Schaum Mathematical Formulas and Tables PDF

Preview

Summary

P r e f The pur-pose of this handbook is to supply a collection of mathematical formulas and tables which will prove to be valuable to students and research workers in the fields of mathematics, physics, engineering and other sciences. TO accomplish this, tare has been taken to include those formul...

Description

P

r

e

f

The pur-pose of this handbook is to supply a collection of mathematical formulas and tables which will prove to be valuable to students and research workers in the fields of mathematics, physics, engineering and other sciences. TO accomplish this, tare has been taken to include those formulas and tables which are most likely to be needed in practice rather than highly specialized results which are rarely used. Every effort has been made to present results concisely as well as precisely SOthat they may be referred to with a maximum of ease as well as confidence. Topics covered range from elementary to advanced. Elementary topics include those from algebra, geometry, trigonometry, analytic geometry and calculus. Advanced topics include those from differential equations, vector analysis, Fourier series, gamma and beta functions, Bessel and Legendre functions, Fourier and Laplace transforms, elliptic functions and various other special functions of importance. This wide coverage of topics has been adopted SOas to provide within a single volume most of the important mathematical results needed by the student or research worker regardless of his particular field of interest or level of attainment. The book is divided into two main parts. Part 1 presents mathematical formulas together with other material, such as definitions, theorems, graphs, diagrams, etc., essential for proper understanding and application of the formulas. Included in this first part are extensive tables of integrals and Laplace transforms which should be extremely useful to the student and research worker. Part II presents numerical tables such as the values of elementary functions (trigonometric, logarithmic, exponential, hyperbolic, etc.) as well as advanced functions (Bessel, Legendre, elliptic, etc.). In order to eliminate confusion, especially to the beginner in mathematics, the numerical tables for each function are separated, Thus, for example, the sine and cosine functions for angles in degrees and minutes are given in separate tables rather than in one table SOthat there is no need to be concerned about the possibility of errer due to looking in the wrong column or row. 1 wish to thank the various authors and publishers who gave me permission to adapt data from their books for use in several tables of this handbook. Appropriate references to such sources are given next to the corresponding tables. In particular 1 am indebted to the Literary Executor of the late Sir Ronald A. Fisher, F.R.S., to Dr. Frank Yates, F.R.S., and to Oliver and Boyd Ltd., Edinburgh, for permission to use data from Table III of their book S T tf B a Aao i b a gtMy o R l n ir e l e e d si d 1 also wish to express my gratitude to Nicola Menti, Henry Hayden and Jack Margolin for their excellent editorial cooperation. M. R. SPIEGEL Rensselaer Polytechnic Institute September, 1968

o

s s

tc i

CONTENTS

Page 1.

Special

Constants..

.............................................................

1

2. Special Products and Factors ....................................................

2

3. The Binomial Formula and Binomial Coefficients .................................

3

4. Geometric Formulas ............................................................

5

5. Trigonometric Functions ........................................................

11

6. Complex Numbers ...............................................................

21

7. Exponential and Logarithmic Functions .........................................

23

8. Hyperbolic Functions ...........................................................

26

9. Solutions of Algebraic Equations ................................................

32

10. Formulas from Plane Analytic Geometry ........................................ ...................................................

34 40

11.

Special Plane Curves........~

12.

Formulas from Solid Analytic Geometry ........................................

46

13.

Derivatives .....................................................................

53

14.

Indefinite Integrals ..............................................................

57

15.

Definite Integrals ................................................................

94

16.

The Gamma

Function .........................................................

..10 1

17.

The Beta Function ............................................................

18.

Basic Differential Equations and Solutions .....................................

19.

Series of Constants..............................................................lO

20.

Taylor Series...................................................................ll

21.

Bernoulliand

22.

Formulas from Vector Analysis..

23.

Fourier Series ................................................................

..~3 1

24.

Bessel Functions..

..13 6

2s.

Legendre Functions.............................................................l4

26.

Associated Legendre Functions .................................................

.149

27. 28.

Hermite Polynomials............................................................l5 Laguerre Polynomials ..........................................................

1 .153

29.

Associated Laguerre Polynomials ................................................

30.

Chebyshev Polynomials..........................................................l5

Euler Numbers ................................................. .............................................

............................................................

..lO 3 .104

7 0 ..114 ..116

6

KG

7

Part

I

FORMULAS

THE

GREEK

Greek

name

G&W

ALPHABET

Greek name

Greek Lower case

tter Capital

Alpha

A

Nu

N

Beta

B

Xi

sz

Gamma

l?

Omicron

0

Delta

A

Pi

IT

Epsilon

E

Rho

P

Zeta

Z

Sigma

2

Eta

H

Tau

T

Theta

(3

Upsilon

k

Iota

1

Phi

@

Kappa

K

Chi

X

Lambda

A

Psi

*

MU

M

Omega

n

1.1 1.2

= natural

base of logarithms

1.3

fi

=

1.41421

35623 73095 04889..

1.4

fi

=

1.73205

08075 68877 2935.

1.5

fi

=

2.23606

79774

1.6

h

=

1.25992

1050..

.

1.7

&

=

1.44224

9570..

.

1.8

fi

=

1.14869

8355..

.

1.9

b

=

1.24573

0940..

.

1.10

eT = 23.14069

26327 79269 006..

.

1.11

re = 22.45915

77183 61045 47342

715..

1.12

ee =

22414

.

1.13

logI,, 2

=

0.30102

99956 63981 19521

37389.

..

1.14

logI,, 3

=

0.47712

12547

19662 43729

50279..

.

1.15

logIO e =

0.43429

44819

03251 82765..

1.16

logul ?r =

0.49714

98726

94133 85435 12683.

1.17

loge 10

In 10

1.18

loge 2 =

ln 2

=

0.69314

71805

59945 30941

1.19

loge 3 =

ln 3 =

1.09861

22886

68109

1.20

y =

1.21

ey =

1.22

fi

=

1.23

6

=

15.15426

=

0.57721

56649

1.78107

r(&)

99789 6964..

79264

2.30258

=

190..

12707

6512.

9852..

00128 1468..

1.77245

F is the gummu 2.67893

85347 07748..

.

1.25

r(i)

3.62560

99082 21908..

.

1-26

1 radian

1.27

1”

=

180°/7r

~/180

radians

.

.

= =

..

57.29577 0.01745

7232.

.

..

69139 5245..

.

.. = Eukr's co%stu~t

[see 1.201

.

38509 05516

II’(&) =

=

.

02729

~ZLYLC~~OTZ [sec pages

1.24

=

.

50929 94045 68401 7991..

01532 86060

24179 90197

1.64872

where

=

..

8167..

95130 8232.. 32925

.

101-102).

.O

19943 29576 92.

1

..

radians

THE

4

BINOMIAL

FORMULA

PROPERTIES

OF

AND

BINOMIAL

BINOMIAL

COElFI?ICIFJNTS

COEFFiClEblTS

3.6 This

leads

to Paseal’s

[sec page 2361.

triangk

3.7

(1)

+

(y)

+

(;)

+

...

3.8

(1)

-

(y)

+

(;)

-

..+-w(;)

3.10

(;)

+

(;)

+

(7)

+

.*.

=

2n-1

3.11

(y)

+

(;)

+

(i)

+

..*

=

2n-1

+

(1)

=

27l

=

0

3.9

3.12

3.13

-d

3.14

MUlTlNOMlAk

3.16

(zI+%~+...+zp)~ where

q+n2+

the

mm,

...

denoted

+np =

72..

by

2,

=

FORfvlUlA

~~~!~~~~~..~~!~~1~~2...~~~

is taken over

a11 nonnegative

integers

% %,

. . , np fox- whkh

1

4

GEUMElRlC

FORMULAS &

RECTANGLE

4.1

Area

4.2

Perimeter

OF LENGTH

b AND

WIDTH

a

= ab = 2a + 2b b

Fig. 4-1

PARAllELOGRAM

4.3

Area

=

4.4

Perimeter

bh =

OF ALTITUDE

h AND

BASE b

ab sin e

= 2a + 2b 1 Fig. 4-2

‘fRlAMf3i.E

Area

4.5

=

+bh

OF ALTITUDE

h AND

BASE b

= +ab sine

*

ZZZI/S(S - a)(s - b)(s - c) where s = &(a + b + c) = semiperimeter

b Perimeter

4.6

n_

L,“Z

.,

= u+ b+ c

Fig. 4-3

:

‘fRAPB%XD

.,,

4.7

Area

4.8

Perimeter

C?F At.TlTUDE

fz AND

PARAl.lEL

SlDES u AND

b

= 3h(a + b) = =

/c-

a + b + h

Y&+2 sin 4 C a + b + h(csc e + csc $)

1 Fig. 4-4

5 / -

GEOMETRIC

6

REGUkAR

4.9

Area

= $nb?- cet c

4.10

Perimeter

=

POLYGON

inbz-

FORMULAS

OF n SIDES EACH CJf 1ENGTH

b

COS(AL)

sin (~4%)

= nb

7,’ 0.’ 0 Fig. 4-5

CIRÇLE OF RADIUS

4.11

Area

4.12

Perimeter

r

= & =

277r

Fig. 4-6

SEClOR

4.13 4.14

Area

=

&r%

OF CIRCLE OF RAD+US Y

[e in radians]

T

Arc length s = ~6 A

8

0 T Fig. 4-7

RADIUS

OF C1RCJ.E INSCRWED

r=

4.15 where

&$.s-

tN A TRtANGlE *

OF SIDES a,b,c

U)(S Y b)(s -.q) s

s = +(u + b + c) = semiperimeter

Fig. 4-6

RADIUS- OF CtRClE

R=

4.16 where

CIRCUMSCRIBING

A TRIANGLE

OF SIDES a,b,c

abc 4ds(s - a)@ -

b)(s - c)

e = -&(a.+ b + c) = semiperimeter

Fig. 4-9

G

4

A

=.

4

P

.

&

sr s =

2e

s

1=

n +

1

=

FE

3 ise n

7

r n

OO

6

ni a

2 nr s i y 8

2r

RM

0

n

n ri i n

M7E

UT

°

2

r mn z

e

t

e

!

?

Fig. 4-10

4

A

=.

4

P

.

= 1 n r t a eL T n

t rZ n

n =

2e

2

t

9 r 2 a n a! 0

2 nr t a

=

2

n

n ri a n

T

!

I : e?

r m nk

T

t

e

0 F

SRdMMHW W

4

o .s

A

f=2 h +

pr

( -ae s

C%Ct&

e) 1 a r

e

OF RADWS

ra i

d2

4

i

-

g

1

T

tn

e T

e

d r

tz!? Fig. 4-12

4

A

=.

4

P

.

r

r

2

a

e

2

2 4 1 - kz rs

e c3

b

a

7r/2

=

e 5 4a

ii

m +

l

e

@

t

e

0 =

w

k = ~/=/a.h

4

A

4

A

l

[

27r@sTq See

p

e254 f

=.

$ab

r

2

.

ABC

r = e -&2dw

a

n a

e

r

to

4

c +n E5

p

u g

e

ar

p

m e

b F

r

4e

l

i

-r

o e g

a 4

gl 1

a )

tn

+

h

AOC

@

T

b Fig. 4-14

- f

1i

GEOMETRIC

8

RECTANGULAR

4.26

Volume

=

4.27

Surface

area

PARALLELEPIPED

FORMULAS

OF

LENGTH

u, HEIGHT

r?, WIDTH

c

ubc Z(ab + CLC + bc)

=

a Fig. 4-15

PARALLELEPIPED

4.28

Volume

=

Ah

=

OF CROSS-SECTIONAL

AREA

A AND

HEIGHT

h

abcsine

Fig. 4-16

SPHERE

4.29

Volume

=

OF RADIUS

,r

+

1 ---x

,-------

4.30

Surface

area

=

4wz

@ Fig. 4-17

RIGHT

4.31

Volume

4.32

Lateral

=

CIRCULAR

CYLINDER

OF RADIUS

T AND

HEIGHT

h

77&2

surface

area

=

h

25dz

Fig. 4-18

CIRCULAR

4.33

Volume

4.34

Lateral

=

m2h

surface

area

CYLINDER

=

OF RADIUS

r AND

SLANT

HEIGHT

2

~41 sine =

2777-1 =

2wh

z

=

2wh csc e Fig. 4-19

.

GEOMETRIC

CYLINDER

=

OF CROSS-SECTIONAL

4.35

Volume

4.36

Lateral surface area

Ah

FORMULAS

9

A AND

AREA

SLANT

HEIGHT

I

Alsine

=

=

pZ =

GPh

--

ph csc t
...