Infinite Series and Differential Equations Nguyen Thieu Huy Hanoi University of Science and Technology Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 12 Fourier Series Let f (x) be defined on R, periodic with period 2L, and piecewise continuous on (−L, L) ∞ a0 nπx Is there any series of the form + an cos nπx such L + bn sin L n=1 ∞ a0 nπx an cos nπx + ∀x ∈ (−L, L) ? that f (x) = L + bn sin L n=1 Suppose such a series as above exists, is it unique? Definition Let f (x) be as above Then, the series of trigonometric functions L ∞ a0 nπx + b sin in which a = + an cos nπx f (x) cos nπx n n L L L L dx for n=1 −L all n = 0, 1, 2, · · · ; bn = L L −L f (x) sin nπx L dx for all n = 1, 2, · · · is called the Fourier Series of f The above-defined numbers an , bn are called Fourier Coefficients of f Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 12 Dirichlet’s Theorem Let f (x) be periodic with period 2L, and piecewise continuous on (−L, L) Suppose that f (x) exists and is piecewise continuous on (−L, L) Then the Fourier series of f is convergent for all x ∈ R to the following sum ∞ a0 nπx = an cos nπx + L + bn sin L n=1 f (x) if f is continuous at x, = if f isn’t continuous at x, (f (x + 0) + f (x − 0)) where an and bn are Fourier coefficients defined as above Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 12 Dirichlet’s Theorem Let f (x) be periodic with period 2L, and piecewise continuous on (−L, L) Suppose that f (x) exists and is piecewise continuous on (−L, L) Then the Fourier series of f is convergent for all x ∈ R to the following sum ∞ a0 nπx = an cos nπx + L + bn sin L n=1 f (x) if f is continuous at x, = if f isn’t continuous at x, (f (x + 0) + f (x − 0)) where an and bn are Fourier coefficients defined as above Remark 1) For even function f , the function f (x) cos nπx L is even, and function nπx f (x) sin L is odd, therefore, L an = L2 f (x) cos nπx L dx ∀n = 0, 1, 2, · · · ; bn = ∀n = 1, 2, · · · Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 12 Dirichlet’s Theorem Let f (x) be periodic with period 2L, and piecewise continuous on (−L, L) Suppose that f (x) exists and is piecewise continuous on (−L, L) Then the Fourier series of f is convergent for all x ∈ R to the following sum ∞ a0 nπx = an cos nπx + L + bn sin L n=1 f (x) if f is continuous at x, = if f isn’t continuous at x, (f (x + 0) + f (x − 0)) where an and bn are Fourier coefficients defined as above Remark 1) For even function f , the function f (x) cos nπx L is even, and function nπx f (x) sin L is odd, therefore, L an = L2 f (x) cos nπx L dx ∀n = 0, 1, 2, · · · ; bn = ∀n = 1, 2, · · · 2) For odd function f , the function f (x) cos nπx L is odd, and function f (x) sin nπx is even, therefore, a = ∀n = 0, 1, 2, · · · n L L nπx bn = L f (x) sin L dx ∀n = 1, 2, · · · Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 12 L 3) For 2L-periodic function g we have −L g (x)dx = any real constant c Therefore, c+2L an = L1 c f (x) cos nπx L dx ∀n = 0, 1, 2, · · · c+2L nπx bn = L c f (x) sin L dx ∀n = 1, 2, · · · for any real constant c Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq c+2L g (x)dx c for / 12 L 3) For 2L-periodic function g we have −L g (x)dx = any real constant c Therefore, c+2L an = L1 c f (x) cos nπx L dx ∀n = 0, 1, 2, · · · c+2L nπx bn = L c f (x) sin L dx ∀n = 1, 2, · · · for any real constant c Example Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq c+2L g (x)dx c for / 12 L 3) For 2L-periodic function g we have −L g (x)dx = any real constant c Therefore, c+2L an = L1 c f (x) cos nπx L dx ∀n = 0, 1, 2, · · · c+2L nπx bn = L c f (x) sin L dx ∀n = 1, 2, · · · for any real constant c Example Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq c+2L g (x)dx c for / 12 Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 12 Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 12 Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 12 Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 12 Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 12 Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 12 Definition [Half-range Fourier Series] A half range Fourier sine or cosine series is a series in which only sine terms or only cosine terms are present, respectively Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 12 Definition [Half-range Fourier Series] A half range Fourier sine or cosine series is a series in which only sine terms or only cosine terms are present, respectively When a half range series corresponding to a given function is desired, the function is generally defined in the interval (0, L) [which is half of the interval (−L, L), thus accounting for the name half range] Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 12 Definition [Half-range Fourier Series] A half range Fourier sine or cosine series is a series in which only sine terms or only cosine terms are present, respectively When a half range series corresponding to a given function is desired, the function is generally defined in the interval (0, L) [which is half of the interval (−L, L), thus accounting for the name half range] Then the function is specified as odd or even, so that it is clearly defined in the other half of the interval, namely, (−L, 0) Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq / 12 Half-range Fourier Sine Series Let f be defined on (0, L) and satisfying Dirichlet’s conditions Extend f to an odd function f˜ on (−L, L) by putting f (x) if x ∈ (0, L), f˜(x) := −f (−x) if x ∈ (−L, 0) Outside (−L, L), f˜ is periodically extended to R Then, Fourier coefficients of f˜ (hence of f ) are an = ∀n = 0, 1, 2, · · · ; L˜ L f (x) sin nπx dx = f (x) sin nπx dx ∀n = 1, 2, · · · bn = L Nguyen Thieu Huy (HUST) L L L Infinite Series and Diff Eq / 12 Half-range Fourier Sine Series Let f be defined on (0, L) and satisfying Dirichlet’s conditions Extend f to an odd function f˜ on (−L, L) by putting f (x) if x ∈ (0, L), f˜(x) := −f (−x) if x ∈ (−L, 0) Outside (−L, L), f˜ is periodically extended to R Then, Fourier coefficients of f˜ (hence of f ) are an = ∀n = 0, 1, 2, · · · ; L˜ L f (x) sin nπx dx = f (x) sin nπx dx ∀n = 1, 2, · · · bn = L L L L Half-range Fourier Cosine Series Let f be as above Extend f to an even function f˜ on (−L, L) by putting f (x) if x ∈ (0, L), f˜(x) := f (−x) if x ∈ (−L, 0) Outside (−L, L), f˜ is periodically extended to R Then, Fourier coefficients L˜ L of f˜ (hence of f ) are an = f (x) sin nπx dx = f (x) cos nπx dx L L ∀n = 0, 1, 2, · · · ; bn = ∀n = 1, 2, · · · Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq L L / 12 Example Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 10 / 12 Example Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 10 / 12 Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 11 / 12 Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 11 / 12 Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 11 / 12 Remarks Let f satisfy Dirichlet’s conditions on (−L, L), an , bn be Fourier coefficients of f , then we have Parseval identity: Differentiation and integration of Fourier series can be justified by using the previous theorems, which hold for series in general So, we can integrate and differentiate the Fourier series of f term by term on (−L, L) Complex Notation: Let f satisfy Dirichlet’s conditions on (−L, L) L − inπx L dx, then the Fourier series in complex Putting cn := 2L −L f (x)e form of f is ∞ cn e inπx L = n=−∞ Nguyen Thieu Huy (HUST) f (x) (f (x + 0) + f (x − 0)) Infinite Series and Diff Eq if f is continuous at x, if f isn’t continuous at x, 12 / 12