[Go to the Notes on Diffy Qs home page]

Every book (no matter how much you paid for it) has errors and typos, especially text that is new in a given edition. I try to be as transparent as possible about any errors found, and I try to fix them as quickly as possible. Please do let me know if you find any errors or typos, so that they may be fixed. A free book relies on readers sending in corrections.

This page lists actual math errors in the various editions. Simple typos, misspellings, and such are not listed here. Also not listed are things that were correct but simply deserved improvement. Of course, older editions suffer from recently discovered errata as well.

**November 7th 2019 edition (version 6.0):**

No known errata.

**March 4th 2019 edition (version 5.5):**

- On page 114, section 3.2: The definition of reduced row echelon form does not include the ordering of the rows. It may be fine for our purposes, but that's no excuse. It should include the ordering of the pivot entries: a pivot entry in any row is strictly to the right of a pivot entry of the row above, and the zero rows come last.
- On page 159, example 3.9.2: In the final answer, when the answer is expressed as a single vector, in the $x_2$, the $2e^{t}$ should come with a minus sign. That is, $x_2 = \frac{e^{-2t}+e^{4t}-2e^t}{3}+\frac{4t-5}{16}$. Thanks to William Meisel.
- In the answer in the back for Exercise 2.4.102 for part b and c, the dependent variable should be $I$ not $x$.
- In the answer in the back for Exercise 3.9.101, replace $e^{5t}$ by $e^{9t}$, also there is a missing 4. Best to just take the solution to be $x(t) = C_1 e^{9t} + 4C_2 e^{4t} - \frac{18t+5}{54}$, $y(t) = C_1 e^{9t} - C_2 e^{4t} + \frac{t}{6}+ \frac{7}{216}$. Thanks to Tom Wan and Shishir Agrawal.
- In the answer in the back for Exercise 4.5.102, the coefficient is $\frac{e^{-n}}{3-(2n)^2}$ and not $\frac{e^{-n}}{3-2n}$.

**October 11th 2018 edition (version 5.4):**

- On page 62, Exercise 1.8.102: There is a typo, the coefficient of $dy$ should be $\frac{x}{y^2}$. That is, the equation is $\bigl( \frac{\cos(x)}{y^2} + \frac{1}{y} \bigr) \, dx + \frac{x}{y^2} \, dy = 0$. The solution in the back is correct, although it is missing parentheses. In a related typo, some of the $dx$s in the solution should be $dy$s. Thanks to Sonmez Sahutoglu.
- On page 173, the units of the angular velocity are given as radians per second squared, they should be radians per second.
- On page 248 on first line, the solution should be $T_n(t)X_n(x)$ not $T_n(t)X(x)$.
- In section 5.2, the discussion of fixed and free ends is missing, even though these are used in the exercises. So it should say the following. Thinking of the left end, if it is hinged then $u(0,t) = u_{xx}(0,t) = 0$. If the end is free, then $u_{xx}(0,t) = u_{xxx}(0,t) = 0$. If the end is fixed, then $u(0,t) = u_{x}(0,t) = 0$.
- Solution to Exercise 5.2.101 should be $\sin(\pi x) \cos(2 \pi^2 t)$.

**February 22nd 2018 edition (version 5.3):**

- On page 118, Example 3.3.1: In the linearly independent example when we write down the linear combination we write $c_3 t^3$ when we mean $c_3 t^2$ in the first component. Though since we immediately find $c_3=0$ it makes no difference for the rest of the computation.

**November 1st 2017 edition (version 5.2):**

- On page 194, Exercise 4.4.2, for the odd extension one needs to assume $f(0) = f(L) = 0$ otherwise it's not actually odd.
- On page 205, when plugging $x_p$ into the equation, the second $p$ subscript is missing. It should be $2x_p'' + 18\pi^2 x_p$ not $2x_p'' + 18\pi^2 x$.

**March 21st 2017 edition (version 5.1):**

- On page 157, at the bottom. It says take $\vec{c} = E^{-1} \vec{b}$, but it should be $\vec{a} = E^{-1} \vec{b}$ as then we use $a_1,\ldots,a_n$.
- On page 157, in the boxed equation. The integral should be with respect to $ds$ not with respect to $dt$.
- On page 174, top of page, it says the whirling string crosses the axis $k$ times, but it crosses the axis $k-1$ times between the endpoints.
- On the bottom of page 175 and the top of 176, the two eigenvectors must correspond to different eigenvalues. The way it is stated would not work if $A$ has repeated eigenvalues.
- On page 334, At the bottom of the page it says to integrate from 0 to $\omega_0$, when that should be $\theta_0$ (which is the integral we actually compute on the next page).
- On page 342, in PoincarĂ¨-Bendixson, we say "spirals towards" which is not the right wording if in fact the periodic solution is just a non-spiraling critical point. Better say "tends towards". Just below the theorem we also forget to mention the possibility of a critical point.
- On page 343, Example 8.4.3: the computation of $\frac{\partial f}{\partial x}+ \frac{\partial g}{\partial y}$ is wrong, it is $-2x+2y$. Therefore the two regions are different and the argument for why there is no limit cycle needs to change. Basically it follows because on the line where $-x+y=0$, the velocity $(x',y')$ always points into the set $-x+y \geq 0$. So the second paragraph of the example needs to be entirely replaced.
- On page 344, Exercise 8.4.103: The computation of the interval in the back was wrong, it is in fact $-1 < x < 1$. Both the problem and the answer in the back needs to also be adjusted. (Issue is that $f_x = 0$ not $1$).
- On Page 363, Solution to Exercise 7.2.101 (page 309). The recurrence relation is actually $a_k = \frac{-2a_{k-5}}{k(k-1)}$ (the denominator was missing), so the series given in the back must be adjusted for both parts a) and b). Thanks to Wesley Snider.

**December 27th 2016 edition (version 5.0):**

- On page 13 the second order derivatives in the PDE examples have the lower exponents (the 2) in the wrong places or missing.
- On page 56, In Theorem 1.8.1 the condition for exactness wrongly does $\frac{\partial N}{\partial y}$ instead of $\frac{\partial N}{\partial x}$. Similarly on page 57 on top of the page we do $N_y$ when we should do $N_x$. Thanks to Erik Boczko.
- On page 93, Exercise 2.5.7 seems misleading as both solutions end up the same (it is a typo even though it is not technically incorrect). In order not to break existing homework, I will leave this exercise as is (I will delete the "What's going on?"). I will add exercise 2.5.10 using $y''-y=e^x$ which does end up with different solutions. Thanks to Asif Shakeel for pointing this out.
- On page 153, Exercise 3.8.4, the initial condition is $\vec{x} = ...$, when clearly it should be $\vec{x}(0) = ...$. Thanks to Chris Peterson.
- On Page 254, Exercise 5.1.102 has a typo in part b. The $-2x$ should be $+2x$. The solution in the back is then correct. With a minus sign the problem is solvable, but harder than intended. Thanks to Nicholas Hu.
- On Page 363, Solution to Exercise 7.2.102 (page 309). The recurrence relation is actually $a_k = \frac{a_{k-3}+1}{k(k-1)}$ (the denominator was missing), so the series given in the back must be adjusted for both parts a) and b). Thanks to Nicholas Hu.

**October 20th 2014 edition:**

- On page 136, when showing the equality $Pe^{tP}=e^{tP}P$ there are two issues. Firstly, the expansion of $e^{tP}$ has an extra $I+$ (on both lines). Secondly, the final thing on the right hand side should be $e^{tP}P$ and not $Pe^{tP}$. Thanks to Barry Conrad.
- On page 139, The derivative $\frac{d}{dx}$ should be $\frac{d}{dt}$ about a third of the page from the bottom (thanks to Barry Conrad).
- On page 140, bottom of the page (the long two-line equation), the $g_2$ gets expanded as $e^t-t$ while it really is $e^t+t$. Thanks to Barry Conrad.
- On page 141, the fourth line from top, the $\xi_2(0) = -1/16$, the minus was missing (thanks to Barry Conrad).
- On page 145, top of the page, the second equation should be $\vec{x}''=A\vec{x}+\vec{F}_1 \cos(\omega_1 t)$ (in other words, $\vec{F}_0$ should be $\vec{F}_1$). Thanks to Barry Conrad.
- On page 175, Exercise 4.3.102, the range should of course be $-\lambda < t \leq \lambda$ (the minus sign is missing).
- On page 210, Example 4.8.1, the definition of $f(x)$ is wrong. The first interval is $0 \leq x < 0.45$, second is $0.45 \leq x < 0.5$, third is $0.5 \leq x < 0.55$, fourth is $0.55 \leq x \leq 1$. (the second and third intervals of definition are wrong, the graph on the following page is correct).
- On page 304 in the caption to Figure 8.2 and in the last paragraph of the page it says $(0,1)$ is a critical point when the point is $(1,0)$. Thanks to Wesley Snider.

**December 18th 2013 edition:**

No known errata.

**April 29th 2013 edition:**

- On page 21, in problem 1.2.8, the initial condition is, of course, $y(0) = 0$ (and not $x(0)=0$, which is nonsense).
- On page 82, in the expression for $C'(\omega)$, the -4 in the numerator should be -2. (Thanks to Simon Tse and Andrew Browning)
- On page 262, exercise 6.2.101: All the x's should be t's. (Thanks to Simon Tse and James Choi)
- On page 266, in the expression for x(t) after reversing the order $\sin(\omega_0 t)$ should be $\sin(\omega_0 \tau)$ (Thanks to John Marriott)
- On page 266, when we set $f(t)$ to $\cos(\omega_0 t)$ the computation reverses sine and cosine, and although it is not incorrect, it is certainly not natural and not what was intended (Thanks to John Marriott)
- On page 293, end of page, the constant term (the one with $a_0$) is not the constant term it should have $x^r$. That this is the $x^r$ term is even mentioned in the next paragraph. (Thanks to Dusty Grundmeier)

**December 17th 2012 edition:**

- On page 232, near the top the approximation 0.86 is for $\sqrt{\lambda_1}$, not for $\lambda_1$. Same for the $\lambda_2$.
- On page 283, in the second line in the computation on top the series should be squared.

**October 1st 2012 edition:**

The references to Edwards and Penney sections in chapter 7 are wrong. It is chapter 8 not chapter 3.

**July 2nd 2012 edition:**

The paragraph about decay rates of Fourier series could be misread. The specific rates mentioned will only work for the specific types of functions we looked at and would of course be possibly lower for more general functions.

**April 8th 2012 edition:**

- Problem 1.1.8 is missing a second initial condition. The problem should be:
*"Solve $y'' = \sin x$ for $y(0)=0$, $y'(0) = 2$."*(Thanks to Pete Peterson) - In problems 1.2.1, 1.2.2, and 1.2.3 we mention "direction field" instead of "slope field" (the terms are somewhat interchangeable, but we defined the term "slope field" for this, so we should stick to it)
- In exercise 2.1.4, it should say "Show that $y=y_1+y_2$ solves ..." (Thanks to Pete Peterson)
- The answer to problem 3.5.101 part e is a saddle as the eigenvalues are actually 5 and -3. (Thanks to Prentiss Hyde)
- Problems 6.3.103 and 6.3.104 are identical (I will replace 104 with a different problem in the next revision). (Thanks to Pete Peterson)
- On page 264, it says that analytic functions are those represented by a "series", where it should say "power series" of course. The following paragraph does set things straight.

**December 25th 2011 edition:**

- Answer to problem 1.4.102 was missing a +1. It should be $y=2e^{\cos(2x)+1}+1$. (Thanks to Thomas Gresham)
- In the answer to problem 1.4.104 gave the correct value of P(10) and even identified it as such. However the problem asks for P(5). (Thanks to Prentiss Hyde)
- The answer to 1.3.102 referred to the answer as y rather than x (Thanks to Jai Welch)
- The answer to 2.5.101 is $y=\frac{-16\sin(3x)+6\cos(3x)}{73}$ (there should be a plus sign in front of the 6). (Thanks to Pete Peterson)

**October 24th 2011 edition:**

No known errata.

**June 17th 2011 edition:**

- The answer to exercise 1.4.103 is 250g. (Thanks to Ziad Adwan)
- The answer to exercise 2.2.103 is $e^{-x/4}\cos(\frac{\sqrt{7}}{4}x)-\sqrt{7}e^{-x/4}\sin(\frac{\sqrt{7}}{4}x)$. (Thanks to Ziad Adwan)
- In exercise 0.2.11 those are second derivatives in c) and d), not first derivatives. Notice we have two initial conditions. (Thanks to Martin Weilandt who is also preparing a partial Portuguese translation)

**December 9th 2010 edition:**

- In exercise 1.4.12 it should be explicitly stated that the incoming concentration is constant. (thanks to Paul Pearson)
- In exercise 4.5.6, the answer is not a 'Fourier' series, it should be given simply as a 'series'.
- In example 5.1.1, there is a missing equals sign in the computation (after the second closing parenthesis)

**November 15th 2010 edition:**

- On bottom of page 233, when expanding in partial fractions, the $A(s^2-1)+s(Bs+C)$ should be $A(s^2+1)+s(Bs+C)$. The result is correct.
- On page 239, when using the second shifting property, the $\mathcal{L}^{-1}$ is missing. The expression should be ${\mathcal{L}}^{-1} \{ e^{-s} \mathcal{L} \{ 1 - \cos t \} \}$. Similarly for the line just below.
- Also on page 239, when giving the solution the 3's somehow managed to become 2's. The solution should be, $x(t) = ( 1 - \cos (t-1) ) \, u(t-1) - ( 1 - \cos (t-3) ) \, u(t-3)$.
- On page 240, the plot of x(t) is not correct (I will change the problem in the next version such that the plot is correct, it is correct if the 3s become 5s. That's a change I wanted to make originally but obviously I stopped short of actually changing the text)
- On page 256, the explicit expression for a
_{5}was wrong, it should be $a_5 = \frac{a_1}{(5)(4)(3)(2)}$ (thanks Gladys Cruz) - On page 260, the a
_{2}expression on the top was missing a negative sign (thanks Jonathan Gomez)

**November 1st 2010 edition:**

- Page 100, when computing the determinant of the matrix, it is $(2-\lambda)((2-\lambda)^2-1)$. There was an extra power of 2. The final expression on the right hand side is correct.
- In Example 3.4.4, the general solution was missing the arbitrary
constants c
_{1}, c_{2}, and c_{3}. Furthermore, the general solution had the wrong eigenvector for $\lambda=2$. - On top of page 108 when showing the spiral sink, we meant to take real and imaginary parts of $\vec{v}e^{(-1-2i)t}$. The displayed equations are correct.
- On bottom of page 121, when the method is being generalized, the equations have 3 in them where there should be $\lambda$.

**October 13th 2010 edition:**

- Page 58, bottom, end of example 2.3.1. When we "divide by $e^x$ again and differentiate" we of course get $2c_3 e^x = 0$ not $4 c_3 e^{2x} = 0$, though the conclusion that $c_3=0$ is still the same. Thanks to Joanne L. Shin.
- On page 84, in the figure, $m_1$ was marked as $m_2$. That is, the left mass should be $m_1$ of course.
- On page 85, Example 3.1.2, we said "differentiate the first equation" when we meant "differentiate the second equation."

**October 3rd 2010 edition:**

- The example on page 38 with h=1.6 gives the wrong critical point. The critical point is 4 as is clear from the graph on page 39.
- Exercise 1.4.9 part a) asked for the concentration as a function of time in seconds, while before, everything was given in terms of hours. That's a typo, just drop the "(in seconds)". I did not mean for this to be a trick question, time in hours hours is good enough.
- On top of page 97, $\vec{x}_p$ was missing the arrow indicating a vector.
- Theorem 3.7.1 had a typo where instead of "then" there was an "and".
- On top of page 120, it said the "number of eigenvalues" was equal to the number of free variables in $A\vec{v} = \lambda \vec{v}$, when of course it is the "number of eigenvectors".
- When discussing variation of parameters, we say that the equations given work for any second order $Ly=f(x)$. Of course, here we must have L of the form $y''+p(x)y'+q(x)y$.

**July 16th 2010 edition:**

- Example 1.1.5 solution was missing a factor of 2. It should be $x(t) = 2 e^{t/2} - 2$. Thanks to Wing Yip Ho.

**July 1st 2010 edition:**

No known errata.

**May 18th 2010 edition:**

No known errata.

**April 28th 2010 edition:**

- On page 113, when writing the solution in terms of amplitudes and phase shifts, α
_{1}and α_{2}were swapped. - Exercise 3.6.4 on page 118 wrongly asked for
*m*_{1}instead of*m*_{2}which is the unknown. - On page 198 the derivation of the change of variables had
*a*^{2}where there should just be*a*in $\frac{d}{dt}$. Thanks to Shawn White.

**April 13th 2010 edition:**

- On page 194 in April 13th version (or 193 in all others), the sentences about the conditions for $T(0)=0$ and $T'(0)=0$ were swapped. The final solution is OK, it is only the one sentence together with the choice of $T_n$ (in case of the solution of z).
- On page 174 in April 13th version (173 in others) the formula near the bottom should have a -4. That is, f(t) is $\frac{1}{2} + \sum_{{n=1\\n~\text{odd}}}^\infty \frac{-4}{\pi^2 n^2} \, \cos (n \pi t)$. Thanks to Sean Robinson.
- On page 207 in April 13th version (206 in others) The formula $u_n(x,0) = X_n(x)Y_n(0) = \sin \left( \frac{n \pi}{n} x \right)$ should be $u_n(x,0) = X_n(x)Y_n(0) = \sin \left( \frac{n \pi}{w} x \right)$ (note the denominator). Thanks to Sean Robinson.

**April 6th 2010 edition:**

- On bottom of page 173, there is the constant term of 1/4 missing in the Fourier series for the solution $x$.
- On page 237, Heaviside function definition is off it should be 1 for all $t ≥ 0$.
- On page 239 near middle of page, there is an $e^{-2s}$ that should be $e^{-3s}$ though the inverse transform was then done correctly. Thanks to Sean Raleigh and Jessica Robinson for spotting this one.
- On page 240, when applying the Laplace transform to solve an integral equation, there was a missing equals sign and $x(t)$ was mistakenly called $f(t)$
- On page 241, exercises 6.2.3, 6.2.4, and 6.2.5 were trivial as given ($x=0$ is the solution). What I meant was to make the initial conditions arbitrary, i.e. $x(0)=a$ and $x'(0)=b$. Thanks to Sean Raleigh and Jessica Robinson for spotting this one.
- On page 242, exercise 6.3.2, there should be no $\omega_0$, it should just be $\omega$. Thanks to Sean Raleigh and Jessica Robinson for spotting this one.

**March 7th 2010 edition:**

- Exercise 4.1.4 on page 145 had forgotten applications of chain rule in the computation. The final result is not changed. Thanks to Michael Angelini for pointing this out.
- On top of page 146, the eigenfunctions given should be cos kt, not sin kt.
- On page 163 top, in the definition of piecewise continuous the condition that one sided limits exist at all endpoints was written wrong, as k already referred to the number of points.
- On page 168 of course it is the cosine terms disappear for an odd extension, not the sine terms.
- On page 170 on the top, there is a plus sign missing in the series after $\frac{a_0}{2}$.
- On page 212 we mistakenly said that eigenvalues are lambdas with no nontrivial solutions, of course it is just the opposite (thanks to Sean Raleigh / Jessica Robinson).
- On page 217, the integral that is the inner product of the sine with itself should not have f(t) in it, the one that equals $\frac{\pi}{4}$ (thanks again to Sean Raleigh / Jessica Robinson).
- On page 222, Newton's second came out all jumbled up. Of course it is "force equals mass times acceleration" (and again: Sean Raleigh / Jessica Robinson).

**February 17th 2010 edition:**

- Exercise 2.4.5 is not possible. There is a typo in the frequencies. If you replace 0.8 with 1.1 and 0.39 with 0.8 the exercise will be possible. Also what was not noted is that you always get two possibilities for the mass in the formulas.
- On page 54 about mid page, the formula for the complex roots is slightly
off. If we want to take out an
*i*, then we must negate what's under the square root sign (the discriminant). That is, the roots should be: $\frac{-b}{2a} \pm i \frac{\sqrt{4ac-b^2}}{2a}$. - On page 60, in the solution corresponding to complex roots of multiplicity
k, the x
^{k}should be x^{k-1}. - On page 60, the factorization and completing the square of the characteristic equation in example 2.3.5 has two typos in it, even though the roots are given correctly. The procedure should proceed as $(r^2-2r+2)^2=0$ and $((r-1)^2+1)^2=0$.
- Theorem 4.1.2 talks about p and q which do not appear in the theorem (they do appear in the more complicated version from chapter 5). Thanks to Sean Raleigh and Jessica Robinson for this.
- IODE Project III has changed (there was always some confusion about what was Project III) and now fits more after section 2.3, rather than before 2.6.
- On page 75 on top, there is a typo in the expression for $u_1'$, the $x$ should be an argument to the cosine.
- In example 2.6.1, when stating the parameters, it should say m=0.5 (the example is computed correctly, so no other changes needed).

**February 8th 2010 edition:**

- Example 1.3.4 has a mistake in the integration of the left hand side. The solution should end up being $y=\frac{6}{x^2+2C}$. Thanks to Leonardo Gomes for noticing this.
- Example 2.4.1 (page 65) has a missing 2 in the derivation. Thus B is off, and so is C and gamma. Figure 2.2 is therefore also not correct. Also the discussion below the example is off by $\omega_0$ (of course $x'(0) = \omega_0 B$).
- Example on bottom of page 72 and top of page 73 is off. The particular solution should have 1/6 not a 1/4.
- The example on page 73 has a $y$ missing in the equation, it should of course be $y''-6y'+9y=e^{3x}$.
- In Example 6.1.7 (page 234), the inverse transform is off by an extra 1/2.
- In Example 6.3.3 (page 243), the convolution is not computed correctly.
- Figure on page 37 (bottom of page) has $y=5$ and $y=0$, when it should be $x=5$ and $x=0$. Thanks to Jeff Winegar for pointing it out.
- Exercise 2.3.4 is hard to read as "equation" refers to the ODE (it is correct, but can be hard to understand).

**January 10th 2010 edition:**

*Mildly ordered by severity:*

- Example on page 34 uses the wrong integrating factor. It is off by a constant. The correct integrating factor would be $x^-4 e^{-4x+4}$. Thanks to Sean Raleigh and Jessica Robinson for pointing this out.
- Example 1.1.3 on page 15 has the wrong constant in front when we solve for $|y|$. It should be $\frac{1}{|k|} e^{-C'k}$. It doesn't (of course) change the final solution as C' was arbitrary.
- Example 1.3.3 had a minus sign error in the derivation, though it does not affect the final answer (and is not technically wrong if we allow complex logarithms...). Also first we implicitly give time in seconds and then we use minutes (again computations were OK).
- Example 1.4.1 was missing a minus sign in the solution. It the solution should be $y=e^{x-x^2}-2e^{-x^2}$. Thanks to Ian Simon for pointing it out.
- Example on page 103 has the signs wrong. The expression for $x_1$ in the middle of the page should have $e^t \cos(t) - i e^t \sin(t)$. The rest of the example also needs to have the signs fixed. Thanks to Sean Raleigh and Jessica Robinson for noticing.
- Exercise 1.1.7 has a typo making the solution very hard (without a computer algebra system). The right hand side should be a $\frac{1}{y+1}$ not $\frac{1}{y^2+1}$.
- Exercise 1.3.7 was impossible to solve as stated. Ignore this exercise (due to a copy/paste error, the equation happens to be the same as in 1.3.6, but with an impossible initial condition). It will be replaced with a different exercise in the next version of the notes.
- There was a forgotten initial condition in Exercise 0.2.9 (x'(0)=0). The exercise is possible as is, but it was not what was intended.
- In exercise 1.4.9, in part b, the target concentration should be 0.001, not 0.01 (obviously it can't be 0.01)
- Exercise 1.5.6 is way too difficult, ignore. It will be replaced.
- Exercise 1.5.1 was actually worked out as an example. It will be replaced.
- Caption on Figure 1 gives the wrong equation.
- Captions on Figures 1.7-1.10 have an extraneous negative sign, replace -0.1 by 0.1. The vertical scale on 1.6-1.10 is improperly rounded.

**November 25th 2009 edition:**

Examples on the first page of section 2.1 are wrong. The $k$ in the equations should be $k^2$. Thanks to Thomas Wicklund for noticing.

**October 23rd 2009 edition:**

Exercise 2.1.8 has a missing minus sign. It should look like:

**Exercise 2.1.8:** Suppose
$y_1$ is a solution to $y''+p(x)y'+q(x)y=0$.
Show that

$y_2(x)=y_1(x)\int \frac{e^{-\int p(x)dx}}{(y_1(x))^2} dx$

is also a solution.

**September 29th 2009 edition:**

Exercise 4.6.4 on page 188 is not correct. At least not what was intended. The side condition should be $u(x,0) = 3 \cos x + \cos 3x$.

**May 12th 2009 edition:**

Example 1.4.2 is wrong. There was an error with integration and there is a factor of 2 missing in the 5th line of the computation on page 30. Here is an updated page 29 and 30. Thanks to Thomas Wicklund for spotting this error.