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“We use Fourier series to derive

1 1 1 1 1 π2 + 2 + 2 + 2 + 2 +
= . 2 6 1 2 3 4 5 and other famous sums.”

Fascinating Fourier Series Abhisek Ukil ABB Corporate Research, Switzerland

F

ourier series, named after its originator, the French mathematician and physicist Jacques Fourier (1768–1830), is a key technique in signal processing. Jacques Fourier introduced this concept in his memorable work Theorie Analytique de la Chaleur (The Analytical Theory of Heat) published in 1822, in which he developed the theory of heat conduction. From then on Fourier series had had a deep influence in the development of mathematics, mathematical physics, as well as in many engineering problems, especially in the field of signal processing—the study of periodic phenomena and acoustics. Fourier series can be used to derive some other fascinating mathematical series, typically derived by other methods.

Review of Fourier Series Any periodic nonsinusoidal waveform which is single valued and continuous except for a finite number of finite discontinuities and which does not contain an infinite number of maxima or minima in the neighborhood of any point, can be expressed as the sum of a number of sinusoids of different frequencies. Let f(x) be a single valued periodic function of x. Euler showed that f ( x) =

A0 2

∞

( )

∞

( )

+ ∑ An cos nx + ∑ Bn sin nx , n =1

n =1

α + 2π

∫

cos(nx )sin( mx )dx = 0;

1 An = π 1 Bn = π

α α + 2π

∫

cos2 (nx )dx = π ;

α α + 2π

sin(nx )sin( mx ) = 0, [ n ≠ m ] ;

∫

sin(nx )dx = 0;

∫

sin(nx ) cos(nx )dx = 0;

α α + 2π

∫

α

α + 2π

α

∫

sin 2 (nx )dx = π .

α

Proofs of the Euler’s formulae can be referred to in the book by M.R. Spiegel. Using these formulae, any periodic function can be expressed in terms of its Fourier series expansion. We use these definitions to deduce some interesting mathematical series in the following sections. SERIES 1 :

π2 1 1 1 1 1 1 1 − 2 + 2 − 2 + 2 − 2 + 2 +
= . 2 2 3 4 5 6 7 1 12

Using the Fourier series to represent f(x) = x(1 – x) in the interval [–π, π], we have ∞

( x − x ) = A2 + ∑ A 2

0

n

n =1

π

(

)

π

(

)

∞

cos(nx ) + ∑ Bn sin(nx ), whhere n =1

π

f ( x ) dx.

f ( x ) cos( nx ) dx , and

An =

1 x − x 2 cos(nx ) dx π −∫π

α α + 2π

∫

α + 2π

x3 ⎤ 1 1 ⎡x 2π 2 2 = − = − , and x x dx − ⎢ ⎥ π −∫π π⎣2 3⎦ 3 −π

α α + 2π

∫

α

A0 =

α + 2π

∫

∫

cos(nx ) cos( mx )dx = 0, [ n ≠ m ] ;

∫

α α + 2π

wheree 1 A0 = π

α + 2π

cos(nx )dx = 0;

f ( x ) sin( nx ) dx.

α

These relations can be established with the help of some standard definite integrals, like,

30 NOVEMBER 2007

(

2

)

π

⎡ ⎛ sin(nx ) ⎞ ⎤ ⎛ sin(nx ) ⎞ ⎛ cos(nx ) ⎞ = ⎢ x − x2 ⎜ ⎟ ⎟⎠ − (1 − 2 x ) ⎜⎝ − ⎟⎠ + (−2 ) ⎜⎝ − 2 ⎝ n n n 3 ⎠ ⎥⎦ −π ⎣ =

−4 (−1)n . n2

Thus, A1 = 4/12, A2 = –4/22, A3 = 4/32, A4 = –4/42, and so on. Likewise,

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MATH HORIZONS

π

Bn =

(

)

1 1 1 1 1 π2 π2 π2 + + + + +
= + = . 8 24 6 12 2 2 32 4 2 5 2

1 x − x 2 sin(nx ) dx π −∫π

(

π

)

⎡ ⎛ cos(nx ) ⎞ ⎤ ⎛ sin(nx ) ⎞ ⎛ cos(nx ) ⎞ = ⎢ x − x2 ⎜ − ⎟⎠ + (−2 ) ⎜⎝ ⎟⎠ − (1 − 2 x ) ⎜⎝ − ⎟ 2 ⎝ n n n 3 ⎠ ⎥⎦ −π ⎣ −2(−1)n . n ∴ B1 = 2 1 , B2 = −2 2 , B3 = 2 3 , B4 = −2 4 , and so on. =

Altogether, we have

( x − x ) = − π3 + 4 ⎡⎢⎣ cos(1 x) − cos(2 2 x) + cos(3 3x) −
⎤⎥⎦ 2

2

2

2

2

⎡ sin( x ) sin(2 x ) sin( 3x ) ⎤ +2 ⎢ − + −
⎥ . 2 3 ⎣ 1 ⎦

BONUS PROBLEM: Using the Fourier series for ⎧ 1, f (x) = ⎨ ⎩−1,

Letting x = 0, we get 0=−

π2 1 1 1 ⎡1 ⎤ + 4 ⎢ 2 − 2 + 2 − 2 +
⎥ , and therefoore 3 2 3 4 ⎣1 ⎦

π2 1 1 1 1 − + − +
= . 12 12 2 2 32 4 2 π 1 1 1 1 + 2 + 2 + 2 +
= . 2 8 1 3 5 7 2

SERIES 2 :

Proof: The proof is similar to the proof of Series 1, and we leave it as an exercise, where this time you need to find the Fourier series for ⎧ x, f ( x) = ⎨ ⎩2π − x ,

for 0 ≤ x < π , for π ≤ x ≤ 2π .

Subtracting Series 1 from Series 2 produces, 1 22

+

1 42

+

1 62

+

1 82

+

1 102

Adding this to Series 2, we have

+
=

This is Euler’s summation of ∑ 1 n 2 , which is a famous problem in number theory known as the ‘Basel problem’ solved by Euler in 1735. The name of the problem originated from Euler’s hometown Basel in Switzerland. Solution of this problem by Euler was the starting point for the famous zeta function introduced by Bernhard Riemann in his seminal 1859 eight-page paper entitled “Über die Anzahl der Primzahlen unter einer gegebenen Größe” (“On the Number of Primes Less Than a Given Magnitude”).

π2 π2 π2 − = . 8 12 24

for for

0≤ x <π, π ≤ x ≤ 2π ,

derive the series

π 1 1 1 1 1 1− + − + − +
= . 3 5 7 9 11 4 This is the Leibniz–Gregory–Nilakantha series. This particular series had been proven independently and differently by Indian mathematician and astronomer Nilakantha Somayaji (1444–1544?), Scottish mathematician James Gergory (1638–1675) and German mathematician Gottfreid Wilheim Leibniz (1646–1716) as a series formula for π.

Further Reading For more information on Fourier series, see M.R. Spiegel, Schaum’s Outline of Fourier Analysis with Applications to Boundary Value Problems, and Applied Complex Analysis with Partial Differential Equations by N.H. Asmar. See also R. Roy, “The discovery of the series formula for π by Leibniz, Gregory and Nilakantha,” by R. Roy, Mathematics Magazine, vol. 63, pp. 291–306, 1990.

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