2280 Lecture Record Week 15 S2016
Last Modified: April 12, 2016, 12:30 MDT.    Today: September 24, 2018, 01:06 MDT.
Week 15: Sections 9.1, 9.2, 9.3, 9.4
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)
Monday and Tuesday: Fourier Series. Section 9.1, 9.2, 9.3
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.
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 + 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. Let n=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
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 = 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. 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
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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