From: 3blue1brown
The concept of complex Fourier series [00:00:08], where rotating vectors sum to draw out complex shapes [00:00:15], is a generalized form of how Fourier series are often described as sums of sine waves [00:01:56]. This fundamental idea, particularly its application in decomposing functions into simple oscillations [00:06:01], originated from Fourier’s efforts to solve the heat equation [00:02:11].
The Heat Equation and its Properties
The heat equation is an equation that describes how temperature distribution on a rod evolves over time [00:03:36]. It also applies to many other phenomena unrelated to heat [00:02:40].
Linearity of the Heat Equation
The heat equation is a “linear equation” [00:03:10]. This means:
- If two solutions are known, their sum is also a new solution [00:03:15].
- Solutions can be scaled by a constant, providing “dials to turn” to construct custom functions [00:03:20].
This linearity is crucial because it allows for the combination of simple solutions to address more complex initial conditions [00:03:41].
Simple Solutions: Cosine Waves
A simple solution to the heat equation exists if the initial temperature distribution looks like a cosine wave, with its frequency tuned so it’s flat at each endpoint [00:02:53]. When graphed over time, these cosine waves simply get scaled down exponentially [00:03:01]. Higher frequency waves experience a faster exponential decay [00:03:04].
When combining these waves, the higher frequency components decay faster, causing the overall sum to smooth out over time as these terms quickly approach zero [00:03:54]. This difference in decay rates for various frequency components captures the complexity in the evolution of heat distribution [00:04:12].
Fourier’s Bold Idea: Decomposing Arbitrary Distributions
Despite the convenience of solving the heat equation for initial conditions that are simple waves or sums of waves, most real-world temperature distributions do not resemble this [00:04:28]. For example, the initial temperature distribution of two rods at different uniform temperatures brought into contact would be a discontinuous “step function” [00:04:54]. This function is flat, not wavy, and discontinuous [00:05:05].
Fourier famously asked what seemed an absurd question at the time: How can any initial distribution, even a discontinuous one like a step function, be expressed as a sum of sine waves [00:05:17]? This sum must also be constrained to waves that satisfy specific boundary conditions, such as cosine functions whose frequencies are whole number multiples of a base frequency [00:05:24].
Infinite Sums and Approximations
Any finite sum of sine waves will only be an approximation, never perfectly flat or discontinuous [00:06:33]. Fourier, however, considered infinite sums [00:06:42]. For the step function example (temperature 1 degree for the left rod, -1 degree for the right), it equals an infinite sum with specific coefficients (e.g., 1, -1/3, +1/5, -1/7 for odd frequencies, scaled by 4/pi) [00:06:51].
An infinite sum means that as more terms are added, the sequence of partial sums approaches a specific value [00:07:30]. For functions, this means that for a given input, the sum of the scaled functions will approach the target function’s value [00:07:55]. This allows an infinite sum of wavy, continuous functions to equal a discontinuous, flat function [00:08:40].
Mathematical Rigor and Real Analysis
While practical, this approach introduces technical nuances:
- How does prescribing a value at a discontinuity affect the heat flow problem [00:08:59]?
- What does it mean to solve a partial differential equation (PDE) with a discontinuous initial condition [00:09:06]?
- Is the limit of solutions to the heat equation also a solution [00:09:09]?
- Do all functions have a Fourier series [00:09:13]?
These questions are addressed by real analysis [00:09:19]. The key takeaway is that summing all (infinitely many) heat equation solutions associated with these cosine waves does yield an exact solution for how a step function (representing the initial heat distribution) will evolve over time [00:09:37].
Complex Fourier Series and Coefficients
To find the coefficients of the Fourier series, a more general approach involving functions with complex number outputs is beneficial [00:10:00]. Functions with real number outputs (like temperature functions) are a special case of this broader view [00:11:03], where sine waves correspond to pairs of vectors rotating in opposite directions [00:12:14].
This broader context simplifies computations and provides a better foundation for understanding concepts like the Laplace transform [00:10:24]. The heart of Fourier series lies in the complex exponential [00:12:58], which describes a value moving around the unit circle as its input progresses [00:13:04].
The coefficients () for each rotating vector (represented by ) [00:15:25] are determined by multiplying the original function by and then taking its average (integral) over the interval [00:20:05]. This effectively “kills” all moving vector terms, leaving only the desired coefficient [00:20:23].
For the step function example (modeling heat dissipation), this calculation of coefficients allows the sum of rotating vectors to approximate the initial temperature distribution, quickly jumping between values as required [00:22:46]. Computing this integral allows for an exact answer for the coefficients, which can then be related back to the concept of cosine waves [00:23:13].
Conclusion
Fourier’s breakthrough, the idea of breaking down functions into simple oscillations, proved incredibly important and far-reaching [00:06:05]. Its origin in the physics of heat flow [00:06:13], seemingly unrelated to frequencies, highlights the general applicability of Fourier series [00:06:23].
This decomposition of a function into a combination of exponentials, used to solve differential equations like the heat equation, is a recurring and vital concept in mathematics [00:24:16].