# Errata for Standard Mathematical Tables and Formulae (30th 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

This is incorrect. It should be

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

This is incorrect. It should say (note )

6.
On page 38, second line of first displayed equation, replace (x-[x])'' with ''

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 , change to

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, exists. ...
This is incorrect. It should say
...That is, 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 ) For every real number x there is an integer N such that x < n.
This is incorrect. It should say
(Archimedean property of ) For every real number x there is an integer N such that x < N.

14.
On page 68, line18, '' with ''.

15.
On page 70, section 1.6.9, line 10, we presently have
is An orthonormal set ...
This is incorrect. It should say
is 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 is ... expressed as .
This is incorrect. It should say
...series is ... expressed as .

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 258,716 2756839-1 227,832 65,087 2216091-1 65,050

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 378,632 2859433 - 1 258,716 2756839-1 227,832 65,087 2216091-1 65,050

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: (with 4,932 digits), (with 4,030 digits), and (with 3,439 digits).
This is incorrect. It should say
The largest known twin primes are: (with 11,713 digits), (with 5,129 digits), and (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:

This is incorrect. It should say

30.
On page 128:

(a)
line 6, replace '' with ''
(b)
On page 128, item number 2, first displayed equation, now reads:

This is incorrect. It should say

(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 '' with ''

31.
On page 131, top line, we presently have
The characteristic polynomial has...
This is incorrect. It should say (note the upper limit)
The characteristic polynomial has...

32.
Page 131, note number 4, replace eigenvector that'' with eigenvector (different from ) that''

33.
Page 133, section 2.5.15, note 3, replace u(x)'' with ''

34.
On page 135, note 4, line 3, replace '' with ''

35.
On page 136, first displayed equation, replace '' with
''.

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 '' with ''.

41.
On page 174, line 11, the last expression is

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

42.
On page 174, section 3.2.9, first paragraph, we presently have
The Stirling cycle number, , 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 )
The Stirling cycle number, , 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

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

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 is a degree sequence of a graph if there is some ordering of the vertices for which di is the degree of vi for each i.

46.
On page 189,
(a)
In the definition of 2-switch, the last line now has edges and is''. This should be changed to edges and 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 '', this should be changed to ''.
(b)
We now have '', this should be changed to ''.

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, '' should be replaced by '' (that is, the second occurrence of '' should be replaced).
(b)
lines 11-12 of the table, replace an'' with an''.

53.
On page 234, the Logistic Equation'' section,

(a)
second line of equation (3.9.24), replace '' with ''.

(b)
first line after equation (3.9.24), replace '' with ''.

(c)
item number 1, replace When r=k=2'' with When r=k=4''.

(d)
equation (3.9.25), replace xn+1=2xn-xn2'' with xn+1=4xn-xn2''.

(e)
first line after equation (3.9.25), replace '' with ''.

(f)
equation (3.9.26), replace '' with '' (once in each line of this equation).

54.
On pages 234 and 235, the Logistic Equation'' section, replace all six occurrences of '' with ''

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)
This is incorrect. It should say

57.
On page 237, line -3 replace '' with ''.

58.
On page 237, line -2 replace '' with ''.

59.
On page 241, middle of page, replace kth'' with ''.

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 '' with ''.

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): '' should be ''.

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 .

70.
Page 309, line -2 (for regular dodecahedron) the '' should be ''.

71.
Page 314, equation (4.18.1), now has
This is incorrect. It should say

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 Ck regular parametric representation if an only ...
This is incorrect. It should say
...is equivalent to a Ck regular parametric representation if and only ...

74.
On page 321, fifth paragraph, we have
A regular curve C of class Ck is an equivalence
This is incorrect. It should say
A regular curve C of class Ck 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 '' with ''.

77.
On page 327, line 8, in the Gauss-Mainardi-Codazzi equations
This is incorrect. It should say (Note indices in first )

78.
On page 335, formulae 11, we presently have
This is incorrect. It should say

79.
On page 337, second equation from bottom of page, we presently have

This is incorrect. It should say

80.
On page 341, last expression before section 5.2, we presently have
This is incorrect. It should say

81.
On page 347, note 4, we now have
This is confusing. It should be replaced by

82.
On page 349, Useful transformations, number 4 and 5, the transformation is now written as when ''. This is incorrect, it should be when ''.

83.
On page 360, make the following statement the second bullet on the page:
Logarithmic expressions are to base , 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 and ...
This is incorrect. It should say
If q=0 then and ...

85.
On page 390, rule number 479 has the integral
...
This is incorrect. It should say
...

86.
On page 410, constant coefficient equation, the equation is represented as

This is incorrect. It should be

87.
On page 410, line -2, replace '' with ''.

88.
On page 413, section 5.6.8, equation number 4 (Bessel equation transformed) the solution is missing the statement: .

89.
On page 419, last equation on page, replace '' with ''.

90.
On page 423 the first equation (last row of the matrix, first term) now is '' This is incorrect. It should be ''.

91.
On page 428, note 3, replace the three occurrences of L'' with ''.

92.
On page 429, note 4, replace the three occurrences of F'' with ''.

93.
On page 432, item number 5, in the displayed equation, the right-most term in the r factors'' is now ''. This is incorrect, it should say (note the tilde) ''.

94.
On page 432, equation (5.10.4) reads

This is incorrect. It should read (Note font size in )

95.
On page 432, equation (5.10.6) reads

This is incorrect. It should read (Note subscript positioning and )

96.
On page 433, equation (5.10.8) (line 2) presently has

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

97.
On page 433, equation (5.10.9) presently has

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

98.
On page 433, item number 4, line 3, we now have
with respect to the condition
This is incorrect. It should read
with respect to the connection

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

This is incorrect. It should read (note indices)

102.
On page 435, item number 3, lines 1-2, now reads
This is incorrect. It should read (note subscript j in R)

103.
On page 435, item number 3, line 4, we now have
This is incorrect. It should read (note subscript k in second term)

104.
On page 435, equation (5.10.16), we now have
This is incorrect. It should read (note subscripts on first ,and change of sign)

105.
On page 435, equations (5.10.17) and (5.10.18) presently read

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

106.
On page 436, item number 3, we now have
The Laplacian of a scalar invariant f is given by
This is incorrect. It should read (note the subscript on the second )
The Laplacian of a scalar invariant f is given by

107.
On page 436, equation (5.10.19), we now have

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

108.
On page 437, equation (5.10.24), we now have

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

109.
On page 439, note number 11, lines 1 and 3 presently have
x2+y2+z2+=r2 ......
These are incorrect. They should be
x2+y2+z2=r2 ......

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), , , and should be replaced with , , and .

112.
On page 443: equation (5.11.17), , , and should be replaced with , , and .

113.
On page 443, equation (5.11.20), the Ar should be replaced with Er.

114.
On page 456, second table, heading, third column, replace cotx'' with ''.

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
is incorrect. It should say

117.
On page 488, line -4, replace if and '' with if and ''

118.
On page 488, line -2, delete the line with ''.

119.
On page 492, section 6.10.9, change the notation. Instead of , , ...on the left hand side of the equations, use , , ....

120.
Page 499, section 6.13.5, last line
This is incorrect. It should say

121.
Page 507, section 6.16.1, the present title is
DEFINITION OF THE -FUNCTION
This is incorrect. It should say
DEFINITION OF THE F-FUNCTION

122.
Page 520, line -3, replace '' with ''.

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)=eat is equivalent to .

126.
Page 583, equation (7.2.17), last line. We presently have
This is incorrect (notice the t in the denominator). It should say

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 '' with ''.

129.
Page 628, section 7.10.2,

• Line 4, change '' to ''.
• Equation (7.10.7), the left hand sides of the equations should be '', '', and '', not '', '', and ''.
• In note numbers 3 and 4, change the three occurrences of '' to ''.
• In note number 6, change '' to '' and change '' to ''.

130.
Page 629, item number 10,

• Line 1, change x0'' to ''.
• Line 6, change all three occurrences of '' to ''.
• Line 6, replace '' with ''.

131.
Page 651, section 7.12.6, first equation on page is now
which is incorrect. It should be

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

where K(z) is a filter. The mean square error is ...
This is incorrect. It should say (note ):
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

where K(z) is a filter. The mean square error is ...

133.
Page 675, first line, we now have '' which is incorrect. This should be ''.

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 an'' with '' and ''.
• Statement 4: replace u0'' with ''.

136.
Page 688, middle of page, we now have

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

137.
Page 710:

• Second line after equation (8.3.2): we currently have ''. This is incorrect, it should be ''.
• Fifth line after equation (8.3.2): we currently have yj=b+jk''. This is incorrect, it should be yj=c+jk''.

138.
Page 711:

• First equation on the page, the right hand side is presently h2f(xi,yj)''. This is incorrect, it should be -h2f(xi,yj)''.
• First line after the first equation on the page contains ''. This is incorrect, it should be ''.
• Last line before the large displayed equation contains wi=u(Pi)''. This is incorrect, it should be ''.
• Large displayed equation is presently

This is incorrect. It should say:

139.
Page 720, last line before Present Value, presently has 9''. This is incorrect, it should be ''.

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 Feet Meters 3.280840 Yards Meters 1.093613 Miles Kilometers 0.6213712 Ounces Grams Pounds Kilograms 2.204623 Gallons (US) Liters 0.2641721 Fluid ounces Milliliters (cc) Square inches Square centimeters 0.1550003 Square feet Square meters 10.76391 Square yards Square meters 1.195990 Cubic inches Milliliters (cc) 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 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 ''. This should be replaced by ''.

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