CRC standard mathematical tables and formulae, thirtieth edition

1.1 CONSTANTSNatural numbers The natural numbers are customarily denoted by N.. Rational numbers The rational numbers are customarily denoted byQ.. Two fractions p q andr Real numbers ar

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Rensselaer Polytechnic Institute

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AT&T Bell Laboratories

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Pennsylvania State University

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George K Tzanetopoulos

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Probability and Statistics

William C Rinaman, Christopher Heil, Michael T Strauss, Michael Mascagni, and Mike Sousa

Chapter 1

Analysis

1.1 CONSTANTS

1.1.1 Types of numbers 1.1.2 Representation of numbers 1.1.3 Decimal multiples and prefixes 1.1.4 Roman numerals

1.1.5 Decimal equivalents of common fractions 1.1.6 Hexadecimal addition and subtraction table 1.1.7 Hexadecimal multiplication table

1.1.8 Hexadecimal–decimal fraction conversion table

1.2 SPECIAL NUMBERS

1.2.1 Positive powers of 2 1.2.2 Negative powers of 2 1.2.3 Powers of 16 in decimal scale 1.2.4 Powers of 10 in hexadecimal scale 1.2.5 Special constants

1.2.6 Factorials 1.2.7 Important numbers in different bases 1.2.8 Bernoulli polynomials and numbers 1.2.9 Euler polynomials and numbers 1.2.10 Fibonacci numbers

1.2.11 Powers of integers 1.2.12 Sums of powers of integers 1.2.13 Negative integer powers 1.2.14 Integer sequences 1.2.15 de Bruijn sequences

1.3 SERIES AND PRODUCTS

1.3.1 Definitions 1.3.2 General properties 1.3.3 Convergence tests 1.3.4 Types of series 1.3.5 Summation formulae 1.3.6 Improving convergence: Shanks transformation 1.3.7 Summability methods

1.3.8 Operations with series 1.3.9 Miscellaneous sums and series 1.3.10 Infinite series

1.3.11 Infinite products 1.3.12 Infinite products and infinite series

1.4 FOURIER SERIES

1.4.1 Special cases 1.4.2 Alternate forms 1.4.3 Useful series 1.4.4 Expansions of basic periodic functions

1.5 COMPLEX ANALYSIS

1.5.1 Definitions 1.5.2 Operations on complex numbers 1.5.3 Powers and roots of complex numbers 1.5.4 Functions of a complex variable 1.5.5 Cauchy–Riemann equations 1.5.6 Cauchy integral theorem 1.5.7 Cauchy integral formula 1.5.8 Taylor series expansions 1.5.9 Laurent series expansions 1.5.10 Zeros and singularities 1.5.11 Residues

1.5.12 The argument principle 1.5.13 Transformations and mappings 1.5.14 Bilinear transformations 1.5.15 Table of transformations 1.5.16 Table of conformal mappings

1.6 REAL ANALYSIS

1.6.1 Relations 1.6.2 Functions (mappings) 1.6.3 Sets of real numbers 1.6.4 Topology

1.6.5 Metric space 1.6.6 Convergence in R

1.1 CONSTANTS

Natural numbers

The natural numbers are customarily denoted by N They are the set {0, 1, 2, }.

Many authors do not consider 0 to be a natural number

Integers

The integers are customarily denoted by Z They are the set {0, ±1, ±2, }.

Rational numbers

The rational numbers are customarily denoted byQ They are the set {p q | p, q ∈

Z, q = 0} Two fractions p q andr

Real numbers are often divided into two subsets One subset, the algebraic

numbers, are real numbers which solve a polynomial equation in one variable with

integer coefficients For example; √1

2 is an algebraic number because it solves thepolynomial equation 2x2− 1 = 0, and rational numbers are algebraic Real numbers

that are not algebraic numbers are called transcendental numbers Examples of

transcendental numbers includeπ and e.

Complex numbers

The complex numbers are customarily denoted byC They are numbers of the form

The sum of two complex numbersa + bi and c + di is a + c + (b + d)i The

product of two complex numbersa + bi and c + di is ac − bd + (ad + bc)i The

reciprocal of the complex numbera +bi is a

a2+b2− b

a2+b2i If z = a +bi, the complex

conjugate ofz is z = a − bi Properties include: z + w = z + w and zw = z w.

1.1.2 REPRESENTATION OF NUMBERS

Numerals as usually written have radix or base 10, because the numerala n a n−1

a2a1a0represents the numbera n10n +a n−110n−1 +· · ·+a2102+a110+a0 However,

other bases can be used, particularly bases 2, 8, and 16 (called binary, octal, and

hexadecimal, respectively) When another base is used, it is indicated by a subscript:

5437 = 5 × 72+ 4 × 7 + 3 = 276,

101112 = 1 × 24+ 0 × 23+ 1 × 22+ 1 × 2 + 1 = 23,

A316 = 10 × 16 + 3 = 163.

When writing a number in baseb, the digits used can range from 0 to b−1 If b > 10,

then the digit A stands for 10, B for 11, etc

The above algorithm can be used to convert a numeral from baseb to base 10 To

convert a numeral from base 10 to baseb, divide the numeral by b, and the remainder

will be the last digit Then divide the quotient by b, using the remainder as the

previous digit Continue dividing the quotient byb until a quotient of 0 is arrived at.

For example, to convert 574 to base 12, divide, yielding a remainder of 10 and

a quotient of 47 Hence, the last digit of the answer isA Divide 47 by 12, giving a

remainder of 11 again and a quotient of 3 Divide 3 by 12, giving a remainder of 3and a quotient of 0 Therefore, 57510= 3BA12

In general, to convert from baseb to base r, it is simplest to convert to base 10

as an intermediate step However, it is simple to convert from baseb to base b a For

example, to convert 1101111012 to base 16, group the digits in fours (because 16

is 24), yielding 1 1011 1101, and then convert each group of 4 to base 16 directly,yielding 1BD16

1.1.3 DECIMAL MULTIPLES AND PREFIXES

The prefix and symbols below are taken from Conference G´en´erale des Poids etMesures, 1991 The common names are for the U.S

The major symbols in Roman numerals are I= 1, V = 5, X = 10, L = 50, C = 100,

D= 500, and M = 1,000 The rules for constructing Roman numerals are

1 A symbol following one of equal or greater value adds its value (for example,

1/64 0.0156251/32 2/64 0.031253/64 0.0468751/16 2/32 4/64 0.0625

5/64 0.0781253/32 6/64 0.093757/64 0.1093751/8 4/32 8/64 0.125

9/64 0.1406255/32 10/64 0.1562511/64 0.1718753/16 6/32 12/64 0.1875

13/64 0.2031257/32 14/64 0.2187515/64 0.2343751/4 8/32 16/64 0.25

17/64 0.2656259/32 18/64 0.2812519/64 0.2968755/16 10/32 20/64 0.3125

21/64 0.32812511/32 22/64 0.3437523/64 0.3593753/8 12/32 24/64 0.375

25/64 0.39062513/32 26/64 0.4062527/64 0.4218757/16 14/32 28/64 0.4375

29/64 0.45312515/32 30/64 0.4687531/64 0.4843751/2 16/32 32/64 0.5

33/64 0.51562517/32 34/64 0.5312535/64 0.5468759/16 18/32 36/64 0.5625

37/64 0.57812519/32 38/64 0.5937539/64 0.6093755/8 20/32 40/64 0.625

41/64 0.64062521/32 42/64 0.6562543/64 0.67187511/16 22/32 44/64 0.6875

45/64 0.70312523/32 46/64 0.7187547/64 0.7343753/4 24/32 48/64 0.75

49/64 0.76562525/32 50/64 0.7812551/64 0.79687513/16 26/32 52/64 0.8125

53/64 0.82812527/32 54/64 0.8437555/64 0.8593757/8 28/32 56/64 0.875

57/64 0.89062529/32 58/64 0.9062559/64 0.92187515/16 30/32 60/64 0.9375

61/64 0.95312531/32 62/64 0.9687563/64 0.9843751/1 32/32 64/64 1

1.1.6 HEXADECIMAL ADDITION AND SUBTRACTION TABLE

A= 10, B = 11, C = 12, D = 13, E = 14, F = 15.

Example: 6+ 2 = 8; hence 8 − 6 = 2 and 8 − 2 = 6

Example: 4+ E = 12; hence 12 − 4 = E and 12 − E = 4

1.1.8 HEXADECIMAL–DECIMAL FRACTION CONVERSION

TABLE

Hex Decimal Hex Decimal Hex Decimal Hex Decimal.00 0 40 0.250000 80 0.500000 C0 0.750000.01 0.003906 41 0.253906 81 0.503906 C1 0.753906.02 0.007812 42 0.257812 82 0.507812 C2 0.757812.03 0.011718 43 0.261718 83 0.511718 C3 0.761718.04 0.015625 44 0.265625 84 0.515625 C4 0.765625.05 0.019531 45 0.269531 85 0.519531 C5 0.769531.06 0.023437 46 0.273437 86 0.523437 C6 0.773437.07 0.027343 47 0.277343 87 0.527343 C7 0.777343.08 0.031250 48 0.281250 88 0.531250 C8 0.781250.09 0.035156 49 0.285156 89 0.535156 C9 0.785156.0A 0.039062 4A 0.289062 8A 0.539062 CA 0.789062.0B 0.042968 4B 0.292968 8B 0.542968 CB 0.792968.0C 0.046875 4C 0.296875 8C 0.546875 CC 0.796875.0D 0.050781 4D 0.300781 8D 0.550781 CD 0.800781.0E 0.054687 4E 0.304687 8E 0.554687 CE 0.804687.0F 0.058593 4F 0.308593 8F 0.558593 CF 0.808593.10 0.062500 50 0.312500 90 0.562500 D0 0.812500.11 0.066406 51 0.316406 91 0.566406 D1 0.816406.12 0.070312 52 0.320312 92 0.570312 D2 0.820312.13 0.074218 53 0.324218 93 0.574218 D3 0.824218.14 0.078125 54 0.328125 94 0.578125 D4 0.828125.15 0.082031 55 0.332031 95 0.582031 D5 0.832031.16 0.085937 56 0.335937 96 0.585937 D6 0.835937.17 0.089843 57 0.339843 97 0.589843 D7 0.839843.18 0.093750 58 0.343750 98 0.593750 D8 0.843750.19 0.097656 59 0.347656 99 0.597656 D9 0.847656.1A 0.101562 5A 0.351562 9A 0.601562 DA 0.851562.1B 0.105468 5B 0.355468 9B 0.605468 DB 0.855468.1C 0.109375 5C 0.359375 9C 0.609375 DC 0.859375.1D 0.113281 5D 0.363281 9D 0.613281 DD 0.863281.1E 0.117187 5E 0.367187 9E 0.617187 DE 0.867187.1F 0.121093 5F 0.371093 9F 0.621093 DF 0.871093

Hex Decimal Hex Decimal Hex Decimal Hex Decimal.20 0.125000 60 0.375000 A0 0.625000 E0 0.875000.21 0.128906 61 0.378906 A1 0.628906 E1 0.878906.22 0.132812 62 0.382812 A2 0.632812 E2 0.882812.23 0.136718 63 0.386718 A3 0.636718 E3 0.886718.24 0.140625 64 0.390625 A4 0.640625 E4 0.890625.25 0.144531 65 0.394531 A5 0.644531 E5 0.894531.26 0.148437 66 0.398437 A6 0.648437 E6 0.898437.27 0.152343 67 0.402343 A7 0.652343 E7 0.902343.28 0.156250 68 0.406250 A8 0.656250 E8 0.906250.29 0.160156 69 0.410156 A9 0.660156 E9 0.910156.2A 0.164062 6A 0.414062 AA 0.664062 EA 0.914062.2B 0.167968 6B 0.417968 AB 0.667968 EB 0.917968.2C 0.171875 6C 0.421875 AC 0.671875 EC 0.921875.2D 0.175781 6D 0.425781 AD 0.675781 ED 0.925781.2E 0.179687 6E 0.429687 AE 0.679687 EE 0.929687.2F 0.183593 6F 0.433593 AF 0.683593 EF 0.933593.30 0.187500 70 0.437500 B0 0.687500 F0 0.937500.31 0.191406 71 0.441406 B1 0.691406 F1 0.941406.32 0.195312 72 0.445312 B2 0.695312 F2 0.945312.33 0.199218 73 0.449218 B3 0.699218 F3 0.949218.34 0.203125 74 0.453125 B4 0.703125 F4 0.953125.35 0.207031 75 0.457031 B5 0.707031 F5 0.957031.36 0.210937 76 0.460937 B6 0.710937 F6 0.960937.37 0.214843 77 0.464843 B7 0.714843 F7 0.964843.38 0.218750 78 0.468750 B8 0.718750 F8 0.968750.39 0.222656 79 0.472656 B9 0.722656 F9 0.972656.3A 0.226562 7A 0.476562 BA 0.726562 FA 0.976562.3B 0.230468 7B 0.480468 BB 0.730468 FB 0.980468.3C 0.234375 7C 0.484375 BC 0.734375 FC 0.984375.3D 0.238281 7D 0.488281 BD 0.738281 FD 0.988281.3E 0.242187 7E 0.492187 BE 0.742187 FE 0.992187.3F 0.246093 7F 0.496093 BF 0.746093 FF 0.996093

1.2.3 POWERS OF 16 IN DECIMAL SCALE

1.2.5 SPECIAL CONSTANTS

The transcedental numberπ is defined as the ratio of the circumference of a circle to

the diameter It is also the ratio of the area of a circle to the square of the radius (r)

and appears in several other formulas from elementary geometry (see Section 4.6)

3πr3,

One method of computingπ is to use the infinite series for the function tan−1x

and one of the identities

To 50 decimal places:

π/20 ≈ 0.15707 96326 79489 66192 31321 69163 97514 42098 58469 96876 π/15 ≈ 0.20943 95102 39319 54923 08428 92218 63352 56131 44626 62501 π/12 ≈ 0.26179 93877 99149 43653 85536 15273 29190 70164 30783 28126 π/11 ≈ 0.28559 93321 44526 65804 20584 89389 04571 67451 97218 12501 π/10 ≈ 0.31415 92653 58979 32384 62643 38327 95028 84197 16939 93751 π/9 ≈ 0.34906 58503 98865 91538 47381 53697 72254 26885 74377 70835 π/8 ≈ 0.39269 90816 98724 15480 78304 22909 93786 05246 46174 92189 π/7 ≈ 0.44879 89505 12827 60549 46633 40468 50041 20281 67057 05359 π/6 ≈ 0.52359 87755 98298 87307 71072 30546 58381 40328 61566 56252 π/5 ≈ 0.62831 85307 17958 64769 25286 76655 90057 68394 33879 87502 π/4 ≈ 0.78539 81633 97448 30961 56608 45819 87572 10492 92349 84378 π/3 ≈ 1.04719 75511 96597 74615 42144 61093 16762 80657 23133 12504 π/2 ≈ 1.57079 63267 94896 61923 13216 91639 75144 20985 84699 68755

The functione xis defined bye x =
∞n=0 x n

n! The numberse and π are related by the

formulae πi= −1

of every other integer: n!! = n · (n − 2) · (n − 4) · · · , where the last element in the

product is either 2 or 1, depending on whethern is even or odd The generalization of

the factorial function is the gamma function (see Section 6.11) Whenn is an integer,

(n) = (n − 1)!.

The shifted factorial (also called the falling factorial and Pochhammer’s symbol)

is denoted by(a) n(sometimesa n) and is defined as

Theq-shifted factorial is defined as

1.2.7 IMPORTANT NUMBERS IN DIFFERENT BASES

The Bernoulli polynomialsB n (x) are defined by the generating function

The Bernoulli numbers are the Bernoulli polynomials evaluated at 0: B n =

n=0 B n n! t n = t

fraction of integers:B n = Nn /D n Note thatB2m+1 = 0 for m ≥ 1.

The Euler polynomialsE n (x) are defined by the generating function

Alternating sums of powers can be computed in terms of Euler polynomials

An exact formula is available:F n= √ 1

5

1 +√5 2

n

− 1 −√5 2

n Note that lim

Defines k (n) = 1 k+ 2k + · · · + n k=
n m=1 m k Properties include:

• s k (n) = (k + 1)−1

B k+1 (n + 1) − B k+1 (0)

(where theB k are Bernoulli polynomials, see Section 1.2.8)

• Writing sk (n) as
k+1 m=1 a m n k−m+2there is the recursion formula:

n
n k=1 k
n

Riemann’s zeta function isζ(n) =
∞k=1 1

k n Related functions are

Number of ways to cover ann set, n ≥ 1

22567393309593600 Coefficients of the modular functionj

For more information about all of these sequences including formulae and

ref-erences, see N.J.A Sloane and S Plouffe, Encyclopedia of Integer Sequences,

Aca-demic Press, 1995, where over 5000 other sequences are also described

1.2.15 DE BRUIJN SEQUENCES

A sequence of lengthq nover an alphabet of sizeq is a de Bruijn sequence if every

possible n-tuple occurs in the sequence (allowing wraparound to the start of the

sequence) There are de Bruijn sequences for anyq and n The table below gives

some small examples

• S N is theNthpartial sum ofS.

• The series is said to converge if the limit exists and diverge if it does not

• For an infinite series: S = limN→∞ S N=
∞n=1 a n(when the limit exists)

• If an = bn x n, whereb nis independent ofx, then S is called a power series.

• If a n = (−1) n |a n |, then S is called an alternating series.

• If
|a n| converges, then the series converges absolutely

• If S converges, but not absolutely, then it converges conditionally.

For example, the harmonic seriesS = 1+1

2+1

3+ diverges The corresponding

alternating series (called the alternating harmonic series)S = 1 − 1

3 A conditionally convergent series can be made to converge to any value bysuitably rearranging its terms.

4 If the component series are convergent, then

1 LetT be the alternating harmonic series S rearranged so that each positive term

is followed by the next two negative terms By combining each positive term