Lecture notes · Differential Equations · Lecture 15

Introduction to Fourier Series

MIT 18.03 Differential Equations, lecture 15, explained in plain English

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Part 1 · 0:00–10:00

This part introduces Fourier series, a way to break any repeating input into simple sines and cosines so we can find the response to it.

Key ideas

So far the inputs to the equations were very special, like exponentials, sines, and cosines. The lecture says those are not as special as they look. 2:23
Any reasonable periodic function, meaning one that repeats itself, can be written as an infinite sum of sines and cosines plus a constant term. 2:45
If you know the response to each sine and each cosine, you can find the response to a sine or cosine multiplied by a number by multiplying that response by the same number. 7:02
The plan is to find the Fourier series of the periodic input, find the response to each piece, and add those responses together to get the response to the whole input. 8:40
This adding works only because the equation is linear, which is the idea behind the superposition principle. 9:38

Words to know

periodic: Repeating the same pattern over and over, with a fixed amount of time before it starts again.
response: The output of the system, meaning the solution you get when a particular input is put into the equation.
linear equation: An equation where you can multiply by a number or add results together, and the rules still hold. The lesson says this is what makes adding responses work.
superposition principle: The rule that if one input gives one response, then a sum of inputs gives the sum of the matching responses.

Example

Think of a song made of many separate notes played together. If you know how a single note sounds, you can work out the full sound by adding up the notes. This is only a comparison to help the idea click, and the lesson itself does not use this example.

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Part 2 · 10:00–20:00

This part introduces orthogonality, the special rule about certain pairs of functions that makes it possible to calculate the coefficients of a Fourier series.

Key ideas

The lecturer's goal is to find the coefficients an and bn of a Fourier series, and the method depends on rules called orthogonality relations. 11:33
Two functions are orthogonal on the interval from minus pi to pi if the integral (a way of adding up a continuous quantity) of their product over that interval is zero. 13:39
Any two different members of the family of sine nt and cosine mt functions are orthogonal on this interval. For example, sine 3t times cosine 4t integrates to zero. 15:21
If you multiply a function by itself, the integral is never zero, because a square is never negative. The integral of sine squared nt (or cosine squared nt) over the interval is pi. 17:09
The lecturer says there are several ways to prove this, including using trig identities (rules that rewrite products of sines and cosines) or complex exponentials. 17:51

Words to know

orthogonal: A fancy word for perpendicular. Here it means two functions give a total of zero when you multiply them together and add up the results over one full cycle.
integral: A way of adding up a continuously changing quantity over a range, such as adding up all the tiny values of a function from one point to another.
coefficients: The numbers in front of each part of a sum. In a Fourier series, the coefficients an and bn are the numbers in front of the sine and cosine parts.
periodic: Repeating in the same pattern over and over. A function with period two pi repeats every two pi units of time.

Example

Think of two different musical notes played for one full cycle and mixed together. Over the whole cycle, their up-and-down motions cancel out, which is like how the integral of two different sine and cosine functions comes to zero. But a note multiplied by itself always pushes in one direction, so it never cancels out.

3 checked practice questions for this part are in the app.

Part 3 · 20:00–30:00

This part shows how a math teacher starts proving that sine and cosine waves are orthogonal (they don't interfere with each other), using the differential equation they all solve.

Key ideas

The lecturer uses the differential equation that the functions satisfy, not trig identities. This method works for many other functions too, not just sines and cosines. 20:18
All these functions satisfy the same kind of equation, u double prime plus n squared times u equals zero. The only thing that changes from one function to the next is the frequency n. 21:38
The lecturer uses integration by parts to rewrite an unbalanced expression so that the two functions are treated the same way. In this step, the boundary terms come out to zero for sines and cosines. 25:12
Next, the lecturer replaces the double derivative with minus n squared times the function. This turns the integral into a form that leads to the orthogonality result. 29:15

Words to know

orthogonal: Two functions are orthogonal when their product, added up over the interval, comes out to zero. They don't overlap or interfere with each other.
differential equation: An equation that describes how a function changes, using its derivatives (rates of change), instead of listing its values directly.
integration by parts: A calculus trick for turning one hard integral into an easier one by moving the work of differentiating from one part of the expression to another.
frequency: How often a wave repeats over time. A higher frequency means a faster wave.

Example

Think of two swings that move back and forth. They follow the same basic rule of motion, and only how fast each one swings is different. The lecturer's point is similar: the rule stays the same, and only the frequency changes from one function to the next.

3 checked practice questions for this part are in the app.

Part 4 · 30:00–40:00

This part shows why certain sine and cosine products cancel out over one full cycle, and uses that to find the formula for the coefficient an in a Fourier series.

Key ideas

Cosine and sine terms with different numbers, like cos(nt) and cos(mt) with m not equal to n, have an integral of zero from minus pi to pi. The lecturer calls these the orthogonality relations. 32:38
Cosine times sine with the same number is also zero over that range, because the product can be rewritten as half of a sine of 2nt, and that integrates to zero. 33:34
To find an, multiply the whole series by cos(nt) and integrate from minus pi to pi. Because of the orthogonality relations, every term disappears except the one with an. 37:55
The only term left is an times the integral of cos squared(nt), which equals pi. Dividing by pi gives the formula an = (1/pi) times the integral of f(t) cos(nt) dt. 39:34
The formula for bn works the same way, so once you know how to find an, you can find bn too. 39:54

Words to know

orthogonality: A fancy word for pairs of functions that cancel each other out when you multiply them and add up the results over one full cycle, giving zero.
integral: The total area under a curve between two points. Here it is the running total of a function from minus pi to pi.
period: The length of one full cycle before a repeating pattern starts over. Here the function repeats every 2 pi.
paradox: A situation where two statements seem to contradict each other. The lecturer shows that a paradox can only be resolved if one expression is zero.

Example

Think of two people walking in a circle at different speeds, one going around 2 times and the other 3 times. Over one full loop, the ups and downs of their paths average out to nothing, which is like how the sine and cosine waves cancel each other out over one period. Only the wave that matches your measuring tool leaves a result behind, just as only the an term survives the integration.

3 checked practice questions for this part are in the app.

Part 5 · 40:00–49:22

This part shows how to handle the special constant term in a Fourier series and walks through a full example, a square wave, to find its sine-term coefficients.

Key ideas

The constant term is the exception. To find it, you integrate over one full period and divide by 2 pi, not by pi as you do for the other coefficients. 42:21
There are two ways to write the constant term. You can use one formula for all coefficients and then write the constant as a zero over two, or you can learn a special formula for the first term. 43:02
For a square wave, the cosine coefficients all come out to zero because the function is odd, so the whole series is made of sine terms. 45:56
The square wave is discontinuous, meaning it jumps from minus one to plus one. So the sine integral is split into two parts, one from minus pi to zero and one from zero to pi. 46:14
The sine coefficients work out to four over n for odd n and zero for even n. The final series is the sum of those coefficients times the matching sine terms. 48:34

Words to know

constant term: The part of the series that is a fixed number with no wave in it. It is the flat, steady level the wave sits on.
periodic: Repeating the same pattern over and over, again and again, with no change.
discontinuous: Having a break or a sudden jump, so the function cannot be drawn without lifting your pen.
odd function: A function that goes equally above and below the middle line, so the positive and negative parts balance out.

Example

Think of a ticking metronome that clicks loud, then soft, then loud again in a steady repeating pattern. A Fourier series breaks that repeating pattern into simple waves, just as you can break the pattern into a set of plain sine tones added together. Each tone has its own strength, and together they rebuild the original pattern.

3 checked practice questions for this part are in the app.

← Lecture 13: Finding Particular Solutions to Inhomogeneous ODEs · Lecture 19: Introduction to the Laplace Transform →

Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. MIT OpenCourseWare · CC BY-NC-SA 4.0 · not affiliated with or endorsed by MIT. These notes are shared under the same license.

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