(x)}{f_2(x)} $$
The derivative (using the quotient rule):
$$ g'(x)=\frac{f_2(x)f_1'(x)-f_2'(x)f_1(x)}{(f_2(x))^2} $$
Replace derivatives with the values calculated previously:
$$ g'(x)=\frac{f_2(x)if_1(x)-if_2(x)f_1(x)}{(f_2(x))^2}=0 $$
NB: This assumes $f_1(x) \ne0$, which is valid for all $x$.
Since the derivative of $g$ is 0, it means that function is a constant: