Analysis II: Fourier Series

Transcription

1 .... Analysis II: Fourier Series Kenichi Maruno Department of Mathematics, The University of Texas - Pan American May 3, 011 K.Maruno (UT-Pan American) Analysis II May 3, / 16

2 Fourier series were introduced by Joseph Fourier ( ) for the purpose of solving the heat equation in a metal plate. The heat equation u t = u α x. Jean Baptiste Joseph Fourier (1 March May 1830) was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. Fourier is also generally credited with the discovery of the greenhouse effect. K.Maruno (UT-Pan American) Analysis II May 3, 011 / 16

3 To solve a problem in the heat equation, Fourier needed to express a function f as an infinite series of sine and cosine functions: f(x) = a 0 + (a n cos nx + b n sin nx) = a 0 + a 1 cos x + a cos x + a 3 cos 3x + +b 1 sin x + b sin x + b 3 sin 3x + This is called a trigonometric series or Fourier series. Expressing a function as a Fourier series is sometimes more advantageous than expanding it as a power series. Applications: astronomical phenomena, heartbeats, tides, vibrating strings, ocean waves, sound waves, music, etc.(periodic phenomena) K.Maruno (UT-Pan American) Analysis II May 3, / 16

4 Let f(x) be a continuous function on [, π]. Then assume that we can express f(x) by the trigonometric series f(x) = a 0 + (a n cos nx + b n sin nx), x π We must determine a n and b n! Integrate the above expression: f(x)dx = = πa 0 + a 0 dx + (a n cos nx + b n sin nx)dx a n cos nx dx + b n sin nx dx. K.Maruno (UT-Pan American) Analysis II May 3, / 16

5 and cos nx dx = 1 n sin nx] π = 1 [sin nπ sin( nπ)] = 0, n sin nx dx = 1 n cos nx] π = 1 [cos nπ cos( nπ)] = 0. n So Thus f(x)dx = πa 0. a 0 = 1 f(x)dx π K.Maruno (UT-Pan American) Analysis II May 3, / 16

6 To determine a n for n 1, we multiply both sides of the equation by cos mx where m is an integer and m 1: f(x) cos mx dx = = a 0 [ a 0 + ] (a n cos nx + b n sin nx) cos mx dx cos mx dx + a n cos nx cos mx dx + b n sin nx cos mx dx K.Maruno (UT-Pan American) Analysis II May 3, / 16

7 (Orthogonality) So Thus cos mx dx = 0 for all m sin nx cos mx dx = 0 for all n and m cos nx cos mx dx = a m = 1 π { 0 for n m π for n = m f(x) cos mx dx = a m π. f(x) cos mx dx. K.Maruno (UT-Pan American) Analysis II May 3, / 16

8 To determine a n for n 1, we multiply both sides of the equation by sin mx where m is an integer and m 1: f(x) sin mx dx = = a 0 [ a 0 + ] (a n cos nx + b n sin nx) sin mx dx sin mx dx + a n cos nx sin mx dx + b n sin nx sin mx dx K.Maruno (UT-Pan American) Analysis II May 3, / 16

9 (Orthogonality) So Thus sin mx dx = 0 for all m cos nx sin mx dx = 0 for all n and m sin nx sin mx dx = b m = 1 π { 0 for n m π for n = m f(x) sin mx dx = b m π. f(x) sin mx dx. K.Maruno (UT-Pan American) Analysis II May 3, / 16

10 Definition: Fourier Series Let f be a piecewise continuous function on [, π]. Then the Fourier series of f is the series a 0 + (a n cos nx + b n sin nx) where the coefficients a n and b n in this series are defined by a 0 = 1 π a n = 1 π b n = 1 π and are called the Fourier coefficients of f. f(x)dx, f(x) cos nx dx, f(x) sin nx dx. K.Maruno (UT-Pan American) Analysis II May 3, / 16

11 Euler formula So So e iθ = cos θ + i sin θ. cos θ = eiθ + e iθ, sin θ = eiθ e iθ. i e nix = cos nx + i sin nx. So cos nx = einx + e inx, sin nx = einx e inx. i K.Maruno (UT-Pan American) Analysis II May 3, / 16

12 a n = 1 π = 1 π f(x) einx + e inx dx f(x)e inx dx + 1 π f(x)e inx dx, b n = 1 π = i π f(x) i einx + e inx dx f(x)e inx dx + i π f(x)e inx dx, So a n + ib n = 1 f(x)e inx dx, π a n ib n = 1 f(x)e inx dx. π K.Maruno (UT-Pan American) Analysis II May 3, / 16

13 where a 0 + (a n cos nx + b n sin nx) ( ) = a 0 + a n einx + e inx + b n i einx + e inx ( an ib n = a 0 + e inx + a ) n + ib n e inx ( = c 0 + cn e inx + c n e inx) = c n e inx c n = a n ib n an + ibn c n = c 0 = a 0 = 1 n= = 1 f(x)e inx dx, π = 1 f(x)e inx dx, π f(x)dx, π K.Maruno (UT-Pan American) Analysis II May 3, / 16

14 Definition: Complex Form of Fourier Series Let f be a piecewise continuous function on [, π]. Then the Fourier series of f is the series c n e inx n= where the Fourier coefficients c n in this series are defined by c n = 1 π f(x)e inx dx, for all integers n. K.Maruno (UT-Pan American) Analysis II May 3, / 16

15 We can generalize Fourier series for functions of period L. Definition: Complex Form of Fourier Series Let f be a piecewise continuous function on [ L/, L/]. Then the Fourier series of f is the series c n e π L inx n= where the Fourier coefficients c n in this series are defined by c n = 1 L L L f(x)e π L inx dx, for all integers n. K.Maruno (UT-Pan American) Analysis II May 3, / 16

16 Fourier Transform The Fourier transform is a generalization of the complex Fourier series in the limit as L. Fourier transform of f(x): F (k) = f(x)e πikx dx The Fourier transform has many applications. Any field of physical science that uses sinusoidal signals, such as engineering, physics, applied mathematics, and chemistry, will make use of Fourier series and Fourier transforms. K.Maruno (UT-Pan American) Analysis II May 3, / 16