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\begin{document}
\section{Errata for ``Principles of Applied Mathematics; Transformation and
Approximation'', second edition}


There are two lists here.  In the first list are those errors that appear
in the first printing but were corrected for the second printing.  The
second list are those errors that appear in both the first and second
printing.

\noindent
List 1: (Corrected for the second printing)

\begin{enumerate}
\item
pg. 7; The law of cosines (line 15) should read

$||x+y||^2 = ||x||^2 +2 ||x||\cdot||y|| \cos \theta + ||y||^2$.
\item
pg 12;  line -3 of footnote.  ``vetor" should be ``vector".
\item
pg. 15 line 3; $M$ lies in a $k$-dimensional space,
\item
pg. 25 line 14; the null space is spanned by $(2,-1)^T$.
\item
pg 35; Caption to figure 1.8; ``rotations" should be ``reflections".

\item
pg. 49; Problem 1.3.c; n must be greater than 1.
\item
pg. 65
\begin{enumerate}
\item line 17; property (3) should be property (4)
\item
 line 23;  $\ldots = (||x||+||y||)^2$.
\end{enumerate}
\item
pg. 67, last line;  components (delete ``of'')
\item
pg 72
\begin{enumerate}
\item line -6;  The coefficient for the scale factor for the
Legendre polynomial should be $2^n$  rather than $2_n$.
\item
line -3  $\omega(x)$ should be $w(x)$.
\end{enumerate}
\item pg 73
\begin{enumerate}
\item
 lines 1, 4, 6, 8, 13;  $\omega(x)$ should be $w(x)$.

\item line 8, $w(x) = (1-x)^\alpha(1+x)^\beta$

\item
line 10, $p_n^{\alpha\beta}(x) = {(-1)^n\over
2^nn!}(1-x)^{-\alpha}(1+x)^{-\beta}
\cdots$.
\end{enumerate}
\item pg. 80, line 12; ``basis functions'' rather than ``basis
function''.
\item
pg. 93, line 15, Ingrid
\item
pg. 94, Exercise 8. (a):  Use property 5 \ldots, (b) Use property 6 \ldots
\item
pg. 96, Exercise 8 e), $T_n(x) = {(-1)^n2^n\over
(2n)!}(1-x^2)^{1/ 2}\cdots$.
 \item
pg. 109, line -11:  In the string of (in)equalities,
change
                   $|\!<\!\!Lu,v\!\!>\!| = \norm{Lu}\cdot\norm{v}$ to  $\le$
                   instead of $=$ (using the Schwartz inequality).
\item
pg. 116, line -9; $K_n u$ rather than $K u$
 \item 
pg. 125, line -3; $I = \int_0^{\pi/2} 2 d\theta = \pi.$
 \item pg. 126, 
\begin{enumerate}
\item
line 6; Integrand is missing an $r$.
  \item
line 11; $g_r(r)$ should be $g_n(r)$
 \item
 line 13; In denominator of integrand,
$\sqrt{\rho^2-x^2}$
                  should be $\sqrt{\rho^2-z^2}$
\item
line -1;  This is not incorrect, although with the change of variables
suggested, one finds $I_0 = \sqrt{a^2-b^2}\int_0^{\pi}{d\theta\over
a+b\cos \theta}$, which because of symmetry of $\cos \theta$ is the
same as shown. 
\end{enumerate}

\item
pg. 129, line 2;  $dy$ rather than $dt$
\item
pg. 141, Example 1 should read $\inprod{H'}{\phi} = -\int_0^\infty
\phi'(x) dx = ...$
\item
pg. 142, Example.  As $n\rightarrow \infty$, ...

 \item
pg. 150, line 2: $a_2(x)h(x,x)f'(x)$...  (i.e. lower
case h)
 
    \item
pg. 155, line 3: $g^*(x,y)$ should be $g^*(y,x)$, and
$J$ should be  $J_y$


    \item
pg. 164, line -9: $S_{n+1/2}(\xi-x)$
should be $S_{n+1/2}(2\pi(\xi-x))$


    \item
pg. 168, line 11; in definition
of the
        Jacobi polynomials, $(01)^\alpha$ should be $ (-1)^\alpha$ 
\item
pg. 171, Problem 1a.  second line of definition should be for $|x| \ge 1$.
\item
pg. 175 Problem 4.3.9;  $-\beta$ should be replaced by $= - \beta $
\item
pg. 180 line -2 Replace $F+\lambda G$ by $F-\lambda G$ (twice)
\item
pg. 213, Caption to Figure 6.2,  $e^{(\theta_1\theta_2)/2}$ replaced
by $e^{i(\theta_1+\theta_2)/2}$.
\item pg. 215
\begin{enumerate}

    \item line 11; $f(z) = u(x,y)+iv(x,y)$
\item
line 17; $\partial
v\over \partial z$ should be $\partial v\over
\partial x$.
\end{enumerate}
\item pg. 216
\begin{enumerate}
\item
 line 2, $\cdots =u_x(x_0,y_0) + \cdots$

\item
line -1 should read $\frac{e^{z+\Delta z} - e^z}{\Delta
z} = \cdots$
\end{enumerate}
\item 
pg. 223 line -3 the denominator is missing a factor of $r$, and should
be $1-2r\cos(\phi-\theta) + r^2$.

    \item
pg. 235, line -7: $\cdots = -2\pi i \rho A V$

    \item
pg. 242, line 7: ...with angle of attack $\alpha$ in
the
$z$-plane.

    \item
pg. 248, line -3; $\cdots = \left.{2\pi i\over
2(z+i)}\right|_{z=i}
\cdots$

    \item
pg. 253, line -12; $I(R) =  \int_{\Gamma(R)}
e^{iaz}g(z)dz$. 
  \item
pg. 254, line 13, $I=2\pi i\left. {e^{iaz}\over
z+i}\right|_{z=i}$

    \item pg. 259, line 6: $=$ should be $+$

\item
pg. 276
\begin{enumerate}
\item
 Problem 6.2.9; The contour should be $|z| = 1$.
\item
 Problem 6.3.5;  The ranges for $z$ should be strict
inequalities, that is, Im $z < 0$ and Im $z > 0$.
\end{enumerate}


    \item pg. 286, line 15; $=\ldots
\left.\right|_{\lambda=k^2\pi^2}$
 

    \item
pg. 298, line -8; $\ldots = {1\over n!}{d^n\over
dz^n}$

\item
pg. 299, line -5 should begin $={1\over 2\pi} \sum_{k=-\infty}^\infty
\cdots$
\item
pg. 308 line 4, singularities of $F(s)$ ...
\item
pg. 309, line 6,  becomes $Lu = -d^2u/dt^2 - \lambda u$
 \item
pg. 310, line -2; $(\Delta u)_n + \lambda u_n = 0$
\item
pg. 311 \begin{enumerate}
\item line 13 $\lambda = 2 - \xi - 1/\xi$.

    \item line -8; $\delta_{nj} = \frac{1}{2\pi i}\int_{C'}
        z^{|n-j|-1}dz$
\end{enumerate}
\item
 pg. 314, line 2,  ``to be real'' should  be
"to be nonzero and real''.
\item
pg. 316, last line;
$R_R(k)=-R_L(-k)$ is not correct. The correct statement is
$R_R(k)/T_R(k)=-R_L(-k)/T_L(-k)$ for real, nonzero $k$.
\item
 pg. 317, line 3: $e^{ik_0}$ should
be replaced by
$e^{ik_0x}$.
\item
pg. 328, last line, delete ``$\alpha$ real''.
\item
pg. 330, Exercise 11.  Reference should be to Theorem 7.3.
\item
pg. 361, line 6 ...if and only if $J_n(\sqrt{\lambda}R) = $
\item
pg. 363, (8.34) $\ldots \left(\sin \phi {\partial U_m\over \partial
\phi}\right)  \ldots$
\item
pg. 382, line -4 should read ``$\nabla^2\phi_j = v_j$ in $\Omega$ and
${\bf n}\cdot\nabla\phi_j = 0$ on $\partial \Omega$, for $j = 1, 2$.''
 \item pg. 386, line -4; $u(R,t) = T_1$
\item
pg. 396, line -1 insert ``to'' before solve.
\item
pg. 408, last line should end with ``?''.
\item
pg. 409, Problem 8.4.1, the equation should read $u_{n_t} = {1\over
h^2}(u_{n+1} - 2u_n+u_{n-1})$.
\item pg. 411, line 2: {\bf scattering transform} should 
be {\bf inverse scattering transform}.  (However, the index was {\bf not}
changed.)
\item pg. 412
\begin{enumerate}
\item
line 5; For reasons that are somewhat subtle, it is not
sufficient to assume that $q(x)$ is absolutely integrable.  Instead,
one must make a stronger assumption, such as $\int_{-\infty}^\infty
(1+|x|)|q(x)|dx <\infty$.
\item
 line -7, Im $k\ge 0$ should be Im $k>0$.
\end{enumerate}
\item
pg. 426
  \begin{enumerate}
\item
line -12 ${dq_n\over dt}$ should be replaced by ${d^2q_n\over
dt^2}$.
\item
 line -3  ${dv_n\over dt}$ should be replaced by ${d^2v_n\over
dt^2}$.
\end{enumerate}

    \item
pg.~441, line 3; In Table
10.1


    \item
pg.~463, prob 10.2.1; $E_n(x) = \int_1^\infty
{e^{-xt}\over t^n}dt.$
\item pg. 540, line 16; The equation for $y$ should read $L^{-1}
  y_0'' - vy_0'-{1\over \epsilon^2}y_0f(\theta_0) = 0$
\item
pg. 542, line 17 insert ``('' (open parenthesis) before cardiac.
\item
pg. 567, 1.1.2 should be 1.1.3.
\item
 pg. 568; The answer to 1.2.1 is not computed correctly.  The matrices
 $C$ and $D$ are correct, but the representation of $A$ should be
\begin{equation} \left(\begin{array}{ccc}
{53\over6}&-{19\over 3}&-4\\
{13\over 12}&-{5\over 6}& -1\\
{49\over 4}& -{17\over 2}& -5\\
\end{array}\right)
\end{equation}
\item
pg. 574
\begin{enumerate}
\item
3.4.2; Answer (a) is the correct answer for (b).  Answer
(b) is the correct answer for some other unrelated problem.
   \item line -1: $a_n = \frac{2n+1}{2}$.
\end{enumerate}
\item
pg. 575, Problem 3.5.1c; $u(x) = f(x) + \int_0^x\sinh(x-t)f(t) dt$.
\item

\item
pg. 575, Problem 3.5.2; $u(x) = f(x) +{\lambda\over 1-\lambda}
+\int_0^1f(t)dt = x +{\lambda\over 2(1-\lambda)}$ when $f(x) = x$,
provided $\lambda \ne 1$.

    \item pg. 576
\begin{enumerate}
\item
line 5: (answer to 4.1.6) $u(x) = \delta(x) + c_1x +
        c_2$
\item line 18: (answer to  4.2.3) $g(x,y) = {1\over 2\alpha} \cdots$
\item
line -9: (answer to 4.2.7) $u(x) = - {1\over 2\alpha} \cdots$.
\item line -6, (answer to 4.2.9) $u(x) =
\int_0^1g(x,y)f(y) dy -
\lambda \int_0^1g(x,y)u(y)dy + \cdots$
\item
line -4,(answer to 4.2.10)  $u(x) =
\int_0^1g(x,y)f(y) dy -
\lambda \int_0^1g(x,y)u(y)dy + \cdots$

    \item line -2 (answer to 4.2.11): $u(x) =
\int_0^1g(x,y)f(y) dy -
\lambda \int_0^1g(x,y)u(y)dy +\cdots$
\end{enumerate}
\item pg. 577
\begin{enumerate}
\item
line 2 (answer to 4.2.13) $u(x) = \cos x
+\lambda\int_0^xp(\xi)\sin(x-\xi)u(\xi)d\xi$
\item
line 3 (answer to 4.2.14) $u(x) = 1 + \int_0^x\xi\ln({x\over \xi})u(\xi)
d\xi$
\item line -1 (answer to 4.4.7) $\cdots -{\pi\over 3}(\alpha + {\beta\over
  2}).$
\end{enumerate}


\item pg. 580\begin{enumerate}
\item Problem 6.1.3;
\begin{enumerate}
\item[(a)] $z = {\pi\over 2} + 2n\pi -i\ln(2\pm\sqrt{3})$.
\item[(b)]$z = (2n+1)\pi -i\ln(\sqrt{2}+1), z = 2n\pi
-i\ln(\sqrt{2}-1).$
\end{enumerate}
\item
 Problem 6.2.3; $\int_Cz^{-1/3}dz = -3(2)^{1/3}e^{i\pi/6}$.
\end{enumerate}

\item
pg. 581, Problem 6.3.6; (b) $F_x-F_y = -8\rho\pi iA.$  (c)$F_x - iF_y
= \rho\pi(4\gamma A-8A^2i)$.
\item
pg. 586
\begin{enumerate}
\item Problem 8.1.13; $u(r,\theta) = -{2\over \pi}
\int_0^\infty
\ldots$
\item
 Problem 8.1.14; $u(r,\theta)=\sum_{n=-\infty}^\infty a_n \ldots$
\end{enumerate}
\item pg. 589, Problem 10.2.2;  $\sum_{k=0}^\infty ({\pi\over
x})^{2k}$.

\item
pg. 593, problem  12.1.8; $\phi_\tau={1\over 6}A^2$ instead of
$\phi_\tau={1\over 6}^2$.

\end{enumerate}

\noindent
Second list:  Errors in both first and second printings.
\begin{enumerate}
\item pg. 8, line 19, $\gamma = {\langle x,y\rangle\over ||y||^2}$.
\item pg. 37, eqn (1.12) First term should not be squared:  $||Ax-b||
  = $
  \item
pg. 43, line 12.  The sequence defined by (1.16) converges as stated in Theorem 1.14 only if $\lambda_1>0$.  If $\lambda_1<0$ or is complex, then the iterates must be normalized differently to get convergence.  For example, one could use $x_{n+1} = A{x_n\over s_n}$ where $s_n$ is the element of the vector $x_n$ having largest magnitude.  
  \item pg. 49, line -3 should read "normed linear vector space."
\item pg. 52 line -2; should read:  and $A^*\psi = \overline{\lambda}
  \psi$ 
\item  pg. 73, line 15; The normalization constant is not the same as
in other places, such as Hochstadt, The Functions of Mathematical
Physics, pg. 41.

\item pg. 96, Problem 11; $P_n(x) = {(-1)^n\over 2^nn!}{d^n\over
dx^n}\left(\cdots\right.$.

\item pg. 111, line 12; $\sqrt{2}$ should be $\sqrt{2\pi}$.



\item pg. 132, Problem 3.7.2-a; Require $\alpha > 1$.
\item
pg. 153, equations (4.13) and (4.14).  $p(x)$ should be replaced by
$-p(x)$.

\item pg. 184, line 7; $\ldots = {d^{k-1}\eta\over dx^{k-1}}(x_0) =
{d^{k-1}\eta\over dx^{k-1}}(x_1) = 0$.
\item
pg. 211, line 14 should read $\ln z = \ln(re^{i\theta}) = \ln r +
\ln(e^{i\theta}) = \ln r + i\theta$.
\item 
pg. 229, line -6; there is a missing ``='' sign;  $
 =\frac{\partial}{\partial x}{\partial \phi\over
\partial x} +
\frac{\partial}{\partial y}{\partial \phi\over \partial y}=  {\partial^2\phi\over \partial x^2} +{\partial^2\phi\over \partial y^2} = 0,$

\item
pg. 230, line -15;  $\psi = -A\theta$

\item
pg. 240, line -1; 
 $\left( e^{i\alpha} \coth \frac{\pi
e^{i\alpha}}{2z} + e^{-i\alpha} \tanh \frac{\pi e^{i\alpha}}{2z}
\right)$
\item
pg. 271, eqn 6.40   $f(z) = {1\over h}\int_{-\infty}^\infty f(t) {\rm sinc}({z-t\over h}) dt$
\item
pg. 307, last line $f(t) = {1\over 2\pi} \int \cdots$
\item
pg. 308, line 3; $f(t) =  {1\over 2\pi} \int \cdots$
\item pg. 320, line 3 $T = U_0 = \cdots$
\item pg. 320, eqn 7.33, $u_2(x) = e^{-ikx}$ for $x<0$.
\item 
pg. 321, line 4, $u_2(x) = ae^{-ik_1x} + \cdots $ for $0<x<\alpha$,
\item
pg. 321 line 6, $x=0,\alpha$
\item
pg. 323 line 15, $v = u-2(\ln w)'' $ (delete $``)z=0"$).
\item pg. 323, eqn 7.37 $ \cdots -\beta\cdots.$
\item
pg. 323, line 22 $ \cdots -\beta \cdots$

\item pg. 483, line 7. $a=GM/h^2c^c$.

\item pg. 483, line 18, $c$ is the speed of light.  (not $c^2$)
\item pg. 509, eqn (12.7) should be
\beq
u(t) = \rho_0\left({a\over (a-\rho_0^2)e^{-\epsilon a t} +
\rho_0^2}\right)^{1/2} \cos(t + \phi_0) + O(\epsilon)
  \eeq
\item pg. 515 line -4; $u(x) = U_0(y) + \epsilon u_1(y,\sigma) +
  \epsilon^2 u_2(y,\sigma) + \cdots,$
\item pg. 540, line -14, for all $\theta$ with $0\le\theta<1$,
\item
pg. 567 line 12, 13 should read ``approaches 4 as $p$ goes to infinity''

\item pg. 574, Problem 3.4.1-b) should be $f(x) = {2\over \pi^2}
\cdots$.
\item pg. 576, Problem 4.2.7; $u(x) = -{1\over 2\alpha} \cdots$
\item pg. 577, Problem 4.4.2; $g(x,y) = (3y^2-2y^3)(x-{1\over 2}) +
  {1\over 2}y -{1\over 2}x - {1\over 2}x^2 + (y-x)H(y-x)$.
\item pg. 601, line 3, column 2, ``Reisz'' should be ``Riesz''.
\end{enumerate}
Feel free to let me know about any other errors you may find.  I'll
add them to this list.

 

\end{document}
