Math 201 Lecture 30: Fourier Cosine and Sine Series
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1 Math ecture 3: Fourier Cosine and Sine Series Mar. 3, Many examples here are taken from the textbook. The first number in ( refers to the problem number in the UA Custom edition, the second number in ( refers to the problem number in the 8th edition.. Fourier Series and Pointwise Convergence Fourier Series Expansion: Given f(x defined for <x<, its Fourier series reads a n f(x a [ + a n cos ( nx f(x cos ; b n ( ] nx, ( f(x sin ( nx. ( Pointwise Convergence: First extend f periodically (that is into a periodic function period. Then If f is continuous at x, then a + a n cos nx nx } f(x (3 If f has a jump (from f left to f right at x, then Some pitfalls in computing Fourier series. a + a n cos nx nx } [f left+f right ]. ( Example. Find the Fourier series representation of the function f(xcos x, <x<. (5 The following is a wrong solution. Wrong Solution from Exams of Earlier Years. nx nx. f(x a + [a n cos(nx(nx]. (6 a cos x cos x ( +cos(x + cos(x. (7 a n cos x cos(nx +cos(x cos(nx cos(nx+ cos(x cos(nx n sin(nxn + cos( nx+ [cos( nx+cos(+nx] cos(+nx ( n sin( nxn + (+n sin(+nxn. (8
2 Math ecture 3: Fourier Cosine and Sine Series b n for all n because cos x is an even function. So Remark. Recall cos x. (9. Using pointwise convergence to show that the answer is definitely wrong. (Periodic extension of cos x is continuous everywhere so its Fourier series should converge to cos x for all x, so / is definitely wrong. Spot the mistake in the calculation. (Division by n The calculation for a n is wrong for a n. 3. Is there anyway to obtain this particular Fourier expansion superfast? (cos + cos(x is already of the form a + [a n cos(nx(nx] so it has to be the Fourier expansion. Fourier series:. Fourier, Fourier Cosine, Fourier Sine Related to the eigenvalue problem X KX ; X( X(;X ( X (; ( Base functions: ( nx, cos, sin Expansion formulas: f(x a [ + a n cos a n ( nx f(x cos ; Fourier Cosine series: Related to the eigenvalue problem Base functions: Expansion formulas: Fourier Sine series:,,,3, ( b n ( ] nx, ( f(x sin ( nx. (3 X KX ; X (X (. (, cos f(x a + a n cos a n Related to the eigenvalue problem Base functions: Expansion formulas:,,,3, (5 f(x cos, (6 ( nx. (7 X KX; X(X(. (8 sin ( nx,,,3, (9 f(x ( nx b n sin ; (
3 Mar. 3, 3 b n Clearly there should be some connections. In this lecture we will:. Clarify this connection; f(x sin. Discuss pointwise convergence of Fourier Cosine and Sine series.. Basic Information Fourier Cosine and Fourier Sine as special cases of Fourier. ( nx. ( Odd and even extensions. Consider a function f(x defined on <x<. et s define its odd extension and even extension as f o (x f(x <x< f( x <x< ; Note that f o and f e are functions defined on <x<. Fourier series of f o and f e. et s compute the Fourier series for f o and f e. Fourier series for f o : a n [ f o (x a + a n cos ( nx f o (x cos ( nx f o (x cos + ( nx f(x cos f(x <x< f e(x f( x <x<. ( ( ] nx, (3 f o (x cos f( x cos Now to evaluate the nd integral, we do a change of variable: which leads to Thus a n f( x cos f(x cos. ( t x dt (5 ( nx Similarly we can compute b n f o (x sin f(x sin f(x sin ( n( t f(t cos ( nt f(t cos ( nt f(t cos dt ( dt dt. (6 ( nx ( nt f(t cos dt. (7 f( x sin ( nx. (8
4 Math ecture 3: Fourier Cosine and Sine Series Summarizing, we see that the Fourier series for f o is ( nx b n sin b n ( nx f(x sin. (9 This is exactly the Fourier Sine series for f. Theorem 3. et f(x be defined on < x <. et f o (x be its odd extension to <x<. Then the Fourier sine series for f(x is the same as the Fourier series for f o (x (in the sense that they look exactly the same. Fourier series for f e : We have f e (x a [ + a n cos a n and similar calculation gives b n It follows that ( nx f e (x cos ( nx f e (x cos + ( nx f(x cos + f(x cos f e (x sin ( ] nx, (3 f e (x cos f( x cos. (3 ( nx. (3 Theorem. et f(x be defined on < x <. et f e (x be its even extension to <x<. Then the Fourier cosine series for f(x is the same as the Fourier series for f o (x (in the sense that they look exactly the same. Pointwise convergence for Fourier Cosine and Fourier Sine. Fourier Cosine series. Given f(x defined for <x<, to obtain the function that is the pointwise sum of the Fourier Cosine series of f,. Do an even extension of f to f e ;. Obtain the sum of the Fourier cosine series using the pointwise convergence result for the Fourier sereis for f e (Extend f e periodically.... Fourier Sine series. Given f(x defined for <x<, to obtain the function that is the pointwise sum of the Fourier Sine series of f,. Do an odd extension of f to f o ;. Obtain the sum of the Fourier cosine series using the pointwise convergence result for the Fourier sereis for f o (Extend f o periodically, Examples Example 5. Consider the function a Plot the function. x <x f(x <x. (33
5 Mar. 3, 5 b Plot its odd-periodic and even-periodic extensions over ( 6, 6. c Compute its Fourier Cosine and Sine expansions. d Plot the functions to which the Fourier Cosine and Fourier Sine expansions converge to. Solution. a x b Note that the function is only defined for <x so it is not defined outside. Odd-Periodic Even-Periodic c As f(x is defined for <x, we see that. Fourier Cosine: Recalling we have a n First compute a : a n ( nx ( nx f(x cos f(x cos. (3 x <x f(x <x. (35 ( x cos a ( nx + ( x+ cos ( nx. (36 3. (37
6 6 Math ecture 3: Fourier Cosine and Sine Series Next ( nx ( nx a n ( x cos + cos [ ( ] nx ( xd sin + n n sin [ ( nx ( x sin N sin n [ ( ] nx + sin ( n n n sin ( nx n cos N ( n n sin [ ( ] n n cos ( n n sin n ( n n cos ( n n sin N d( x ] + [ sin(n sin n ( ] n. (38 Now since ( n odd ( n even n n cos nk, sin nk+ (39 nk+ nk+3 we have nk n n nk+ a n 8. ( n nk+ n + n nk+3 The Fourier cosine series is a n given as above. Fourier Sine: We have b n f(x sin f(x a + ( nx a n cos ( nx f(x sin ( nx ( nx ( x sin + sin [ ( xd cos nx ] n n cos [ ( nx ( x cos N cos n [ ( ] n cos(n cos n [ ( ] nx + cos n [ ( ] n cos(n cos n [ + ( ] nx n n sin N [ ( ] n cos(n cos n n ( n n sin [ cos(n cos n N d( x ( ] ( ] n. (
7 Mar. 3, 7 Now again using cos together we have ( n The Fourier sine series is b n given by the above formulas. n odd nk, sin nk+ ( n n even nk+ nk+3 (3 cos(n( n, ( nk n b n n n nk+. (5 nk+ n n + n nk+3 f(x b n sin nx d The functions the Fourier cosine and Fourier sine series converge to: in red: Fourier sine: (6 Fourier cosine:
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