Edwards-Penney, sections 9.1 to 9.4 The textbook topics, definitions and theorems

Edwards-Penney BVP 9.1 to 9.4 (0.0 K, txt, 31 Dec 1969)

Chapter 9 Edwards-Penney BVP textbook Fourier Series Methods 9.1 Periodic Functions and Trigonometric Series DEF. Periodic Function. f(t+p)=f(t) for all t. p=period. Orthogonality Relations DEF. Two functions u, v are said to be othogonal on [a,b] provided integral(u*v,a..b)=0. A list of functions is said to be othogonal on [a,b] provided any two of them are orthgonal on [a,b]. THEOREM. The trigonometric list of sin(nt), cos(mt), n=1..infinity, m=0..infinity, is orthogonal on [-Pi,Pi]. The trigonometric list is independent on [-Pi,Pi], because it is a list of Euler atoms. DEF. A Fourier series is a formal sum of trigonometric terms from the trig list. A Fourier sine series is a Fourier series with no cosine terms. A Fourier cosine series is a Fourier series with no sine terms. Fourier Coefficient Formulas Let f(x) be defined on [-Pi,Pi]. Define a[m] = (1/Pi)*integral(f(t)*cos(mt),-Pi..Pi), m=0..infinity b[n] = (1/Pi)*integral(f(t)*sin(nt),-Pi..Pi), n=1..infinity Classical Fourier Series (1/2)*a[0] + SUM( a[m]*cos(m*x), m=1..infinity) + SUM( b[n]*sin(n*x), n=1..infinity) THEOREM. The formulas for a[m], b[n] need not be memorized. They arise from one idea: (1) Start with f(x) = trigonometric series (2) Multiply the equation in (1) by one trigonometric atom. Integrate over [-Pi,Pi]. (3) Orthogonality implies that the integrated series has exactly one nonzero term! Divide to find the corresponding coefficient a[m] or b[n]. DEF:= integral(u*v,a..b). It has these INNER PRODUCT properties. The vector space V together with these properties is called an INNER PRODUCT SPACE. (1) < u,v+w > = < u,w > + < v,w > linear in the first argument (2) < c*u,v > = c*< u,v > (3) < u,v >=< v,u > symmetry (4) < u,u > = 0 if and only if u=0 DEF: On inner product space V, the NORM is defined by |u| =sqrt(< u,u >), or equivalently, |u|^2 = < u,u >. DEF: A VECTOR is a package in a set V. Set V, called a VECTOR SPACE, is equipped with addition and scalar multiplication, such that the two closure laws hold and the 8 properties are valid (group under addition, scalar disribution laws). THEOREM. Letn=1..infinity be a list of orthogonal functions on [a,b]. Let f = SUM(c[n]*f[n],n=1..infinity). Then c[n] = / = integral(f*f[n],a..b) / integral(f[n]*f[n],a..b) EXAMPLE 1. Find a[m], b[n] for the square wave f(x) = -1 on (-Pi,0), f(x) = 1 on (0,Pi), f(x)=0 for x=-Pi,0,Pi. Plot the Fourier series F(x) of f(x) on -2Pi to 2Pi. ANSWER. a[m]=0 for all m, because f(x) is odd. b[n] = 4/(n*Pi) for n odd b[n] = 0 for n even GIBB's OVERSHOOT. At discontinuities of f(x), F(x) has a strange behavior, called Gibb's Overshoot. This can be seen by plotting a truncated Fourier series near discontinuities of f(x). EXAMPLE 2. Find the Fourier series of f(x) on [-Pi,Pi], where f(x) = x*pulse(x,0,Pi) except that f(x)=Pi/2 at x=Pi and x=-Pi. ANSWER. a[0] = Pi/2 a[m] = [ (-1)^m - 1] /( m^2*Pi^2 ) for m=1..infinity b[n] = (-1)^n (-1) / n for n=1..infinity 9.2 Fourier Convergence Theorem DEF. Piecewise continuous. Piecewise smooth. DEF. Periodic function of period 2L. Half-period L. THEOREM. Let f(x) be smooth on [-Pi,Pi] and F(x) its formal Fourier series, built with the Fourier coefficient formulas. Then f(x) = F(x) for all x in [-Pi,Pi]. THEOREM. The convergence theorem above continues to hold if f(x) is only piecewise smooth, but the equation f(x) = F(x) only holds at points of continuity of f(x). At other points, there is the equation (f(x+)+f(x-))/2 = F(x). THEOREM. The series convergence is uniform if f(x) is smooth. It is not uniform for the Gibb's example. 9.3 Fourier sine and cosine series. DEF. Even function. Odd function. THEOREM. (odd)(odd)=(even) (even)(odd)=(odd) (even)(even)=(even) THEOREM. On a symmetric interval [-L,L]: 1. Integral (odd) = 0 2. Integral (even) = 2 * Integral over [0,L] DEF. The Fourier Cosine series of f(t) defined only on [0,L] is the full Fourier series on [-L,L] of the even exension of f(t) to the interval [-L,L]. The series has only cosine terms. DEF. The Fourier Sine series of f(t) defined only on [0,L] is the full Fourier series on [-L,L] of the odd exension of f(t) to the interval [-L,L]. The series has only sine terms. EXAMPLE 1. Let f(t)=t on [0,L]. Find the Fourier Sine and Cosine series of f(t). EXAMPLE 2. Find a formal Fourier series solution x(t) for the periodic BVP x'' + 4x = 4t, x(0)=x(1)=0. Choose the Fourier series for the interval [0,1] so that the boundary conditions are automatically satisfied for every term of the Fourier series. Answer: The exact solution is x(t) = t + c*sin(2t) with c chosen to make x(1)=0: c = -1/sin(2). The Fourier series derived for x(t) is x(t) = sum(b[n]*sin(n Pi t),n=1..infinity) THEOREM. Term-wise integration and differentiation of Fourier series. 1. Term-by-term integration usually succeeds, because f(t) only needs to be piecewise continuous. 2. Term-by-term differentiation generally fails. It works in the limited setting where f(t) is continuous and f'(t) is piecewise continuously differentiable. There are a few other exceptions, not covered in Edwrds-Penney. EXAMPLE. The Fourier series for f(t)=t on [-L,L] converges to f(t), but its term-by-term derivative diverges. The problem is caused by the discontinuities of f(t).

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