- [[#Introduction and Motivation]] - [[#The Derivative Method]] - [[#Deriving Other Useful Identities]] - [[#Applications in Digital Signal Processing]] - [[#Connections | Questions:]] ## Introduction and Motivation 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. ## The Derivative Method 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: $\frac{f_1(x)}{f_2(x)}=\text{C}$ The constant can be found by setting $x$ to any value, say 0: $C=\frac{f_1(0)}{f_2(0)}=\frac{e^{i0}}{\cos{0} +i\sin{0}}= \frac{1}{1+0}=1$ Therefore _Euler’s Formula_ is: $e^{ix}=\cos{x}+i\sin{x}$ To get to the _Euler’s Identity_, $x=\pi$ and we move some terms around for ascetics: $e^{i\pi}=\cos{\pi}+i\sin{\pi}=-1+i\ 0$ > [!important] > > $e^{i\pi}+1=0$ ## Deriving Other Useful Identities We can also derive two more useful formulas by first extracting a negative from $x$, since it can be any value: $e^{-ix}=\cos{x}-i\sin{x}$ We then combine the above with Euler’s Formula to get: $e^{ix}+e^{-ix}=(\cos{x}+i\sin{x})+(\cos{x}-i\sin{x})$ Which simplifies to: $\cos x=\frac{e^{ix}+e^{-ix}}{2}$ The same logic can be used by with subtraction to get $\sin x=\frac{e^{ix}-e^{-ix}}{i2}$ These two results are profound in the sense that they show $\cos$ and $\sin$ (an orthogonal basis) as a summation of exponentials raised to a complex number. A fundamental concept in the [[Laplace Transforms]] ## Applications in Digital Signal Processing From Euler’s Formula: $e^{ix}=\cos{x}+i\sin{x}$ In DSP, since signals represent real-valued physical quantities such as voltage or amplitude, we often isolate the real or imaginary part of $e^{ix}$ $\Re \{e^{ix}\}=\cos x, \ \Im \{e^{ix}\}=\sin x$ This separation allows us to express and manipulate signals efficiently, especially when working with complex exponentials in the Fourier domain. ![[Connect#Connect]]