SCHAUM’S® outlines mathematical handbook of formulas and tables, fifth edition

SCHAUM’S
outlines

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Mathematical Handbook of
Formulas and Tables

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SCHAUM’S
outlines

®

Mathematical Handbook of
Formulas and Tables
Fifth Edition

Murray R. Spiegel, PhD
Former Professor and Chairman
Mathematics Department
Rensselaer Polytechnic Institute
Hartford Graduate Center

Seymour Lipschutz, PhD
Mathematics Department
Temple University

John Liu, PhD
Mathematics Department
University of Maryland

Schaum’s Outline Series

New York Chicago San Francisco
Athens London Madrid Mexico City
Milan New Delhi Singapore Sydney Toronto

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Copyright © 2018 by McGraw-Hill Education. All rights reserved. Except as permitted under the United States Copyright Act of
1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval
system, without the prior written permission of the publisher.

ISBN: 978-1-26-001054-1
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SEYMOUR LIPSCHUTZ is on the faculty of Temple University and formally taught at the Polytechnic Institute of Brooklyn.
He received his PhD in 1960 at Courant Institute of Mathematical Sciences of New York University. He is one of Schaum’s most
prolific authors. In particular, he has written, among others, Linear Algebra, Probability, Discrete Mathematics, Set Theory, Finite
Mathematics, and General Topology.
JOHN LIU is presently a professor of mathematics at University of Maryland, and he formerly taught at Temple University. He
received his PhD from the University of California, and he has held visiting positions at New York University, Princeton University, and Berkeley. He has published many papers in applied mathematics, including the areas of partial differential equations and
numerical analysis.
The late MURRAY R. SPIEGEL received the MS degree in physics and the PhD degree in mathematics from Cornell University.
He had positions at Harvard University, Columbia University, Oak Ridge, and Rensselaer Polytechnic Institute, and served as a
mathematical consultant at several large companies. His last position was Professor and Chairman of Mathematics at the Rensselaer Polytechnic Institute, Hartford Graduate Center. He was interested in most branches of mathematics, especially those that
involve applications to physics and engineering problems. He was the author of numerous journal articles and 14 books on various
topics in mathematics.
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Preface
This handbook supplies a collection of mathematical formulas and tables which will be valuable to students and
research workers in the fields of mathematics, physics, engineering, and other sciences. Care has been taken to
include only those formulas and tables which are most likely to be needed in practice, rather than highly specialized results which are rarely used. It is a “user-friendly” handbook with material mostly rooted in university
mathematics and scientific courses. In fact, the first edition can already be found in many libraries and offices,
and it most likely has moved with the owners from office to office since their college times. Thus, this handbook
has survived the test of time (while most other college texts have been thrown away).
This new edition maintains the same spirit as previous editions, with the following changes. First of all,
we have deleted some out-of-date tables which can now be easily obtained from a simple calculator, and we
have deleted some rarely used formulas. The main change is that sections on Probability and Random Variables
have been expanded with new material. These sections appear in both the physical and social sciences, including
education. There are also two new sections: Section XIII on Turing Machines and Section XIV on Mathematical
Finance.
Topics covered range from elementary to advanced. Elementary topics include those from algebra, geometry, trigonometry, analytic geometry, probability and statistics, and calculus. Advanced topics include those
from differential equations, numerical analysis, and vector analysis, such as Fourier series, gamma and beta

functions, Bessel and Legendre functions, Fourier and Laplace transforms, and elliptic and other special functions of importance. This wide coverage of topics has been adopted to provide, within a single volume, most of
the important mathematical results needed by student and research workers, regardless of their particular field
of interest or level of attainment.
The book is divided into two main parts. Part A presents mathematical formulas together with other material, such as definitions, theorems, graphs, diagrams, etc., essential for proper understanding and application of
the formulas. Part B presents the numerical tables. These tables include basic statistical distributions (normal,
Student’s t, chi-square, etc.), advanced functions (Bessel, Legendre, elliptic, etc.), and financial functions (compound and present value of an amount, and annuity).
McGraw-Hill Education wishes to thank the various authors and publishers—for example, the Literary
Executor of the late Sir Ronald A. Fisher, F.R.S., Dr. Frank Yates, F.R.S., and Oliver and Boyd Ltd., Edinburgh,
for Table III of their book Statistical Tables for Biological, Agricultural and Medical Research—who gave their
permission to adapt data from their books for use in several tables in this handbook. Appropriate references to
such sources are given below the corresponding tables.
Finally, I wish to thank the staff of McGraw-Hill Education Schaum’s Outline Series, especially Diane
Grayson, for their unfailing cooperation.
Seymour Lipschutz
Temple University

v

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Contents
Part A

FORMULAS

1

Section I

Elementary Constants, Products, Formulas

3

  1. Greek Alphabet and Special Constants
 2.Special Products and Factors
  3.The Binomial Formula and Binomial Coefficients
  4. Complex Numbers
  5. Solutions of Algebraic Equations
  6. Conversion Factors

Section II

Geometry

16

 7. Geometric Formulas
 8.Formulas from Plane Analytic Geometry

  9. Special Plane Curves
10. Formulas from Solid Analytic Geometry
11. Special Moments of Inertia

Section III

43
53
56

62
67
71
108

Differential Equations and Vector Analysis
19. Basic Differential Equations and Solutions
20. Formulas from Vector Analysis

Section VI

43

Calculus62
15.Derivatives
16. Indefinite Integrals
17. Tables of Special Indefinite Integrals
18. Definite Integrals

Section V

16
22
28
34
41

Elementary Transcendental Functions
12. Trigonometric Functions
13.Exponential and Logarithmic Functions
14. Hyperbolic Functions

Section IV

3
5
7
10
13
15

116
116
119

Series

134

21. Series of Constants

22. Taylor Series
23. Bernoulli and Euler Numbers
24. Fourier Series

134
138
142
144

vii

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vi i i C on t e n t s
Section VII

Special Functions and Polynomials
25. The Gamma Function
26. The Beta Function
27. Bessel Functions
28. Legendre and Associated Legendre Functions
29. Hermite Polynomials
30. Laguerre and Associated Laguerre Polynomials
31. Chebyshev Polynomials
32. Hypergeometric Functions

Section VIII

205

208
208
217
223

Numerical Methods

231
231
235
237
239
241
244

Turing Machines

246

48. Basic Definitions, Expressions
49.Pictures
50. Quintuple, Turing Machine
51. Computing with a Turing Machine
52.Examples

Section XIV

198
203

Probability and Statistics

42.Interpolation
43.Quadrature
44. Solution of Nonlinear Equations
45. Numerical Methods for Ordinary Differential Equations
46. Numerical Methods for Partial Differential Equations
47. Iteration Methods for Linear Systems

Section XIII

198

205
207

39. Descriptive Statistics
40.Probability
41.Random   Variables

Section XII

180
193

Inequalities and Infinite Products

37.Inequalities
38. Infinite Products

Section XI

180

Elliptic and Miscellaneous Special Functions
35. Elliptic Functions
36. Miscellaneous and Riemann Zeta Functions

Section X

149
152
153
164
169
171
175
178

Laplace and Fourier Transforms
33. Laplace Transforms
34. Fourier Transforms

Section IX

149

246
247
248
250
252

Mathematical Finance
53. Basic Probability
54. Interest Rates
55. Arbitrage Theorem and Options

254
254
256
257

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ix

Conte n ts

56. Arbitrage Theorem
57. Black-Scholes Formula
58. The Delta Hedging Arbitrage Strategy

Part B

TABLES

Section I

Logarithmic, Trigonometric, Exponential Functions

258
259
260

263
265

  1.Four Place Common Logarithms log10 N or log N265
  2. Sin x (x in Degrees and Minutes)
267
  3.Cos x (x in Degrees and Minutes)
268
 4.Tan x (x in Degrees and Minutes)
269
 5. Conversion of Radians to Degrees, Minutes,
and Seconds or Fractions of Degrees
270
 6. Conversion of Degrees, Minutes, and Seconds to Radians
271
 7. Natural or Napierian Logarithms loge x or ln x272
  8.Exponential Functions ex274
  9. Exponential Functions e-x275

10. Exponential, Sine, and Cosine Integrals
276

Section II

Factorial and Gamma Function, Binomial Coefficients
11.Factorial n
12. Gamma Function
13. Binomial Coefficients

Section III

277
277
278
279

Bessel Functions

281

14. Bessel Functions J0(x)281
15. Bessel Functions J1(x)281
16. Bessel Functions Y0(x)282
17. Bessel Functions Y1(x)282
18. Bessel Functions I0(x)283
19. Bessel Functions I1(x)283
20. Bessel Functions K0(x)284
21. Bessel Functions K1(x)284
22. Bessel Functions Ber(x)285

23. Bessel Functions Bei(x)285
24. Bessel Functions Ker(x)286
25. Bessel Functions Kei(x)286
26. Values for Approximate Zeros of Bessel Functions
287

Section IV

Legendre Polynomials

288

27. Legendre Polynomials Pn(x)288
28. Legendre Polynomials Pn(cos θ)289

Section V

Elliptic Integrals

290

29. Complete Elliptic Integrals of First and Second Kinds
30. Incomplete Elliptic Integral of the First Kind
31. Incomplete Elliptic Integral of the Second Kind

290
291
291

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x C on t e n t s
Section VI

Financial Tables

292

32. Compound Amount: (1 + r)n292
33. Present Value of an Amount: (1 + r)-n293
(1 + r )n − 1
34. Amount of an Annuity:
294
r
1 – (1 + r ) – n
35. Present Value of an Annuity:
295
r

Section VII

Probability and Statistics

296

36. Areas Under the Standard Normal Curve from -∞ to x296

37. Ordinates of the Standard Normal Curve
297
38. Percentile Values (tp) for Student’s t Distribution
298
2
2
39. Percentile Values (cp) for c (Chi-Square) Distribution
299
40. 95th Percentile Values for the F Distribution
300
41. 99th Percentile Values for the F Distribution
301
42. Random Numbers
302

Index of Special Symbols and Notations

303

Index305

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SCHAUM’S
outlines

®

Mathematical Handbook of
Formulas and Tables

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Part A

FORMULAS

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Section I: Elementary Constants, Products, Formulas

1

   GREEK ALPHABET and SPECIAL CONSTANTS

Greek Alphabet
Greek
name


Greek letter
Lower case

Capital

Alpha

a

A

Beta

b

Gamma

Greek
name


Greek letter
Lower case

Capital

Nu

n

N

B

Xi

x

X

g

G

Omicron

o

O

Delta

d

D

Pi

p

P

Epsilon

e

E

Rho

r

Ρ

Zeta

z

Z

Sigma

s

S

Eta

hH

Tau

t

T

Theta

q



Upsilon

u

Υ

Iota

i

I

Phi

f

F

Kappa

k

K

Chi

c

Χ

Lambda

l

L

Psi

y

Ψ

Mu

m

M

Omega

w

Ω

    

Special Constants
 1.1.  p = 3.14159 26535 89793 …
1
 1.2.  e = 2.71828 18284 59045 … = lim  1 + 

n→∞ 
n

n

 
= natural base of logarithms
 1.3.  g  = 0.57721 56649 01532 86060 6512 … = Euler’s constant
1 1
1
= lim  1 + + +    + − ln n 
n→∞ 

n
  2 3

 1.4.  eγ = 1.78107 24179 90197 9852 … [see 1.3]

3

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4

 1.5. 

G R EEK A L P H A BET a nd S P EC IA L C O NS TA NT S

e = 1.64872 12707 00128 1468 …

 1.6.  π = Γ ( 12 ) = 1.77245 38509 05516 02729 8167 …
where Γ is the gamma function [see 25.1].
 1.7.  Γ ( 13 ) = 2.67893 85347 07748 …
 1.8.  Γ ( 14 ) = 3.62560 99082 21908 …
 1.9.  1 radian = 180°/p = 57.29577 95130 8232 …°
1.10. 1° = p /180 radians = 0.01745 32925 19943 29576 92 … radians

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2



SPECIAL PRODUCTS and FACTORS

 2.1.  ( x + y)2 = x 2 + 2 xy + y 2
 2.2.  ( x − y)2 = x 2 − 2 xy + y 2
 2.3.  ( x + y)3 = x 3 + 3 x 2 y + 3 xy 2 + y 3
 2.4.  ( x − y)3 = x 3 − 3 x 2 y + 3 xy 2 − y 3
 2.5.  ( x + y)4 = x 4 + 4 x 3 y + 6 x 2 y 2 + 4 xy3 + y 4
 2.6.  ( x − y)4 = x 4 − 4 x 3 y + 6 x 2 y 2 − 4 xy3 + y 4
 2.7.  ( x + y)5 = x 5 + 5 x 4 y + 10 x 3 y 2 + 10 x 2 y 3 + 5 xy 4 + y 5

 2.8.  ( x − y)5 = x 5 − 5 x 4 y + 10 x 3 y 2 − 10 x 2 y 3 + 5 xy 4 − y 5
 2.9.  ( x + y)6 = x 6 + 6 x 5 y + 15 x 4 y 2 + 20 x 3 y 3 + 15x 2 y 4 + 6 xy 5 + y6
2.10.  ( x − y)6 = x 6 − 6 x 5 y + 15 x 4 y 2 − 20 x 3 y 3 + 15x 2 y 4 − 6 xy 5 + y6
The results 2.1 to 2.10 above are special cases of the binomial formula [see 3.3].
2.11.  x 2 − y 2 = ( x − y)( x + y)
2.12.  x 3 − y 3 = ( x − y)( x 2 + xy + y 2 )
2.13.  x 3 + y 3 = ( x + y)( x 2 − xy + y 2 )
2.14.  x 4 − y 4 = ( x − y)( x + y)( x 2 + y 2 )
2.15.  x 5 − y 5 = ( x − y)( x 4 + x 3 y + x 2 y 2 + xy 3 + y 4 )
2.16.  x 5 + y 5 = ( x + y)( x 4 − x 3 y + x 2 y 2 − xy 3 + y 4 )
2.17.  x 6 − y6 = ( x − y)( x + y)( x 2 + xy + y 2 )( x 2 − xy + y 2 )

5

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6

S P EC IA L P ROD U C TS and FAC TO R S

2.18.  x 4 + x 2 y 2 + y 4 = ( x 2 + xy + y 2 )( x 2 − xy + y 2 )
2.19.  x 4 + 4 y 4 = ( x 2 + 2 xy + 2 y 2 )( x 2 − 2 xy + 2 y 2 )
Some generalizations of the above are given by the following results where n is a positive integer.
2.20.  x 2 n+1 − y 2 n+1 = ( x − y)( x 2 n + x 2 n−1 y + x 2 n−2 y 2 +  + y 2 n )
2π
4π

= ( x − y)  x 2 − 2 xy cos
+ y 2   x 2 − 2 xy cos
+ y2 



2n + 1
2n + 1
2nπ
 x 2 − 2 xy cos
+ y2 


2n + 1

2.21.  x 2 n+1 + y 2 n+1 = ( x + y)( x 2 n − x 2 n−1 y + x 2 n−2 y 2 −  + y 2 n )
2π
4π
= ( x + y)  x 2 + 2 xy cos
+ y 2   x 2 + 2 xy cos
+ y2 



2n + 1
2n + 1
2nπ
 x 2 + 2 xy cos
+ y2 



2n + 1

2.22.  x 2 n − y 2 n = ( x − y)( x + y)( x n−1 + x n− 2 y + x n−3 y 2 + )( x n−1 − x n−2 y + x n−3 y 2 − )

π
2π
= ( x − y)( x + y)  x 2 − 2 xy cos + y 2   x 2 − 2 xy cos
+ y2 



n
n
 x 2 − 2 xy cos


(n − 1)π
+ y2 

n

π
3π
+ y2 
2.23.  x 2 n + y 2 n =  x 2 + 2 xy cos + y 2   x 2 + 2 xy cos





2n
2n
 x 2 + 2 xy cos


(2n − 1)π
+ y2 

2n

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3

THE BINOMIAL FORMULA and BINOMIAL

COEFFICIENTS

Factorial n
For n = 1, 2, 3, …, factorial n or n factorial is denoted and defined by
3.1.  n! = n(n − 1) ⋅ ⋅ 3 ⋅ 2 ⋅ 1
Zero factorial is defined by
3.2. 0! = 1
Alternately, n factorial can be defined recursively by
0! = 1

and

n! = n ⋅ (n – 1)!

EXAMPLE: 4! = 4 ⋅ 3 ⋅ 2 ⋅ 1 = 24,

5! = 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 5 ⋅ 4! = 5(24) = 120,
6! = 6 ⋅ 5! = 6(120) = 720

Binomial Formula for Positive Integral n
For n = 1, 2, 3, …,
3.3.  ( x + y)n = x n + nx n−1 y +

n(n − 1) n−2 2 n(n − 1)(n − 2) n−3 3
x y +
x y +  + yn
2!
3!

This is called the binomial formula. It can be extended to other values of n, and also to an infinite series
[see 22.4].
EXAMPLE:

(a)  (a − 2b)4 = a 4 + 4 a 3 (−2b) + 6a 2 (−2b) 2 + 4 a(−2b)3 + (−2b) 4 = a 4 − 8a 3 b + 24 a 2 b 2 − 32ab 3 + 16b 4

Here x = a and y = −2b.
(b)  See Fig. 3-1a.

Binomial Coefficients
Formula 3.3 can be rewritten in the form

 n  n−1  n  n−2 2  n  n−3 3
 n  n
n
n
x y+
x y +
x y ++ 
y
3.4.  ( x + y) = x + 



 n 
 1 
 2 
 3 

7

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8

The BIN OM IA L F OR M U L A a nd BIN OM IA L C OEF F I C I ENT S

where the coefficients, called binomial coefficients, are given by

3.5. 

n!
n 
 n = n(n − 1)(n − 2)  (n − k + 1) =
=
 k

−
n
k
k!
k !(n − k )!

EXAMPLE: 

 9 = 9 ⋅ 8 ⋅ 7 ⋅ 6 = 126,
 4 1 ⋅ 2 ⋅ 3 ⋅ 4

12 = 12 ⋅ 11 ⋅ 10 ⋅ 9 ⋅ 8 = 792,
 5
1⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5

10 = 10 = 10 ⋅ 9 ⋅ 8 = 120
 7   3  1⋅ 2 ⋅ 3

n
Note that   has exactly r factors in both the numerator and the denominator.
 r

The binomial coefficients may be arranged in a triangular array of numbers, called Pascal’s triangle, as
shown in Fig. 3-1b. The triangle has the following two properties:
(1) The first and last number in each row is 1.
(2)Every other number in the array can be obtained by adding the two numbers appearing directly above
it. For example
10 = 4 + 6,

15 = 5 + 10,

20 = 10 + 10

Property (2) may be stated as follows:
3.6.   n +  n  =  n + 1
 k  k + 1  k + 1

Fig. 3-1

Properties of Binomial Coefficients
The following lists additional properties of the binomial coefficients:
 n  n  n
 n
n
 3.7.    +   +   +  +   = 2
n
0
1
2
 n  n  n
n  n
 3.8.    −   +   − (−1)   = 0

n
0
1
2
 n  n + 1 +  n + 2 +  +  n + m =  n + m + 1 
 3.9.    + 
 n   n +1 
n
n   n 

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T he B I N O MI AL F O RMULA an d BIN OM IA L C OEF F IC IEN TS

9

n
n
n
3.10.    +   +   +  = 2n−1
 0  2  4
n
n
n
3.11.    +   +   +  = 2n−1
 1  3  5

2

2

2

2

n
n
n
n
2n
3.12.    +   +   +  +   =  
 n
 1
 2
 0
 n
 m  n  m + n
 m  n  m  n 
3.13.   0   p +  1   p − 1 +  +  p  0 =  p 
n
n
n
n
3.14.  (1)   + (2)   + (3)   +  + (n)   = n 2n−1
 n
 1
 2

 3
n
n
n
n
3.15.  (1)   − (2)   + (3)   − (−1)n+1 (n)   = 0
 n
 1
 2
 3

Multinomial Formula
Let n1, n2, …, nr be nonnegative integers such that n1 + n2 +  + nr = n. Then the following expression, called
a multinomial coefficient, is defined as follows:
n
n!


3.16.   n , n , …, n  =
 1 2

n
n
nr!
!
r
1
2!


7



7!

= 210,
EXAMPLE:   2, 3, 2 =

 2!3!2!

8
8!


 4, 2, 2, 0 = 4!2!2!0! = 420

The name multinomial coefficient comes from the following formula:
3.17.

( x1 + x 2 +

n

 n 1 n2
+ x p )n = ∑ 
x x
 n1, n2, …, nr  1 2

n

xr r

where the sum, denoted by Σ, is taken over all possible multinomial coefficients.

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4



COMPLEX NUMBERS

Definitions Involving Complex Numbers
A complex number z is generally written in the form
z = a + bi
where a and b are real numbers and i, called the imaginary unit, has the property that i2 = -1. The real numbers a and b are called the real and imaginary parts of z = a + bi, respectively.
The complex conjugate of z is denoted by z ; it is defined by
a + bi = a − bi

Thus, a + bi and a – bi are conjugates of each other.

Equality of Complex Numbers
4.1.  a + bi = c + di    if and only if   a = c and b = d

Arithmetic of Complex Numbers

Formulas for the addition, subtraction, multiplication, and division of complex numbers follow:
4.2.  (a + bi ) + (c + di ) = (a + c) + (b + d )i
4.3.  (a + bi ) − (c + di ) = (a − c) + (b − d )i
4.4.  (a + bi )(c + di ) = (ac − bd ) + (ad + bc)i
4.5. 

a + bi a + bi c − di ac + bd  bc − ad 
=
•
=
+
i
c + di c + di c − di c 2 + d 2  c 2 + d 2 

Note that the above operations are obtained by using the ordinary rules of algebra and replacing i 2 by -1
wherever it occurs.
EXAMPLE:  Suppose z = 2 + 3i and w = 5 - 2i. Then

z + w = (2 + 3i ) + (5 − 2i ) = 2 + 5 + 3i − 2i = 7 + i
zw = (2 + 3i )(5 − 2i ) = 10 + 15i − 4i − 6i 2 = 16 + 11i
z = 2 + 3i = 2 − 3i and w = 5 − 2i = 5 + 2i



w 5 − 2i (5 − 2i )(2 − 3i ) 4 − 19i 4 19
=
=
=
= − i
z 2 + 3i (2 + 3i )(2 − 3i )

13
13 13

10

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11

COM PLEX N UMB ERS

Complex Plane
Real numbers can be represented by the points on a line, called the real line, and, similarly, complex numbers can be represented by points in the plane, called the Argand diagram or Gaussian plane or, simply, the
complex plane. Specifically, we let the point (a, b) in the plane represent the complex number z = a + bi. For
example, the point P in Fig. 4-1 represents the complex number z = -3 + 4i. The complex number can also
be interpreted as a vector from the origin O to the point P.
The absolute value of a complex number z = a + bi, written | z |, is defined as follows:
4.6.  | z | = a 2 + b 2 = zz
We note | z | is the distance from the origin O to the point z in the complex plane.


Fig. 4-1

Fig. 4-2

Polar Form of Complex Numbers

The point P in Fig. 4-2 with coordinates (x, y) represents the complex number z = x + iy. The point P can
also be represented by polar coordinates (r, q ). Since x = r cos q and y = r sin q , we have
4.7.  z = x + iy = r (cosθ + i sin θ )
called the polar form of the complex number. We often call r =  | z |  = x 2 + y 2 the modulus and q the
amplitude of z = x + iy.

Multiplication and Division of Complex Numbers in Polar Form
4.8.  [r1 (cosθ1 + i sin θ1 )][r2 (cosθ 2 + i sin θ 2 )] = r1r2 [cos(θ1 + θ 2 ) + i sin(θ1 + θ 2 )]
4.9. 

r1 (cosθ1 + i sin θ1 ) r1
= [cos(θ1 − θ 2 ) + i sin (θ1 − θ 2 )]
r2 (cosθ 2 + i sin θ 2 ) r2

De Moivre’s Theorem
For any real number p, De Moivre’s theorem states that
4.10.  [r (cosθ + i sin θ )] p = r p (cos pθ + i sin pθ )

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12

C OM P L EX N U M B E R S

Roots of Complex Numbers
Let p = 1/n where n is any positive integer. Then 4.10 can be written

4.11.

[r (cosθ + i sin θ )]1/n = r 1/n  cos


θ + 2 kπ 
θ + 2 kπ
+ i sin
n
n 

where k is any integer. From this formula, all the nth roots of a complex number can be obtained by putting
k = 0, 1, 2, …, n – 1.

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