The most universal formula: the Fourier transform equation that "disassembles everything"

There is such a mysterious equation. It looks delicate and elegant, but it is very difficult to learn. Countless students resented its abstractness and difficulty, but were eventually changed by its magical power and worldview.

When it was first invented, it was a method of solving problems, which was intended to simplify the complex mathematical operations in thermal problems. After its popularization, its fame overshadowed the difficult problem it originally solved.

Its applications are so wide that it can process images, interpret the starry sky, help build houses that are not prone to collapse, and deeply participate in financial data analysis. Whether it is a mixed signal or a complex convolution, it can be tamed by its magic and become clear, concise and efficient.

This is the great Fourier transform . If a history of science is written using equations, Fourier transform must have a place, not only because it is brilliant enough in itself, but also because it is closely related to other great equations.

1. Fourier disassembles everything

Why is the Fourier transform so useful? Because it is a great tool for breaking down everything.

Whether exploring things or solving problems, in the field of science and engineering, people often start from the mathematical model of the object of study. Sound, vibration, image, starlight... Finding their functional incarnation means peeling away the appearance and seeing the skeleton.

The Fourier transform breaks down these mathematical bones. The core idea is that any pattern in spacetime can be viewed as a superposition of sinusoidal patterns of different frequencies. Provide a function of how a signal changes over time, and the Fourier transform can find the hidden frequency information in it. You can get the different vibrations in an earthquake, remove the noise from an audio clip, interpret the cosmic microwave background, or process images and complete compression.

Such influence and importance may have shocked even Fourier himself, although he knew he had found a treasure when he proposed this transformation in the early 19th century. At that time, the French Academy of Sciences had a somewhat complicated attitude towards this achievement. They awarded the formula related to it but refused to publish Fourier's memoirs of winning the award.

Furious, Fourier bypassed the censorship and published the Fourier transform in Theory of Heat Analysis in 1822. Two years later, Fourier returned as secretary of the Academy of Sciences and published the rejected memoir in the Academy's prestigious journal.

From its official publication to the present, the Fourier transform has gone through two centuries. From a historical perspective, two centuries is not too long, but new achievements are always based on old achievements. By doing the most direct disassembly of the Fourier transform, we can go beyond the 19th century when it made its debut and trace back to the more distant past.

2. Everyone loves calculus

When talking about Fourier transform, calculus is an unavoidable topic. This is not only because the transform itself involves integration, but also because Fourier originally proposed this transform in order to solve such an equation:

u(x, t) represents the temperature of a metal rod at position x at time t, and the constant α is the thermal diffusivity. It can be seen that this equation focuses on the change in temperature.

From a modern perspective, it is natural to use derivatives to study changes. This is of course because calculus has completely entered our lives. However, for a long time before that, scholars often needed to estimate the average state of different time periods first, and then infer the overall change law of the state of the object.

Calculus was born in the 17th century, coinciding with the rise of the Age of Reason. A great scientist ushered in his miracle year. At that time, Newton, who was hiding from the plague, completed several world-shaking physics studies on his hometown farm. In the process of solving these problems, he found an advanced mathematical tool to introduce the idea of ​​limit into the expression and calculation of changes.

After the publication, calculus was also mired in controversy, but this time, the biggest focus of the debate was not "whether this works" but "who invented it". At the same time, another great scientist, Leibniz, found the same method from another way. From then on, "who is the father of calculus" almost became the fuse of a powder keg, which could easily lead to fierce arguments.

However, if we continue to learn about the previous research of the two, we will find that human beings have a long-standing interest in infinity and limits. By 1656, Wallis's "Arithmetic of Infinites" had already proposed the predecessor of calculus, and Fermat proposed important problems closely related to calculus in "On Tangents to Curves" in 1679. The birth of calculus is imminent, and history may have chosen Newton and Leibniz at the same time. This is a coincidence, but also a necessity.

3. Unconventional Imaginary Numbers

In Fourier transform, another important component is the imaginary unit i——

In many people's impressions, imaginary numbers are a relatively new concept. After all, its definition reveals an unconventional punk style - this thing is actually the product of taking the square root of a negative number.

In fact, imaginary numbers are like a ghost concept in history. Scholars discovered long ago that if negative numbers can also be squared, some dead-end equations can find a way out. Later, people began to try to use imaginary numbers for practicality and curiosity. However, early mathematicians, including Descartes and Newton, interpreted imaginary numbers as a sign that there was no solution to the problem. Even Leibniz, who had high hopes for imaginary numbers, did not know what it was.

In the 17th to early 19th centuries, the situation gradually changed. Mathematicians proposed the complex plane, where imaginary numbers and real numbers appeared on the same graph, and it was no longer an intangible concept.

Of course, there is something more important, which is Euler's formula published in the mid-18th century.

When z=π, the formula is even more amazing.

In this way, the imaginary number i combines the two most famous numbers in mathematics, e and π, into an elegant equation. Everything suddenly becomes clear, and later physicist Feynman called it: "our treasure" and "the most extraordinary formula in mathematics."

After floating in the background of the scientific world for hundreds of years, imaginary numbers were finally accepted by the mainstream and began to show their strength in the 19th century. When used in conjunction with calculus, they can produce unexpected results. Today, complex variable functions and integral transformations have become a headache for many science students and a must-learn magic weapon.

4. Ancient and young trigonometric functions

There is another part in Fourier transform that is as important as imaginary numbers and calculus. What is it? It is trigonometric functions that even elementary school students know.

Back to the core idea of ​​Fourier transform: any pattern in space-time can be regarded as a superposition of sinusoidal patterns of different frequencies. Trigonometric functions act as disassembled components here, small and exquisite, concise and clear. Trigonometric functions are of course also a treasure in the world of mathematics, with a long history and never outdated.

Let's first talk about why there are trigonometric functions: because right triangles follow this rule:

This is the Pythagorean theorem , and it's at least two thousand years old. There's even evidence that it was passed down among craftsmen before scholars summarized and wrote it down. It's the foundation of trigonometry, the foundation of trigonometric functions.

The ancients were keen on studying trigonometry because they saw how it could help estimate giant objects, with typical applications in astronomy, surveying, and navigation. From ancient Greece and India to the Arab world and Europe, trigonometry has witnessed the rise and fall of civilizations in its dissemination and development.

However, traditional applications often limit trigonometry to specific geometric problems. If trigonometric functions are to be used in a wider range of fields, they need a more flexible definition.

With the emergence of analytic geometry and analysis, people's perspective began to change. Finally, in the 18th century, Euler published "Introduction to Infinite Analysis", proposing to redefine trigonometric functions using the unit circle in the rectangular coordinate system. Next, these long-standing concepts began to partner with complex numbers and appear in series, so the story returns to the Euler formula mentioned above, and of course the Fourier transform that we can't stop talking about.

There are many interesting things to say about Fourier transform. For example, Fourier's heat equation and d'Alembert's wave equation are very similar but very different; for example, different forms of Fourier transform; for example, Fourier and wavelets, and so on.

In fact, every equation is a pearl strung on the veins of history. They are all treasures, and they are also hints for finding other treasures. The development of science is closely linked, and many equations that have changed the world are inextricably linked. The process of understanding them is like reading a story, and also like solving a case. When the clues come together and point to the future, you realize the unspeakable splendor -