Errata for
Standard Mathematical Tables and Formulae (30^th edition)
(First Printing)

by Daniel Zwillinger (Editor-in-chief)
(Boca Raton, Florida: CRC Press, 1996) x+812pp. ISBN 0-8493-2479-3

NOTE THAT THERE MAY EXIST ERRORS IN THIS PRINTING THAT WERE NOT CORRECTED IN THE NEXT PRINTING. PLEASE LOOK AT THE LIST OF ERRATA FOR LATER PRINTINGS.

1.

Among the pages before the numbered pages, one contributor is listed as ``Lawrence M. Glasser'', this should be changed to ``M. Lawrence Glasser''.

2.

On page 10, the powers of 2 from 41-50 are missing. Add the following to the bottom of the table:

n	2ⁿ	n	2ⁿ

41	2199023255552	81	2417851639229258349412352
42	4398046511104	82	4835703278458516698824704
43	8796093022208	83	9671406556917033397649408
44	17592186044416	84	19342813113834066795298816
45	35184372088832	85	38685626227668133590597632

46	70368744177664	86	77371252455336267181195264
47	140737488355328	87	154742504910672534362390528
48	281474976710656	88	309485009821345068724781056
49	562949953421312	89	618970019642690137449562112
50	1125899906842624	90	1237940039285380274899124224

3.

On page 19, second displayed equation, we have

$\begin{displaymath} \sum_{n=0}^{\infty} E_n \frac{t^n}{n!}=\frac{2}{e^t+1}\end{displaymath}$

This is incorrect. It should be

$\begin{displaymath} \sum_{n=0}^{\infty} E_n \frac{t^n}{n!}=\frac{2e^t}{e^{2t}+1}\end{displaymath}$

4.

On page 28, sequence number 79, the last entry is now a ``2''. This is incorrect, it should be a ``26''.

5.

On page 36, Other types of series, number 2, we presently have

$\begin{displaymath} \sum_{n=1}^{N}(a+nb)x^n=\frac{a-(a+bN)x^{n+1}}{1-x} +\frac{bx(1-x^n)}{(1-x)^2}\qquad (x\ne1)\end{displaymath}$

This is incorrect. It should say (note $n\to N$ )

$\begin{displaymath} \sum_{n=1}^{N}(a+nb)x^n=\frac{a-(a+bN)x^{N+1}}{1-x} +\frac{bx(1-x^N)}{(1-x)^2}\qquad (x\ne1)\end{displaymath}$

6.

On page 38, second line of first displayed equation, replace ``(x-[x])'' with `` $(x-\left\lfloorx\right\rfloor)$ ''

7.

On page 40, ``Exponential functions'', move the phrase ``(all real values of x)'' from the first equality to the second equality.

8.

On page 43, third line for $\coth x$ , change $\displaystyle 2x\left[ +\frac{1}{\pi^2+x^2}+\dots \right]$ to $\displaystyle 2x\left[ \frac{1}{\pi^2+x^2}+\dots \right]$

9.

On page 46, line -7, replace ``Equation (1.3.1)'' with ``Equation (1.4.1)''.

10.

On page 48, line +2, replace ``Equation (1.3.1)'' with ``Equation (1.4.1)''.

11.

On page 52, line -10, we presently have

...That is, $\lim_{h\to0} \frac{f(z_0+h)-f(z)}{h}$ exists. ...

This is incorrect. It should say

...That is, $\lim_{h\to0} \frac{f(z_0+h)-f(z_0)}{h}$ exists. ...

12.

On page 64, line 11, replace ``all x in domain'' with ``all x in the domain''

13.

On page 65, lines 10-11, we presently have

(Archimedean property of $\hbox{$\cal R$}$ ) For every real number x there is an integer N such that x < n.

This is incorrect. It should say

(Archimedean property of $\hbox{$\cal R$}$ ) For every real number x there is an integer N such that x < N.

14.

On page 68, line18, `` $n\in\hbox{$\cal N$}$ '' with `` $n\ge\hbox{$\cal N$}$ ''.

15.

On page 70, section 1.6.9, line 10, we presently have

$\sum_{n=1}^N a_n \phi_n$ is $\frac{1}{b-a} \int_a^b \vert f(x)-\sum_{n=1}^N \phi_n(x)\vert^2 dx.$ An orthonormal set ...

This is incorrect. It should say

$\sum_{n=1}^N a_n \phi_n$ is $\frac{1}{b-a} \int_a^b \vert f(x)-\sum_{n=1}^N a_n\phi_n(x)\vert^2 dx.$ An orthonormal set ...

16.

On page 71, section 1.6.10, item 3, line 4, replace ``use'' with ``used''.

17.

On page 71, section 1.6.10, item 8, lines 2-3, we presently have

...series $\sum_{k=0}^{x_0}a_kg_k(x)$ is ... expressed as $f(x) \sim \sum_{k=0}^{x_0}a_kg_k(x)$ .

This is incorrect. It should say

...series $\sum_{k=0}^{\infty}a_kg_k(x)$ is ... expressed as $f(x) \sim \sum_{k=0}^{\infty}a_kg_k(x)$ .

18.

On page 83, last line of text in section 2.2.3, replace ``given by'' with ``given by the solutions of''.

19.

On page 85, first note, replace `For polynomial'' with ``For the polynomial''.

20.

On page 99, item 2 now has the text

All the known Mersenne primes, ..., 756839, 859433.

This is incorrect. It should say

All the known Mersenne primes, ..., 756839, 859433, 1257787.

21.

On page 98, section 2.3.11, it now says ``September 1995''. This should be changed to ``September 1996''.

22.

On page 98, the table begins

Number	Number of digits
$\relax\ifmmode\copy\bigstrutbox\else\unhcopy\bigstrutbox\fi2^{859433} - 1$	258,716
2⁷⁵⁶⁸³⁹-1	227,832
$391581\cdot2^{216193}-1$	65,087
2²¹⁶⁰⁹¹-1	65,050
$\vdots$

Largest prime of the form 2ⁿ-1, a Mersenne prime.

Largest non-Mersenne prime.

This should be changed to (note new top line, and addition of the word ``known''):

Number	Number of digits
$\relax\ifmmode\copy\bigstrutbox\else\unhcopy\bigstrutbox\fi2^{1257787} - 1$	378,632
2⁸⁵⁹⁴³³ - 1	258,716
2⁷⁵⁶⁸³⁹-1	227,832
$391581\cdot2^{216193}-1$	65,087
2²¹⁶⁰⁹¹-1	65,050
$\vdots$

Largest known prime of the form 2ⁿ-1, a Mersenne prime.

Largest known non-Mersenne prime.

Note that the last, say, three lines of the table should be deleted.

23.

On page 99, fourth bullet, is ``4977!!+1 (largest ...''. This entire line should be deleted.

24.

On page 99, item 3 now has the text

The largest known twin primes are: $697053813\cdot 2^{16352}\pm1$ (with 4,932 digits), $1691232\cdot 1001\cdot 10^{4020}\pm1$ (with 4,030 digits), and $4650828\cdot 1001\cdot 10^{3429}\pm1$ (with 3,439 digits).

This is incorrect. It should say

The largest known twin primes are: $242206083 \cdot 2^{38880} \pm 1$ (with 11,713 digits), $570918348\cdot10^{5120}\pm1$ (with 5,129 digits), and $697053813\cdot 2^{16352}\pm1$ (with 4,932 digits).

25.

On page 101, table heading, replace ``Values of n for'' with ``Values of n or p for''

26.

On page 101, 4th sequence, we presently have the numbers

3, 5, 11, 41, ...

This is incorrect (there is a missing term in the sequence). It should say

3, 5, 11, 13, 41, ...

27.

On page 101, section 2.3.12, line 3, the number ``0.142856 142856...'' should be ``0.142857 142857...''.

28.

On page 119, item 12, second sentence, now reads:

All real matrices are Hermitian.

This is incorrect. It should say

All real symmetric matrices are Hermitian.

29.

On page 126, equation (2.5.19), last line, now reads:

$\begin{displaymath} a_{m1} x_1 +a_{m1} x_2 + \cdots +a_{mn} x_n =c_{m}\end{displaymath}$

This is incorrect. It should say

$\begin{displaymath} a_{m1} x_1 +a_{m2} x_2 + \cdots +a_{mn} x_n =c_{m}\end{displaymath}$

30.

On page 128:

(a): line 6, replace `` ${\bf b}_S={\bf P}_Sv$ '' with `` ${\bf b}_S={\bf P}_S{\bf v}$ ''
(b): On page 128, item number 2, first displayed equation, now reads:

$\begin{displaymath} R({\bf A})=\left\{{\bf y}\mid {\bf A}{\bf y}={\bf x}; \ {\bf x}\in\hbox{$\cal R$}^m \right\}\end{displaymath}$
This is incorrect. It should say

$\begin{displaymath} R({\bf A})=\left\{{\bf y}\mid {\bf A}{\bf y}={\bf x}; \ {\bf x}\in\hbox{$\cal R$}^n \right\}\end{displaymath}$
(c): item number 3: delete the word ``four'' (appears twice, once in each sentence)
(d): Equation (2.5.23): delete the third line of the 4 equations
(e): Equation (2.5.24): delete the fourth line of the 4 equations
(f): Page 128, line -6, replace `` $x_{+}=\tilde{{\bf A}}b$ '' with `` ${\bf x}_{+}=\tilde{{\bf A}}{\bf b}$ ''

31.

On page 131, top line, we presently have

The characteristic polynomial $\det{({\bf A}-\lambda{\bf I})}=\sum_{i=0}^m r_i \lambda^i$ has...

This is incorrect. It should say (note the upper limit)

The characteristic polynomial $\det{({\bf A}-\lambda{\bf I})}=\sum_{i=0}^n r_i \lambda^i$ has...

32.

Page 131, note number 4, replace ``eigenvector ${\bf x}$ that'' with ``eigenvector ${\bf x}$ (different from $\hbox{\bf 0}$ ) that''

33.

Page 133, section 2.5.15, note 3, replace ``u(x)'' with `` $u({\bf x})$ ''

34.

On page 135, note 4, line 3, replace `` $(b_1,\dots,b_n)$ '' with `` $(b_1,\dots,b_n)^{\textrm{T}}$ ''

35.

On page 136, first displayed equation, replace `` $\displaystyle {\bf A}\otimes{\bf B}= \begin{bmatrix} a_{11}B&a_{12}B\\ a_{21}B&a_{22}B\\ \end{bmatrix}$ '' with
`` $\displaystyle {\bf A}\otimes{\bf B}= \begin{bmatrix} a_{11}{\bf B}&a_{12}{\bf B}\\ a_{21}{\bf B}&a_{22}{\bf B}\\ \end{bmatrix}$ ''.

36.

On page 140

(a): note 11, fourth bullet, replace ``page 150'' with ``page 151''.
(b): note 5, last line, replace ``pages 151 and 152'' with ``pages 152 and 152''.
(c): note 6, first line, replace ``groups, called the'' with ``groups, the''.
(d): note 8, replace ``entried'' with ``entries''.

37.

On page 147, top table, ``permuation'' should be ``permutation''.

38.

On page 163, section 3.1.2, after the first sentence insert the following sentence:

The letters P, Q, and R are used to designate compound statements.

39.

On page 164, section 3.1.4, second sentence. If space permits, replace

For example, if P ...

with

For example, suppose that P and Q are compound statements. Then if P ...

40.

On page 171, replace the two occurrences of `` $\dots+=2^{n-1}$ '' with `` $\dots=2^{n-1}$ ''.

41.

On page 174, line 11, the last expression is

$\begin{displaymath} x^{(3)}=x(x-1)(x-2)=2x-3x^2+x^3= \left[ 3 \atop 1\right]x +\left[ 3 \atop 2\right]x^2 +\left[ 3 \atop 2\right]x^3\end{displaymath}$

This is incorrect. It should say (look at the lower value in the last Stirling number)

$\begin{displaymath} x^{(3)}=x(x-1)(x-2)=2x-3x^2+x^3= \left[ 3 \atop 1\right]x +\left[ 3 \atop 2\right]x^2 +\left[ 3 \atop 3\right]x^3\end{displaymath}$

42.

On page 174, section 3.2.9, first paragraph, we presently have

The Stirling cycle number, $\left\{n \atop m\right\}$ , is the number of ways to partition n into k blocks. (Equivalently, it is the number of ways that n distinguishable balls can be placed in k indistinguishable cells, with no cell empty.)

This is incorrect. It should say (note $k\to m$ )

The Stirling cycle number, $\left\{n \atop m\right\}$ , is the number of ways to partition n into m blocks. (Equivalently, it is the number of ways that n distinguishable balls can be placed in m indistinguishable cells, with no cell empty.)

43.

On page 175, line 4, the last expression is

$\begin{displaymath} x^3= \left\{3 \atop 1\right\}x^{(1)} +\left\{3 \atop 2\right\}x^{(2)} +\left\{3 \atop 2\right\}x^{(3)}\end{displaymath}$

This is incorrect. It should say (look at the lower value in the last Stirling number)

$\begin{displaymath} x^3= \left\{3 \atop 1\right\}x^{(1)} +\left\{3 \atop 2\right\}x^{(2)} +\left\{3 \atop 3\right\}x^{(3)}\end{displaymath}$

44.

Page 177-178: these are combinations, not permutations. Hence, these pages should appear after the tables on pages 179-180.

45.

On page 186,

(a): In the displayed equation, there should be a comma after the ``1''.
(b): The definition of degree sequence should be:
A sequence $(d_1,\dots,d_n)$ is a degree sequence of a graph if there is some ordering $v_1, \dots, v_n$ of the vertices for which d_i is the degree of v_i for each i.

46.

On page 189,

(a): In the definition of 2-switch, the last line now has ``edges $\{v,x\}$ and $\{w,y\}$ is''. This should be changed to ``edges $\{v,y\}$ and $\{x,w\}$ is''.
(b): In the definition of thickness, the last line now has ``union of planar graphs''. This should be changed to ``union of k planar graphs''.

47.

On page 193:

(a): We now have `` ${\cal G}_{3,1} = \{K_2\cup \overline{K}_1\}$ '', this should be changed to `` ${\cal G}_{3,1}=\{K_2\cup K_1\}$ ''.
(b): We now have `` ${\cal G}_{4,4}=\{C_4,P_2\cdot K_3\}$ '', this should be changed to `` ${\cal G}_{4,4}=\{C_4,(K_2\cup K_1)+ K_1\}$ ''.

48.

On page 196,

(a): Note 11, insert ``(Appel-Haken)'' before Four-Color Theorem:.
(b): Note 1, part (i), replace ``d(u,v)'' with ``d(u,v)''.

49.

On page 200,

(a): Last line of text (just above bottom table) now says:
The numbers of ...following table:''
This is incorrect and should be replaced by
The following table lists the number of isomorphism classes of graphs of order n and size m.
(b): Bottom table, there are two lines corresponding to m=9. The first line should be removed, it is redundant and wrong.

50.

On page 201, second table, line 4, we presently have the numbers

1 1 2 3 7 16 54 283

This is incorrect. It should say

1 1 2 3 7 16 54 243

51.

On page 219, second line of text, replace ``self-doubt'' with ``self-dual''.

52.

On page 232, left column

(a): line 10 of the table, `` $\sin(\beta n)$ '' should be replaced by `` $\cos(\beta n)$ '' (that is, the second occurrence of `` $\sin(\beta n)$ '' should be replaced).
(b): lines 11-12 of the table, replace ``a_n'' with ``aⁿ''.

53.

On page 234, the ``Logistic Equation'' section,

(a): second line of equation (3.9.24), replace `` $\alpha\in[0,4]$ '' with `` $\alpha\in[0,k]$ ''.
(b): first line after equation (3.9.24), replace `` $rk\le4$ '' with `` $r\le4$ ''.
(c): item number 1, replace ``When r=k=2'' with ``When r=k=4''.
(d): equation (3.9.25), replace ``x_n+1=2x_n-x_n²'' with ``x_n+1=4x_n-x_n²''.
(e): first line after equation (3.9.25), replace `` $\alpha=2\sin^2$ '' with `` $\alpha=4\sin^2$ ''.
(f): equation (3.9.26), replace `` $2\sin^2$ '' with `` $4\sin^2$ '' (once in each line of this equation).

54.

On pages 234 and 235, the ``Logistic Equation'' section, replace all six occurrences of $\pi\theta)$ '' with `` $\theta)$ ''

55.

On page 235, remainder of section 3.9, remove the three occurrences of the number ``4''.

56.

On page 237, equation (3.11.1), we presently have (in part)

$x_j\ge \ \ \hbox{for} \ \ j=1,\dots,n,$

This is incorrect. It should say

$x_j\ge 0 \ \ \hbox{for} \ \ j=1,\dots,n,$

57.

On page 237, line -3 replace `` $(x_1,x_2,\dots,x_n)$ '' with `` $(x_1,x_2,\dots,x_n)^{\textrm{T}}$ ''.

58.

On page 237, line -2 replace `` $(b_1,b_2,\dots,b_m)$ '' with `` $(b_1,b_2,\dots,b_m)^{\textrm{T}}$ ''.

59.

On page 241, middle of page, replace ``kth'' with `` $k^{\textrm{th}}$ ''.

60.

On page 247, reference 3, replace ``to appear'' with ``1996''.

61.

On page 274, line -11 replace ``are parallel (Figure'' with ``are parallel and the diagonals intersect in the middle (Figure''.

62.

On page 274, lines -2 and -3, replace ``The diagonals of a rectangle intersect in the middle.'' with ``The diagonals of a rectangle have the same length.''

63.

On page 275, line 1, replace ``the rhombus or diamond, where opposite sides...The diagonals of a rhombus are perpendicular and intersect in the middle.'' with ``the rhombus or diamond, where adjacent sides ...The diagonals of a rhombus are perpendicular.''.

64.

On page 276, section 4.5.3, first paragraph, we now have

A polygon is regular if all its angles are equal. This also implies that all sides are equal. (However, the sides can be equal even if the polygon is not regular, as in the case of the rhombus).

This is incorrect. It should say

A polygon is regular if all its sides are equal and all its angles are equal. Either condition implies the other in the case of a triangle, but not in general. (A rhombus has equal sides but not necessarily equal angles, and a rectangle has equal angles but not necessarily equal sides.)

65.

Page 288, line -6 replace `` $x=a(1+\sin 2\theta)$ '' with `` $x=a \cos^2 \theta$ ''.

66.

Page 290, caption for figure 4.8.23, replace ``for k=.5a, .9, a, 1.1a and 1.5a'' with ``for k=.5a, .9a, a, 1.1a and 1.5a''.

67.

Page 301 (equation (4.10.3) and last equation on page), and also page 302 (line 5): `` $(x,y)\mapsto$ '' should be `` $(x,y,z)\mapsto$ ''.

68.

Page 305, line 1: ``x and y:'' should be ``x, y, and z:''

69.

Page 309, last expression before section 4.15.1: should be $\sqrt{\vert\Gamma/2\Delta\vert}$ .

70.

Page 309, line -2 (for regular dodecahedron) the `` $2\sqrt{5}$ '' should be `` $2\sqrt{5}$ ''.

71.

Page 314, equation (4.18.1), now has

$\displaystyle S=\left( \sum_{i=1}^n\theta_i(n-2)-\pi \right)r^2$

This is incorrect. It should say

$\displaystyle S=\left( \sum_{i=1}^n\theta_i-(n-2)\pi \right)r^2$

72.

On page 320, table at top,

In the table text we have ``With'', this should be ``with''.
We presently have the numbers
1 1 2 3 7 21 49 166 549
This is incorrect. It should say
1 1 2 3 7 21 49 165 552

73.

On page 321, fourth paragraph, we have

...is equivalent to C^k regular parametric representation $\textbf{g}$ if an only ...

This is incorrect. It should say

...is equivalent to a C^k regular parametric representation $\textbf{g}$ if and only ...

74.

On page 321, fifth paragraph, we have

A regular curve C of class C_k is an equivalence

This is incorrect. It should say

A regular curve C of class C^k is an equivalence

75.

On page 324, first line after the displayed matrix

whose the real ...

This is incorrect. It should say

where the real ...

76.

On page 324, line -1, replace `` $g(u,v)\,du^2$ '' with `` $g(u,v)\,dv^2$ ''.

77.

On page 327, line 8, in the Gauss-Mainardi-Codazzi equations

$b_{\alpha\beta , \gamma} - b_{\alpha\gamma , \beta} + \Gamma_{\alpha\beta}^{\gamma} b_{\delta\gamma} - \Gamma_{\alpha\gamma}^{\delta} b_{\delta\beta} = 0$

This is incorrect. It should say (Note indices in first $\Gamma$ )

$b_{\alpha\beta , \gamma} - b_{\alpha\gamma , \beta} + \Gamma_{\alpha\beta}^{\delta} b_{\delta\gamma} - \Gamma_{\alpha\gamma}^{\delta} b_{\delta\beta} = 0$

78.

On page 335, formulae 11, we presently have

$\displaystyle \frac{ d^{2} }{ d {x}^{2}}\left( f(x) \right)= \frac{ d {f(u)}}{... ...x}^{2}} +\frac{ d^{2} f(u) }{ d {u}^{2}}\left( \frac{ d {u}}{ d {x} } \right)^2$

This is incorrect. It should say

$\displaystyle \frac{ d^{2} }{ d {x}^{2}}\left( f(x) \right)= \frac{ d {f(u)}}{... ...x}^{2}} +\frac{ d^{2} f(u) }{ d {u}^{2}}\left( \frac{ d {u}}{ d {x} } \right)^2$

79.

On page 337, second equation from bottom of page, we presently have

$\begin{displaymath} \frac{\partial F }{\partial x_i}=\frac{\partial f }{\partial... ... \hbox{and} \qquad \frac{\partial F }{\partial \lambda_j}=g_j=0\end{displaymath}$

This is incorrect. It should say

$\begin{displaymath} \frac{\partial F }{\partial x_i}=\frac{\partial f }{\partial... ... \hbox{and} \qquad \frac{\partial F }{\partial \lambda_j}=g_j=0\end{displaymath}$

80.

On page 341, last expression before section 5.2, we presently have

$+(I_p\otimes Y)\frac{\partial Y }{\partial Z}$

This is incorrect. It should say

$+(I_p\otimes X)\frac{\partial Y }{\partial Z}$

81.

On page 347, note 4, we now have

$\int_a^b [af(x)+bg(x)]\,dx=a\int_a^b f(x)\,dx+b\int_a^b g(x)\,dx$

This is confusing. It should be replaced by

$\int_a^b [cf(x)+dg(x)]\,dx=c\int_a^b f(x)\,dx+d\int_a^b g(x)\,dx$

82.

On page 349, Useful transformations, number 4 and 5, the transformation is now written as ``when $z=\tan\frac{x}{a}$ ''. This is incorrect, it should be ``when $z=\tan\frac{x}{2}$ ''.

83.

On page 360, make the following statement the second bullet on the page:

Logarithmic expressions are to base $e=2.71828\dots$ , unless otherwise specified, and are to be evaluated for the absolute value of the arguments involved therein.

84.

On page 367, above rule number 108 is

If q=0 then $X=c\left( x+\frac{b}{2c} \right)$ and ...

This is incorrect. It should say

If q=0 then $X=c\left( x+\frac{b}{2c} \right)^2$ and ...

85.

On page 390, rule number 479 has the integral

$\int \log ax+b \,dx=$ ...

This is incorrect. It should say

$\int \log (ax+b) \,dx=$ ...

86.

On page 410, constant coefficient equation, the equation is represented as

$\begin{displaymath} a_0y^{(n)}+a_1y^{(n-1)}+\dots+a_{n-1}y+a_ny=0\end{displaymath}$

This is incorrect. It should be

$\begin{displaymath} a_0y^{(n)}+a_1y^{(n-1)}+\dots+a_{n-1}y'+a_ny=0\end{displaymath}$

87.

On page 410, line -2, replace `` $M(x,y)+dx+N(x,y)\ne0$ '' with `` $M(x,y)dx+N(x,y)dy\ne0$ ''.

88.

On page 413, section 5.6.8, equation number 4 (Bessel equation transformed) the solution is missing the statement: $q \equiv \sqrt{p^2-\beta^2}$ .

89.

On page 419, last equation on page, replace `` $x_N(s=0)=h_N({\bf t})$ '' with `` $x_N(s=0,{\bf t})=h_N({\bf t})$ ''.

90.

On page 423 the first equation (last row of the matrix, first term) now is `` $\Phi_{31}(u^1)$ '' This is incorrect. It should be `` $\Phi_{31}(u^3)$ ''.

91.

On page 428, note 3, replace the three occurrences of ``L'' with `` ${\cal L}$ ''.

92.

On page 429, note 4, replace the three occurrences of ``F'' with `` ${\cal F}$ ''.

93.

On page 432, item number 5, in the displayed equation, the right-most term in the ``r factors'' is now `` $\frac{\partial \tilde{x}^{i_{r}} }{\partial \tilde{x}^{k_{r}}}$ ''. This is incorrect, it should say (note the tilde) `` $\frac{\partial \tilde{x}^{i_{r}} }{\partial {x}^{k_{r}}}$ ''.

94.

On page 432, equation (5.10.4) reads

$\begin{displaymath} T_{3~~~~~~~~~~~~j_{1} \cdots j_{s} \ell_{1} \cdots \ell_{u}}... ... T_{2_{~~~~~~\ell_{1} \cdots \ell_{u}}}^{~k_{1} \cdots k_{t}} .\end{displaymath}$

This is incorrect. It should read (Note font size in $l_1 \cdots l_u$ )

$\begin{displaymath} T_{3~~~~~~~~~~~~j_{1} \cdots j_{s} \ell_{1} \cdots \ell_{u}}... ...r}} T_{2~~~~~~\ell_{1} \cdots \ell_{u}}^{~k_{1} \cdots k_{t}} .\end{displaymath}$

95.

On page 432, equation (5.10.6) reads

$\begin{displaymath} {\cal A}(T)_{i_{1} \cdots i_{r}} = T_{[i_{1} \cdots i_{r}]} ... ...r}} \mbox{sgn}(\sigma) T_{i_{\sigma(1) \cdots i_{\sigma}(1)}} ,\end{displaymath}$

This is incorrect. It should read (Note subscript positioning and $\sigma (r)$ )

$\begin{displaymath} {\cal A}(T)_{i_{1} \cdots i_{r}} = T_{[i_{1} \cdots i_{r}]} ... ...}} \mbox{sgn}(\sigma) T_{i_{\sigma(1)} \cdots i_{\sigma (r)}} ,\end{displaymath}$

96.

On page 433, equation (5.10.8) (line 2) presently has

$\begin{displaymath} \cdots + \Gamma_{~~\ell k}^{i_{r}} T _{~~~~~~~~~j_1 \cdots... ...^{\ell} T_{~~~~~~j_1 \cdots j_{s-1} \ell}^{i_1 \cdots i_{r}} ,\end{displaymath}$

This is incorrect. It should say (note the first superscript of the first T)

$\begin{displaymath} \cdots + \Gamma_{~~\ell k}^{i_{r}} T _{~~~~~~~~~j_1 \cdots... ...^{\ell} T_{~~~~~~j_1 \cdots j_{s-1} \ell}^{i_1 \cdots i_{r}} ,\end{displaymath}$

97.

On page 433, equation (5.10.9) presently has

$\begin{displaymath} \partial_{k}T_{~~~~~j_i \cdots j_{s} }^{i_1 \cdots i_{r}}= ... ... T_{~~~~~j_1 \cdots j_{s},k}^{i_1 \cdots i_{r}} }{\partial x^k}\end{displaymath}$

This is incorrect. It should say (note the last subscript on the last T)

$\begin{displaymath} \partial_{k}T_{~~~~~j_i \cdots j_{s} }^{i_1 \cdots i_{r}}= ... ...al T_{~~~~~j_1 \cdots j_{s}}^{i_1 \cdots i_{r}} }{\partial x^k}\end{displaymath}$

98.

On page 433, item number 4, line 3, we now have

with respect to the condition $\nabla$

This is incorrect. It should read

with respect to the connection $\nabla$

99.

On page 434, line 1, we now have

is said to an ...

This is incorrect. It should read

is said to be an ...

100.

On page 434, item number 2, line -2, we now have

is said to Lorentzian

This is incorrect. It should read

is said to be Lorentzian

101.

On page 435, equation (5.10.15) currently reads

$\begin{displaymath} \dots = \dots - \frac{1}{2} \left( S_{~[ik}^{\ell} T_{\vert... ... + S_{~[i_{rk}}^{\ell} T_{i_{1} \cdots i_{r-1]} \ell} \right) ,\end{displaymath}$

This is incorrect. It should read (note indices)

$\begin{displaymath} \dots = \dots - \frac{1}{2} \left( S_{~[i_{1}k}^{\ell} T_{\... ... + S_{~[i_{r}k}^{\ell} T_{i_{1} \cdots i_{r-1]} \ell} \right) ,\end{displaymath}$

102.

On page 435, item number 3, lines 1-2, now reads

$\dots = X^{\ell} R_{~\ell_{j} k}^{i}$

This is incorrect. It should read (note subscript j in R)

$\dots = X^{\ell} R_{~\ell j k}^{i}$

103.

On page 435, item number 3, line 4, we now have

$2 \nabla_{[i} \nabla_{j]} Y_{k} + \nabla_{\ell} Y_{i} S_{~ij}^{\ell} = - Y_{\ell} R_{~kij}^{\ell}$

This is incorrect. It should read (note subscript k in second term)

$2 \nabla_{[i} \nabla_{j]} Y_{k} - \nabla_{\ell} Y_{k} S_{~ij}^{\ell} = - Y_{\ell} R_{~kij}^{\ell}$

104.

On page 435, equation (5.10.16), we now have

$2 \nabla_{[i} \nabla_{i]} T_{~\ell m}^{k} + \nabla_{n} T_{~\ell m}^{k} S_{~ij}^{n} = \dots$

This is incorrect. It should read (note subscripts on first $\nabla$ ,and change of sign)

$2 \nabla_{[i} \nabla_{j]} T_{~\ell m}^{k} - \nabla_{n} T_{~\ell m}^{k} S_{~ij}^{n} =\dots$

105.

On page 435, equations (5.10.17) and (5.10.18) presently read

$\begin{displaymath} S_{~(jk)}^{i} = 0 , \quad R_{~j(k \ell)}^{i} = 0 , \quad R_{... ...ll]}^{i} = S_{~[jk; \ell]}^{i} + S_{~m[j}^{i} S_{~k \ell]}^{m},\end{displaymath}$

$\begin{displaymath} R_{~j[kl;m]}^{i}+ R_{~jn[k}^{i} S_{~\ell m]}^{n} = 0 .\end{displaymath}$

They are incorrect. They should read (note the sign changes on the 3rd and 4th equations)

$\begin{displaymath} S_{~(jk)}^{i} = 0 , \quad R_{~j(k \ell)}^{i} = 0 , \quad R_{... ...l]}^{i} = -S_{~[jk; \ell]}^{i} + S_{~m[j}^{i} S_{~k \ell]}^{m},\end{displaymath}$

$\begin{displaymath} R_{~j[kl;m]}^{i}- R_{~jn[k}^{i} S_{~\ell m]}^{n} = 0 .\end{displaymath}$

106.

On page 436, item number 3, we now have

The Laplacian of a scalar invariant f is given by $\Delta f = g^{ij} \nabla_{i} \nabla_{i} f = \dots$

This is incorrect. It should read (note the subscript on the second $\nabla$ )

The Laplacian of a scalar invariant f is given by $\Delta f = g^{ij} \nabla_{i} \nabla_{j} f = \dots$

107.

On page 436, equation (5.10.19), we now have

$\begin{displaymath} \begin{array} {rcl} R_{ijk \ell} & = & [j \ell, i],_{k} - [j... ...~+ g^{mn} ([i \ell, m][jk, n] - [ik,m][i \ell, n]) .\end{array}\end{displaymath}$

This is incorrect. It should read (there are changes in all three lines)

$\begin{displaymath} \begin{array} {rcl} R_{ijk \ell} & = & [j \ell, i],_{k} - [j... ...~+ g^{mn} ([i \ell, m][jk, n] - [ik,m][j \ell, n]) .\end{array}\end{displaymath}$

108.

On page 437, equation (5.10.24), we now have

$\begin{displaymath} \tilde{T}_{i_{1} \cdots i_{r}} = T_{j_{1} \cdots j_{r}} \mbox{O}_{i_{1}j_{r}} \cdots \mbox{O}_{i_{r}j_{r}}\end{displaymath}$

This is incorrect. It should read (notice the second subscript on the first O term)

$\begin{displaymath} \tilde{T}_{i_{1} \cdots i_{r}} = T_{j_{1} \cdots j_{r}} \mbox{O}_{i_{1}j_{1}} \cdots \mbox{O}_{i_{r}j_{r}}\end{displaymath}$

109.

On page 439, note number 11, lines 1 and 3 presently have

x²+y²+z²+=r² ...... $U=(0,\pi)\times(0,\pi)\times(0,2\pi)$

These are incorrect. They should be

x²+y²+z²=r² ...... $U=(0,\pi)\times(0,2\pi)$

110.

On page 439, item number 11, last bullet, the Ricci scalar reads

R=-2r²

This is incorrect. It should read

R=-2r^-2

111.

On page 442: equation (5.11.12), ${\bf e}_x$ , ${\bf e}_y$ , and ${\bf e}_z$ should be replaced with ${\bf a}_x$ , ${\bf a}_y$ , and ${\bf a}_z$ .

112.

On page 443: equation (5.11.17), ${\bf e}_r$ , ${\bf e}_{\theta}$ , and ${\bf e}_z$ should be replaced with ${\bf a}_r$ , ${\bf a}_{\theta}$ , and ${\bf a}_z$ .

113.

On page 443, equation (5.11.20), the A_r should be replaced with E_r.

114.

On page 456, second table, heading, third column, replace ``cotx'' with $\cot x$ ''.

115.

On page 473, first line after equation (6.5.2), replace ``us'' with ``is''.

116.

On page 476, section 6.7.1, the definition

$\displaystyle \tanh z= \frac{e^z-e^{-z}}{e^z+e^{ z}}$

is incorrect. It should say

$\displaystyle \tanh z= \frac{e^z-e^{-z}}{e^z+e^{-z}}$

117.

On page 488, line -4, replace ``if $x\ge 0$ and $\alpha\ge0$ '' with ``if $x\ge 0$ and $\alpha\gt$ ''

118.

On page 488, line -2, delete the line ``with $\alpha\gt-1$ ''.

119.

On page 492, section 6.10.9, change the notation. Instead of $P_0(\alpha,\beta)$ , $P_1(\alpha,\beta)$ , ...on the left hand side of the equations, use $P_0^{(\alpha,\beta)}(x)$ , $P_1^{(\alpha,\beta)}(x)$ , ....

120.

Page 499, section 6.13.5, last line

$\displaystyle \sim \frac{2}{\sqrt{{\pi}}}\,\frac{e^{-z^2}}{2z} \left(1-\frac{1}{z^2}+\frac{6}{z^4}- \frac{15}{8\,z^6}+\ldots\right)$

This is incorrect. It should say

$\displaystyle \sim \frac{2}{\sqrt{{\pi}}}\,\frac{e^{-z^2}}{2z} \left(1-\frac{1}{2z^2}+\frac{3}{4z^4}- \frac{15}{8z^6}+\ldots\right)$

121.

Page 507, section 6.16.1, the present title is

DEFINITION OF THE $\psi$ -FUNCTION

This is incorrect. It should say

DEFINITION OF THE F-FUNCTION

122.

Page 520, line -3, replace $cos(\nu\theta)$ '' with $cos(\nu\theta)$ ''.

123.

Page 533, last line on the page, S(f,t) should be S(f,x).

124.

Page 556, item number 6, left hand column, instead of being ``-1'', this should be ``''.

125.

Page 560-564: In the Laplace Transform tables the s's and t's are swapped. The problem can be fixed by doing a global change from t to s in column 1 and from s to t in column 2, and replacing f(t) at the top of column 1 with F(s), and vice-versa.

For example, number 9 should be f(t)=e^at is equivalent to $F(s)=\frac{1}{s-a}$ .

126.

Page 583, equation (7.2.17), last line. We presently have

$\displaystyle =\frac{2}{(\beta-\alpha)_t}\sinh\left[ \frac{(\beta-\alpha)t}{2} \right] e^{(\alpha+\beta)t/2}$

This is incorrect (notice the t in the denominator). It should say

$\displaystyle =\frac{2}{(\beta-\alpha)t}\sinh\left[ \frac{(\beta-\alpha)t}{2} \right] e^{(\alpha+\beta)t/2}$

127.

Page 596,

(a): replace all bullets on page with enumerated indentation (i.e., ``(a)'', ``(b)'', and ``(c)'' (for notes 1, 2, 3) and ``(a)'' and ``(b)'' (for note 4))
(b): note 4, second bullet (now ``(b)''), replace ``step 1'' with ``step (a)''.

128.

Page 597, note 1, bullet 1, last line, replace `` $\frac{1}{4}<0.4<\frac{1}{2}$ '' with `` $\frac{1}{4}<0.4<\frac{3}{4}$ ''.

129.

Page 628, section 7.10.2,

Line 4, change `` ${\bf y}={\bf X}\beta+\epsilon$ '' to `` ${\bf y}={\bf X}\hbox{\pmb{$\beta$}}+\hbox{\pmb{$\epsilon$}}$ ''.
Equation (7.10.7), the left hand sides of the equations should be `` ${\bf y}$ '', `` $\hbox{\pmb{$\beta$}}$ '', and `` $\hbox{\pmb{$\epsilon$}}$ '', not `` ${\bf y}$ '', `` $\beta$ '', and `` $\epsilon$ ''.
In note numbers 3 and 4, change the three occurrences of `` $\hat\beta$ '' to `` $\hat{\hbox{\pmb{$\beta$}}}$ ''.
In note number 6, change `` $\hat\beta$ '' to `` $\hat{\hbox{\pmb{$\beta$}}}$ '' and change `` ${\bf X}$ '' to `` ${\bf x}$ ''.

130.

Page 629, item number 10,

Line 1, change ``x₀'' to `` ${\bf x}_0$ ''.
Line 6, change all three occurrences of `` ${\bf X}_0$ '' to `` ${\bf x}_0$ ''.
Line 6, replace `` $\hat y={\bf X}^{\textrm{T}}\hat\beta$ '' with `` $\hat{y}={\bf x}^{\textrm{T}}\hat{\hbox{\pmb{$\beta$}}}$ ''.

131.

Page 651, section 7.12.6, first equation on page is now

$\displaystyle B(n,x;,p)=\sum_{k=0}^x {n \choose k}p^k(1=p)^{n-k}$

which is incorrect. It should be

$\displaystyle B(n,x;p)=\sum_{k=0}^x {n \choose k}p^k(1-p)^{n-k}$

132.

Page 660, section 7.13.3, it presently says

Let S(t) represent a signal to be recovered, let N(t) represent noise, and let Y(t)=S(t)+N(t) represent the observable. A prediction of the signal is
$\begin{displaymath} S_p(t)=\int_0^{\infty} K(z) Y(t-z)\,dz\end{displaymath}$
where K(z) is a filter. The mean square error is ${\textrm{E}\left[ (S(t)-S_p(t))^2 \right]}$ ...

This is incorrect. It should say (note $S\to X$ ):

Let X(t) represent a signal to be recovered, let N(t) represent noise, and let Y(t)=X(t)+N(t) represent the observable. A prediction of the signal is
$\begin{displaymath} X_p(t)=\int_0^{\infty} K(z) Y(t-z)\,dz\end{displaymath}$
where K(z) is a filter. The mean square error is ${\textrm{E}\left[ (X(t)-X_p(t))^2 \right]}$ ...

133.

Page 675, first line, we now have `` $\vert p_n-p\vert\le b-a/2^m$ '' which is incorrect. This should be `` $\vert p_n-p\vert\le (b-a)/2^m$ ''.

134.

Page 675, Horner's algorithm, second line, we now have ``z=P'(x₀),'' which is incorrect. This should be ``z=P'(x₀)),''.

135.

Page 681, Algorithm for clamped cubic splines,

Statement 2: replace ``a₀'' and ``a_n'' with `` $\alpha_0$ '' and `` $\alpha_n$ ''.
Statement 4: replace ``u₀'' with `` $\mu_0$ ''.

136.

Page 688, middle of page, we now have

$\begin{displaymath} \left[ \begin{array} {ccccc} a_1 & b_2 & 0 & \cdots & 0\\ ... ...a_{n-1} & b_n\\ 0 & 0 & \dots & b_n & a_n \end{array} \right]\end{displaymath}$

which is incorrect. This should be (note the removal of one set of vertical dots in each matrix)

$\begin{displaymath} \left[ \begin{array} {ccccc} a_1 & b_2 & 0 & \cdots & 0\\ ... ...a_{n-1} & b_n\\ 0 & 0 & \dots & b_n & a_n \end{array} \right]\end{displaymath}$

137.

Page 710:

Second line after equation (8.3.2): we currently have `` $y\in S$ ''. This is incorrect, it should be `` $(x,y)\in S$ ''.
Fifth line after equation (8.3.2): we currently have ``y_j=b+jk''. This is incorrect, it should be ``y_j=c+jk''.

138.

Page 711:

First equation on the page, the right hand side is presently ``h²f(x_i,y_j)''. This is incorrect, it should be ``-h²f(x_i,y_j)''.
First line after the first equation on the page contains `` $j=0,1,\dots,m$ ''. This is incorrect, it should be `` $j=1,2,\dots,m-1$ ''.
Last line before the large displayed equation contains ``w_i=u(P_i)''. This is incorrect, it should be `` $f_{\ell}=f(P_{\ell})$ ''.
Large displayed equation is presently

$\begin{displaymath} \begin{array} {crcl} P_1: & 4w_1-w_2-w_4 &=& w_{0,3}+w_{1,4... ...\ P_{9}: & 4w_{9}-w_{8}-w_{6} &=& w_{3,0}+w_{4,1} \end{array}\end{displaymath}$
This is incorrect. It should say:

$\begin{displaymath} \begin{array} {crcl} P_1: & 4w_1-w_2-w_4 &=& w_{0,3}+w_{1,4... ...: & 4w_{9}-w_{8}-w_{6} &=& w_{3,0}+w_{4,1}-h^2f_9 \end{array}\end{displaymath}$

139.

Page 720, last line before Present Value, presently has ``9 $\to\infty$ ''. This is incorrect, it should be `` $q\to\infty$ ''.

140.

Page 720, line 10, we now have ``on page 726.)''. This is incorrect, it should be ``on page 723.)''.

141.

Page 735, the two tables are presently

1cTo obtain 1cMultiply 1cBy

Inches Centimeters $3.937008\times10^{-1}$

Feet Meters 3.280840

Yards Meters 1.093613

Miles Kilometers 0.6213712

Ounces Grams $3.527396\times 10^{-2}$

Pounds Kilograms 2.204623

Gallons (US) Liters 0.2641721

Fluid ounces Milliliters (cc) $3.381402\times10^{-2}$

Square inches Square centimeters 0.1550003

Square feet Square meters 10.76391

Square yards Square meters 1.195990

Cubic inches Milliliters (cc) $6.102374\times 10^{-2}$

Cubic feet Cubic meters 35.31467

Cubic yards Cubic meters 1.307951

1cTo obtain 1cMultiply 1cBy

Microns Mils 25.4

Centimeters Inches 2.540000

Meters Feet 0.3048000

Meters Yards 0.9144000

Kilometers Miles 1.609344

Grams Ounces 28.34952

Kilograms Pounds 0.4535924

Liters Gallons (US) 3.785412

Milliliters (cc) Fluid ounces 29.57353

Square centimeters Square inches 6.451600

Square meters Square feet 0.09290304

Square meters Square yards 0.8361274

Milliliters (cc) Cubic inches 16.38706

Cubic meters Cubic feet $2.831685\times 10^{-2}$

Cubic meters Cubic yards 0.7645549

These are correct as written, but several users mis-read the column captions. I recommend that the order of the columns be changed, and that the terms in exponential notation be changed, to obtain:

1cMultiply 1cBy 1cTo obtain

Centimeters 0.3937008 Inches

Meters 3.280840 Feet

Meters 1.093613 Yards

Kilometers 0.6213712 Miles

Grams 0.03527396 Ounces

Kilograms 2.204623 Pounds

Liters 0.2641721 Gallons (US)

Milliliters (cc) 0.03381402 Fluid ounces

Square centimeters 0.1550003 Square inches

Square meters 10.76391 Square feet

Square meters 1.195990 Square yards

Milliliters (cc) 0.06102374 Cubic inches

Cubic meters 35.31467 Cubic feet

Cubic meters 1.307951 Cubic yards

1cMultiply 1cBy 1cTo obtain

Mils 25.4 Microns

Inches 2.540000 Centimeters

Feet 0.3048000 Meters

Yards 0.9144000 Meters

Miles 1.609344 Kilometers

Ounces 28.34952 Grams

Pounds 0.4535924 Kilograms

Gallons (US) 3.785412 Liters

Fluid ounces 29.57353 Milliliters (cc)

Square inches 6.451600 Square centimeters

Square feet 0.09290304 Square meters

Square yards 0.8361274 Square meters

Cubic inches 16.38706 Milliliters (cc)

Cubic feet 0.02831685 Cubic meters

Cubic yards 0.7645549 Cubic meters

142.

On page 746, Optimization languages, ``AMPLE'' should be ``AMPL''

143.

On page 752, Maria Agnesi, the equation online 6 is incorrect. Presently we have `` $y=\frac{a\sqrt{a-x}}{\sqrt{a}}$ ''. This should be replaced by `` $y=\pm\frac{a\sqrt{a-x}}{\sqrt{x}}$ ''.

144.

Back cover, last line of text is now ``simple complex mathematical problems''. This should be changed to ``simple or complex mathematical problems''.

Errata for Standard Mathematical Tables and Formulae

1cTo obtain	1cMultiply	1cBy
Inches	Centimeters	$3.937008\times10^{-1}$
Feet	Meters	3.280840
Yards	Meters	1.093613
Miles	Kilometers	0.6213712
Ounces	Grams	$3.527396\times 10^{-2}$
Pounds	Kilograms	2.204623
Gallons (US)	Liters	0.2641721
Fluid ounces	Milliliters (cc)	$3.381402\times10^{-2}$
Square inches	Square centimeters	0.1550003
Square feet	Square meters	10.76391
Square yards	Square meters	1.195990
Cubic inches	Milliliters (cc)	$6.102374\times 10^{-2}$
Cubic feet	Cubic meters	35.31467
Cubic yards	Cubic meters	1.307951

Errata for Standard Mathematical Tables and Formulae (30th edition) (First Printing)

Errata for
Standard Mathematical Tables and Formulae (30^th edition)
(First Printing)