Fourier series

A Fourier series is a way to break down a repeating pattern into a sum of simple wave-like shapes, such as sines and cosines. This helps make many difficult math problems easier because these wave shapes are well understood. It was first used by Joseph Fourier to solve important equations about how heat moves, taking advantage of the easy-to-study patterns of these waves.

Not all patterns can be perfectly described this way, since some need an endless number of these wave shapes. But for smooth, well-behaved patterns, the Fourier series can come very close to matching the original. The amounts of each wave needed in the sum are found using special math tools called integrals.

Fourier series are closely linked to another powerful math tool called the Fourier transform, which works for patterns that do not repeat. Together, these ideas form a big area of math known as Fourier analysis, which helps us understand patterns in many different ways.

History

See also: Fourier analysis § History

The Fourier series is named after Jean-Baptiste Joseph Fourier, who lived from 1768 to 1830. He made big discoveries in studying trigonometric series. Other smart people like Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli had also looked at these ideas before him.

Fourier used these series to solve a big math problem called the heat equation. This helps us understand how heat moves, like in a metal plate. He first shared his ideas in 1807 and later wrote a full book about it in 1822.

Fourier showed that almost any repeating function can be written as a mix of simple wave shapes, like sines and cosines. This made many math and science problems easier to solve. His work has been used in many fields, like electrical engineering, vibration analysis, acoustics, and more.

Beginnings

Fourier wrote a special math formula to describe waves. This formula helps us find the exact numbers needed to build the wave mix. His work was very new and different, and it changed how we solve many math and science problems.

Fourier's motivation

Fourier created his series to solve the heat equation. For example, imagine a square metal plate with one side kept hot and the others cold. The heat pattern on the plate can be very complex, but Fourier's method helps us understand it better, even though the answers might look complicated at first.

Other applications

Fourier's ideas have been used in many other ways, like solving old math puzzles and understanding waves in different areas of science.

Definition

A Fourier series is a way to break down a repeating function into a sum of simple sine and cosine waves. This helps make many math problems easier because sine and cosine waves are well understood.

The Fourier series shows how any repeating pattern can be built using these basic waves. By doing this, complicated patterns can be studied more simply using the properties of sine and cosine functions.

Table of common Fourier series

Some common pairs of repeating patterns and their Fourier series pieces are shown in the table below.

• s ( x ) shows a repeating pattern with a repeat length of P.
• a0, an, bn are the Fourier series pieces for the repeating pattern s ( x ).

Time domain s ( x ) {\displaystyle s(x)}	Plot	Frequency domain (sine-cosine form) a 0 a n for  n ≥ 1 b n for  n ≥ 1 {\displaystyle {\begin{aligned}&a_{0}\\&a_{n}\quad {\text{for }}n\geq 1\\&b_{n}\quad {\text{for }}n\geq 1\end{aligned}}}	Remarks
s ( x ) = A | sin ⁡ ( 2 π P x ) | for  0 ≤ x		a 0 = 2 A π a n = { − 4 A π 1 n 2 − 1 n  even 0 n  odd b n = 0 {\displaystyle {\begin{aligned}a_{0}=&{\frac {2A}{\pi }}\\a_{n}=&{\begin{cases}{\frac {-4A}{\pi }}{\frac {1}{n^{2}-1}}&\quad n{\text{ even}}\\0&\quad n{\text{ odd}}\end{cases}}\\b_{n}=&0\\\end{aligned}}}	Full-wave rectified sine
s ( x ) = { A sin ⁡ ( 2 π P x ) for  0 ≤ x		a 0 = A π a n = { − 2 A π 1 n 2 − 1 n  even 0 n  odd b n = { A 2 n = 1 0 n > 1 {\displaystyle {\begin{aligned}a_{0}=&{\frac {A}{\pi }}\\a_{n}=&{\begin{cases}{\frac {-2A}{\pi }}{\frac {1}{n^{2}-1}}&\quad n{\text{ even}}\\0&\quad n{\text{ odd}}\end{cases}}\\b_{n}=&{\begin{cases}{\frac {A}{2}}&\quad n=1\\0&\quad n>1\end{cases}}\\\end{aligned}}}	Half-wave rectified sine
s ( x ) = { A for  0 ≤ x		a 0 = A D a n = A n π sin ⁡ ( 2 π n D ) b n = 2 A n π ( sin ⁡ ( π n D ) ) 2 {\displaystyle {\begin{aligned}a_{0}=&AD\\a_{n}=&{\frac {A}{n\pi }}\sin \left(2\pi nD\right)\\b_{n}=&{\frac {2A}{n\pi }}\left(\sin \left(\pi nD\right)\right)^{2}\\\end{aligned}}}	0 ≤ D ≤ 1 {\displaystyle 0\leq D\leq 1}
s ( x ) = A x P for  0 ≤ x		a 0 = A 2 a n = 0 b n = − A n π {\displaystyle {\begin{aligned}a_{0}=&{\frac {A}{2}}\\a_{n}=&0\\b_{n}=&{\frac {-A}{n\pi }}\\\end{aligned}}}
s ( x ) = A − A x P for  0 ≤ x		a 0 = A 2 a n = 0 b n = A n π {\displaystyle {\begin{aligned}a_{0}=&{\frac {A}{2}}\\a_{n}=&0\\b_{n}=&{\frac {A}{n\pi }}\\\end{aligned}}}
s ( x ) = 4 A P 2 ( x − P 2 ) 2 for  0 ≤ x		a 0 = A 3 a n = 4 A π 2 n 2 b n = 0 {\displaystyle {\begin{aligned}a_{0}=&{\frac {A}{3}}\\a_{n}=&{\frac {4A}{\pi ^{2}n^{2}}}\\b_{n}=&0\\\end{aligned}}}

Table of basic transformation rules

See also: Fourier transform § Basic properties

This table shows how some math operations in one place change things in the Fourier series. It uses special signs:

• Complex conjugation is shown with an asterisk.
• Special math symbols stand for repeating patterns or parts of patterns.

Property	Time domain	Frequency domain (exponential form)	Remarks
Linearity	a ⋅ s ( x ) + b ⋅ r ( x ) {\displaystyle a\cdot s(x)+b\cdot r(x)}	a ⋅ S [ n ] + b ⋅ R [ n ] {\displaystyle a\cdot S[n]+b\cdot R[n]}	a , b ∈ C {\displaystyle a,b\in \mathbb {C} }
Time reversal / Frequency reversal	s ( − x ) {\displaystyle s(-x)}	S [ − n ] {\displaystyle S[-n]}
Time conjugation	s ∗ ( x ) {\displaystyle s^{*}(x)}	S ∗ [ − n ] {\displaystyle S^{*}[-n]}
Time reversal & conjugation	s ∗ ( − x ) {\displaystyle s^{*}(-x)}	S ∗ [ n ] {\displaystyle S^{*}[n]}
Real part in time	Re ⁡ ( s ( x ) ) {\displaystyle \operatorname {Re} {(s(x))}}	1 2 ( S [ n ] + S ∗ [ − n ] ) {\displaystyle {\frac {1}{2}}(S[n]+S^{*}[-n])}
Imaginary part in time	Im ⁡ ( s ( x ) ) {\displaystyle \operatorname {Im} {(s(x))}}	1 2 i ( S [ n ] − S ∗ [ − n ] ) {\displaystyle {\frac {1}{2i}}(S[n]-S^{*}[-n])}
Real part in frequency	1 2 ( s ( x ) + s ∗ ( − x ) ) {\displaystyle {\frac {1}{2}}(s(x)+s^{*}(-x))}	Re ⁡ ( S [ n ] ) {\displaystyle \operatorname {Re} {(S[n])}}
Imaginary part in frequency	1 2 i ( s ( x ) − s ∗ ( − x ) ) {\displaystyle {\frac {1}{2i}}(s(x)-s^{*}(-x))}	Im ⁡ ( S [ n ] ) {\displaystyle \operatorname {Im} {(S[n])}}
Shift in time / Modulation in frequency	s ( x − x 0 ) {\displaystyle s(x-x_{0})}	S [ n ] ⋅ e − i 2 π x 0 P n {\displaystyle S[n]\cdot e^{-i2\pi {\tfrac {x_{0}}{P}}n}}	x 0 ∈ R {\displaystyle x_{0}\in \mathbb {R} }
Shift in frequency / Modulation in time	s ( x ) ⋅ e i 2 π n 0 P x {\displaystyle s(x)\cdot e^{i2\pi {\frac {n_{0}}{P}}x}}	S [ n − n 0 ] {\displaystyle S[n-n_{0}]\!}	n 0 ∈ Z {\displaystyle n_{0}\in \mathbb {Z} }

Properties

When we break down a complex function into parts, we can see how its different pieces relate to each other. This helps us understand the function better and solve problems more easily.

The Fourier series uses special functions called sines and cosines to build up any repeating function. By studying these building blocks, we can analyze many kinds of patterns and waves, from sound to light to heat. This makes the Fourier series a powerful tool in science and engineering.

Extensions

The Fourier series can be expanded to include more complex functions and situations. One such expansion is called the Fourier-Stieltjes series. This series is useful when dealing with functions that have certain types of measures or variations over intervals.

Another important extension is the Fourier series for functions of two variables, such as those defined on a square grid. This is particularly useful in applications like image compression, where techniques such as the JPEG standard utilize these ideas.

Fourier series also apply to functions that repeat in three dimensions, such as those found in the study of crystals and solid-state physics. These series help describe patterns and properties in materials with repeating structures.

Fourier theorem proving convergence of Fourier series

Main article: Convergence of Fourier series

In engineering, the Fourier series is usually assumed to work well except at sudden jumps in the function. This is because the functions used in engineering are often simpler than those studied in other areas.

If a function is smooth and its rate of change is manageable, then its Fourier series will match the function very closely. For functions that are not too complicated, the Fourier series will come very close to the original function almost everywhere.

There are also special cases where the Fourier series might not match perfectly, but these are less common in everyday engineering problems.