Errata for
by Daniel Zwillinger (Editorinchief)

NOTES:
ERRATA:
This entry updated before 7 March 2001.
Errata for this book will be maintained at
http://errata.mathtable.com
This entry updated before 7 March 2001.
(Thanks to Richard Hughes for this correction.)
This entry last updated 25 Februrary 2002.
A symbol following one of lesser value has the lesser value subtracted from the larger value (for example, IV, IX, VM),This is incorrect, it should have been
A symbol following one of lesser value has the lesser value subtracted from the larger value. An I is only allowed to precede a V or an X, an X is only allowed to precede an L or a C, and a C is only allowed to precede a D or an M. (For example, IV, IX, CD),
This entry updated before 7 March 2001.
(for example, XIV , CIX , DVL ).This is incorrect, it should have been
(for example, XIV , CIX , DXL ).
This entry updated before 7 March 2001.
...(also called the falling factorial ... (sometimes ) ...This is incorrect. It should have been
...(also called the rising factorial ... (sometimes ) ...
[This brings the description into line with, e.g., D. Knuth, Art of Computer Programming, Vol I, 3rd Edition, Section 1.2.5, Permutations and Factorials, page 50.]
(Thanks to David Gehrig for this correction.)
This entry updated before 7 March 2001.
(Thanks to David Cantrell for these corrections.)
This entry updated before 7 March 2001.
This is incorrect. It should have been (note the exponents on the terms)
(Thanks to Alain Boulanger for this correction.)
This entry updated before 7 March 2001.
This is incorrect. It should have been
(Thanks to Richard E. Stone for this correction.)
This entry updated before 7 March 2001.
This is incorrect. It should have been (note the summation should start at 0, not 1)
(Thanks to Greg Iles for this correction.)
This entry updated before 7 March 2001.
(Thanks to Barry Pasternack for this addition.)
This entry updated before 7 March 2001.
This is incorrect, it should have been
This is incorrect, it should have been
This entry updated before 7 March 2001.
This is incorrect and should be:
(Thanks to Kosa Gabor for this addition.)
This entry updated before 7 March 2001.
positive definiteness: for all , in , and if and only if .This is incorrect and should be (note that ``'' is replaced with ``):
positive definiteness: for all , in , and if and only if .
(Thanks to Paul Stanford for this correction.)
This entry updated before 7 March 2001.
for (Fourier sine series)which is incorrect. It should have been
for (Fourier sine series)
This entry last updated 5 October 2001.
.This is incorrect. It should have been
.
(Thanks to Catherine Roberts for this correction.)
This entry updated before 7 March 2001.
(1) 
(2) 
Similarly for Repeated quadratic factors. In other words, the number of equations needed is half the number stated.
(Thanks to David M. Bradley for these corrections.)
This entry updated before 7 March 2001.
The solutions are given by , , and , whereThis is incorrect. It should have been (note the signs)
The solutions are given by , , and , where
This entry updated before 7 March 2001.
This entry last updated 25 Februrary 2002.
This is incorrect. It should have been
(Thanks to See Chew for this correction.)
This entry updated before 7 March 2001.
(Thanks to Paul Stanford for this correction.)
This entry updated before 7 March 2001.
X=(BCE)This is incorrect. It should have been (note the missing ):
X=(BCED)
The inverse of a matrix is as follows (defined when ):
This entry updated before 7 March 2001.
Only if is square and nonsingular, will be unique and . Otherwise, there will exist infinitely many matrices that will satisfy the defining relation.This is incorrect. It should have been
There is a unique pseudoinverse satisfying the conditions in (2.5.26).
If, and only if, is square and nonsingular, then .
(Thanks to Pablo A. Parrilo for this correction.)
This entry updated before 7 March 2001.
(Thanks to Paul Stanford for this correction.)
This entry updated before 7 March 2001.
Suppose that is a permutation with cycles of length , cycles of length , ..., cycles of length in its unique cycle decomposition. Then can be encoded as the expression . Summing these expressions for all permutations in the group , and normalizing by the number of elements in results in the cycle index of the group :
This entry updated before 7 March 2001.
This entry updated before 7 March 2001.
(Thanks to Katherine Jane Harine for this correction.)
This entry last updated 25 Februrary 2002.
There are also several other small errors in sections 3.2.8 and 3.2.9.
(Thanks to David Jeffrey for pointing out many errors.)
This entry last updated 5 October 2001.
This is incorrect. This row should have been (look at second to last element)
This entry updated before 7 March 2001.
(Thanks to Joe Rushanan for these corrections.)
This entry updated before 7 March 2001.
(Thanks to L. W. Maxwell for this correction.)
This entry updated before 7 March 2001.
(All angles are measured in radians.)
This entry updated before 7 March 2001.
This is incorrect. It should have been
(Thanks to Gary E. Young for this correction.)
This entry updated before 7 March 2001.
Note the approximation
(Thanks to David F. Rivera for this addition.)
This entry updated before 7 March 2001.
SettingThis is incorrect, it should have been
Setting
(Thanks to David Cantrell for these corrections.)
This entry updated before 7 March 2001.
This is incorrect. It should say (note )
(Thanks to James Seed for this correction.)
This entry updated before 7 March 2001.
A curve parameterized by arclength and such that the curvature is proportional to the parameter at each point is a Bernoulli spiral.This is incorrect. It should say
A curve parameterized by arclength and such that the radius of curvature is proportional to the parameter at each point is a Bernoulli spiral.
This entry updated before 7 March 2001.
A curve parameterized by arclength and such that the curvature is inversely proportional to the parameter at each point is a Cornu spiral (compare the Bernoulli spiral).This is incorrect. It should say
A curve parameterized by arclength and such that the radius of curvature is inversely proportional to the parameter at each point is a Cornu spiral (compare the Bernoulli spiral).
This entry updated before 7 March 2001.
A rotationreflection (rotation through an angle around a line composed with reflection in a plane perpendicular to ).
(Thanks to Jerry Grossman for these corrections.)
This entry updated before 7 March 2001.
7.663119This is incorrect. It should have been
7.663119
(Thanks to David G. Simpson for this correction.)
This entry updated before 7 March 2001.
Four points not on the same ...This is incorrect. It should have been
Four points not on the same ...
This entry updated before 7 March 2001.
This is correct, but could be clarified by writing it as:
(Thanks to Randolph J. Herber for this clarification.)
This entry updated before 7 March 2001.
This is incorrect. It should have been
(Thanks to Richard W. Johnson for this correction.)
This entry last updated 20 September 2001.
The solutions are: (furthest), and (closest).These are incorrect. They should have been (note the last equation on the first line is missing a ``1'', and a slash is missing in the two solutions):
The solutions are: (furthest), and (closest).
(Thanks to Bruno Van der Bossche for these corrections.)
This entry updated before 7 March 2001.
This is incorrect. It should have been
(Thanks to Pablo A. Parrilo for this correction.)
This entry updated before 7 March 2001.
whenThis is incorrect. It should have been
when
This entry last updated 25 Februrary 2002.
(5.3.21) 
(5.3.21) 
(Thanks to Richard B. Evans for these corrections.)
This entry updated before 7 March 2001.
This is incorrect, it should have been
(Thanks to Richard Finley for this correction.)
This entry updated before 7 March 2001.
If , then and other formulae should be used.This is incorrect, it should have been
If , then and other formulae should be used.
This entry last updated 25 Februrary 2002.
This is incorrect, it should have been
(Thanks to Jonathan Thomas Bartley for these corrections.)
This entry updated before 7 March 2001.
(Thanks to Guizhong Zhang for this correction.)
This entry updated before 7 March 2001.
, .This is incorrect and should be (note the upper limit on the second integral):
, .
, , is an odd integer.This is incorrect and should be (note the lower limit on the second integral):
, , is an odd integer.
, is an integer.This is correct but should have been (note the upper limit on the second integral):
, is an integer.
FOOTNOTE: An alert reader will wonder how these errors could have occurred, since the integrals in the 30th edition of this book have been electronically verified. The error occurred in the typesetting of the integralsnot in the electronic verification of the integrals. In this section on definite integrals, there are 11 integrals in which there were two integrals on the same line; in all cases the limits on the first integral were (sometimes incorrectly) printed as the limits on the second integral. This resulted in the errors listed above.
(Thanks to Michael Strauss for this correction.)
This entry updated before 7 March 2001.
(Thanks to Luke Sweatlock for this correction.)
This entry last updated 25 Februrary 2002.
(Thanks to Luke Sweatlock for this correction.)
This entry last updated 25 Februrary 2002.
This is incorrect. It should have been
,This is incorrect. It should have been
,
This is incorrect. It should have been (note the subscript)
.This is incorrect. It should have been
, and .
This entry updated before 7 March 2001.
(Thanks to Harry Watson of the Naval Warfare Assessment Division (NWAD) for this correction.)
This entry updated before 7 March 2001.
This entry updated before 7 March 2001.
Let be a state vector, let be an observable vector, and let be the control. The vectors , , and have , and components, respectively. If a system evolves as:
then, taking Laplace transforms, where is the transfer function given by .
A system is said to be controllable if and only if for any times and any states there exists a control such that and . The system is controllable if and only if rank .
If, given and on some interval , the value of can be deduced on that interval, then the system is said to be observable. Observability is equivalent to the condition rank .
(Thanks to Pablo A. Parrilo for this correction.)
This entry updated before 7 March 2001.
(Thanks to Andre D. Bandrauk for these corrections.)
This entry updated before 7 March 2001.
(Thanks to Richard F. Stein for this correction.)
This entry updated before 7 March 2001.
(Thanks to Richard Hughes for this correction.)
This entry last updated 25 Februrary 2002.
(Thanks to Donal M. Ragan for this correction.)
This entry last updated 25 Februrary 2002.
This is incorrect. It should have been
(Thanks to William Weintraub for this correction.)
This entry last updated 20 September 2001.
(Thanks to Michael E. Kutz for this correction.)
This entry updated before 7 March 2001.
(Thanks to David W. Cantrell for this correction.)
This entry updated before 7 March 2001.
This is incorrect. It should have been
(Thanks to David Lassonde for this correction.)
This entry updated before 7 March 2001.
This is incorrect. It should have been
74
74
(Thanks to Dale Hinds for this correction.)
This entry last updated 20 September 2001.
A needle of length is placed at random on a plane on which are ruled parallel lines at unit distance apart. Assume that so that only one intersection is possible. The probability that the needle intersects a line is
A needle of length is placed at random on a plane on which are ruled parallel lines a distance apart. If then only one intersection is possible. The probability that the needle intersects a line is
This entry updated before 7 March 2001.
and
They are both incorrect. They should have been
and
(Thanks to Dillard David Ensley for this correction.)
This entry updated before 7 March 2001.
then the variable is said to possess a uniform distributionThis is incorrect. It should have been
then the variable is said to possess a normal distribution
(Thanks to Ian M. Dew of PacificSierra Research Corp. for this correction.)
This entry updated before 7 March 2001.
(Thanks to Paul Stanford for this correction.)
This entry updated before 7 March 2001.
is the probability that a Markov chain in state at time will be in state at time .This is incorrect. It should have been
is the probability that a Markov chain in state at time will be in state at time .
(Thanks to Harry Watson of the Naval Warfare Assessment Division (NWAD) for this correction.)
Define the step transition matrix by as the matrix with entries
Define the step transition matrix by as the matrix with entries
(Thanks to Harry Watson of the Naval Warfare Assessment Division (NWAD) for this correction.)
This entry updated before 7 March 2001.
This is incorrect, it should have been
(Thanks to Dennis J. Day for this correction.)
This entry updated before 7 March 2001.
This should have been (note the missing closing parenthesis for the 3113 entry):
Primitive trinomial exponents (5,2) (7,1) (7,3) (17,3) (17,5) (17,6) (31,3) (31,6) (31,7) (31,13 (127,1) (521,32)
Primitive trinomial exponents (5,2) (7,1) (7,3) (17,3) (17,5) (17,6) (31,3) (31,6) (31,7) (31,13) (127,1) (521,32)
(Thanks to Paul Stanford for this correction.)
This entry updated before 7 March 2001.
This is incorrect. This should have been
This entry updated before 7 March 2001.
...let be the th largest of the values ( ). Hence is the maximum of the values and is the minimum of the values. Then . Hence
...let be the th largest of the values ( ). Hence is the maximum of the values and is the minimum of the values. Then . Hence
This entry updated before 7 March 2001.
...confidence interval for is given by ...This is incorrect. It should have been
...confidence interval for is given by ...
This entry updated before 7 March 2001.
(Thanks to Pablo A. Parrilo for this correction.)
This entry updated before 7 March 2001.
Given the points , , ... the ...This is incorrect. It should have been (notice the subscripts)
Given the points , , ... the ...
(Thanks to Harry Watson of the Naval Warfare Assessment Division (NWAD) for this correction.)
This entry updated before 7 March 2001.
(Thanks to Jeffrey D. Oldham for this correction.)
This entry updated before 7 March 2001.
(Thanks to Pablo A. Parrilo for this correction.)
This entry updated before 7 March 2001.
This entry updated before 7 March 2001.
Double integrals of polynomials over polygons
If the vertices of the polygon are , and we define (with and ) then
(Thanks to Joaquin Marin for this addition.)
This entry updated before 7 March 2001.
This entry updated before 7 March 2001.
Mathematica http://www.wri.comWhile this works, a better reference is
Mathematica http://www.wolfram.com
(Thanks to David Gehrig for this correction.)
This entry updated before 7 March 2001.
American Standard Code for InformationThis is incorrect. It should have been
American Standard Code for Information
(Thanks to Alex Fabrikant for this correction.)
This entry last updated 5 October 2001.
is the floor function (greatest integer less than or equal to the argument)
is the ceiling function (least integer larger than or equal to than the argument)
(Thanks to David Cantrell for this correction.)
This entry last updated 25 Februrary 2002.