(Parte 1 de 3)

Preface

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 SO that they may be referred to with a maxi- mum 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 SO as 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 I 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 sep- arated, Thus, for example, the sine and cosine functions for angles in degrees and minutes are given in separate tables rather than in one table SO that 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 I of their book Statistical Tables foy Biological, Agricultural and Medical Research.

1 also wish to express my gratitude to Nicola Menti, Henry Hayden and Jack Margolin for their excellent editorial cooperation.

Rensselaer Polytechnic Institute September, 1968

2. 3.

4. 5.

6. 7.

9. 10.

1. 12. 13. 14. 15.

16. 17.

18. 19.

20. 21.

23. 24.

2s. 26.

28. 29.

 Special Constants 1 Special Products and Factors 2 The Binomial Formula and Binomial Coefficients 3 Geometric Formulas 5 Trigonometric Functions 1 Complex Numbers 21 Exponential and Logarithmic Functions 23 Hyperbolic Functions 26 Solutions of Algebraic Equations 32 Formulas from Plane Analytic Geometry 34 Special Plane Curves ~ ................................................... 40 Formulas from Solid Analytic Geometry 46 Derivatives 53 Indefinite Integrals 57 Definite Integrals 94 The Gamma Function 10 1 The Beta Function lO 3 Basic Differential Equations and Solutions 104 Series of Constants lO 7 Taylor Series l 0 Bernoulliand Euler Numbers 114 Formulas from Vector Analysis 116 Fourier Series ~3 1 Bessel Functions 13 6 Legendre Functions l4 6 Associated Legendre Functions 149 Hermite Polynomials l5 1 Laguerre Polynomials 153 Associated Laguerre Polynomials KG

Page Chebyshev Polynomials..........................................................l5 7

Part I FORMULAS

Greek name

Alpha Beta Gamma Delta Epsilon Zeta

Eta Theta Iota

Kappa Lambda l? A

Greek name

Nu Xi

Omicron Pi

Rho Sigma Tau

Upsilon Phi

Chi Psi

Omega

Greek

Lower case tter Capital

N sz

0 IT P k @ X

* n

1.2 = natural base of logarithms

 1.4 fi = 1.73205 08075 68877 2935 1.5 fi = 2.23606 79774 99789 6964 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 1.1 re = 2.45915 77183 61045 47342 715 1.12 e = 15.15426 22414 79264 190 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 = 2.30258 50929 94045 68401 7991 1.18 loge 2 = ln 2 = 0.69314 71805 59945 30941 7232 1.19 loge 3 = ln 3 = 1.09861 22886 68109 69139 5245 1.20 y = 0.57721 56649 01532 86060 6512 = Eukr's co%stu~t 1.21 ey = 1.78107 24179 90197 9852 [see 1.201 1.2 fi = 1.64872 12707 00128 1468 1.23 6 = r(&) = 1.77245 38509 05516 02729 8167

1.3 fi = 1.41421 35623 73095 04889.. where F is the gummu ~ZLYLC~~OTZ [sec pages 101-102).

1.25 1-26 1.27

 I’(&) = 2.67893 85347 07748 r(i) = 3.62560 99082 21908 1 radian = 180°/7r = 57.29577 95130 8232 O 1” = ~/180 radians = 0.01745 32925 19943 29576 92 radians

4 THE BINOMIAL FORMULA AND BINOMIAL COElFI?ICIFJNTS PROPERTIES OF BINOMIAL COEFFiClEblTS

3.6 This leads to Paseal’s triangk [sec page 2361.

 3.7 (1) + (y) + (;) + + (1) = 27l

3.8 (1) - (y) + (;) - ..+-w(;) = 0 3.9

3.10 (;) + (;) + (7) + .*. = 2n-1 3.1 (y) + (;) + (i) + ..* = 2n-1

3.12 3.13

-d 3.14

 q+n2+ +np = 72..

MUlTlNOMlAk FORfvlUlA

3.16 (zI+%~+...+zp)~ = ~~~!~~~~~..~~!~~1~~2...~~~ where the m, denoted by 2, is taken over a11 nonnegative integers % %, . . , np fox- whkh

4 GEUMElRlC FORMULAS &

RECTANGLE OF LENGTH b AND WIDTH a

4.1 Area = ab

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

PARAllELOGRAM OF ALTITUDE h AND BASE b

4.3 Area = bh = ab sin e

4.4 Perimeter = 2a + 2b

1 Fig. 4-2

‘fRlAMf3i.E OF ALTITUDE h AND BASE b

4.5 Area = +bh = +ab sine

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

4.6 Perimeter = u + b + c Fig. 4-3 ‘fRAPB%XD C?F At.TlTUDE fz AND PARAl.lEL SlDES u AND b .,,

4.7 Area = 3h(a + b)

4.8 Perimeter = a + b + h C Y&+2 sin 4

= a + b + h(csc e + csc \$) /c- 1

Fig. 4-4

6 GEOMETRIC FORMULAS

REGUkAR POLYGON OF n SIDES EACH CJf 1ENGTH b

4.9 COS (AL) Area = \$nb?- cet c = inbz- sin (~4%)

4.10 Perimeter = nb

Fig. 4-5 CIRÇLE OF RADIUS r

4.1 Area = & 7,’ 0 0.’ 4.12 Perimeter = 277r

Fig. 4-6 SEClOR OF CIRCLE OF RAD+US Y

4.13 Area = &r% [e in radians]

4.14 Arc length s = ~6 0

Fig. 4-7

RADIUS OF C1RCJ.E INSCRWED tN A TRtANGlE OF SIDES a,b,c *

4.15 r= &\$.s - U)(S Y b)(s -.q) s where s = +(u + b + c) = semiperimeter

Fig. 4-6 RADIUS- OF CtRClE CIRCUMSCRIBING A TRIANGLE OF SIDES a,b,c

4.16 R= abc

4ds(s - a)@ - b)(s - c) where e = -&(a. + b + c) = semiperimeter

Fig. 4-9

GEOMETRIC FORMULAS 7

4.17 Area = &nr2 sin s = 360° +nr2 sin n 4.18 Perimeter = 2nr sinz = 2nr sin y

Fig. 4-10

4.19 Area = nr2 tan ZT = nr2 tan L!T!!? n n IT

4.20 Perimeter = 2nr tank = 2nr tan?

Fig. 4-1 SRdMMHW W C%Ct& OF RADWS T

4.21 Area of shaded part = +r2 (e - sin e) e T r tz!? Fig. 4-12

4.2 4.23 Area = rab

5 7r/2 Perimeter = 4a 4 1 - kz si+ e cl@

= 27r@sTq [approximately] where k = ~/=/a. See page 254 for numerical tables. Fig. 4-13

4.24 Area = \$ab

4.25 Arc length ABC = -& dw + Eln 4a+@TTG

1 ) AOC b

Fig. 4-14 f -

8 GEOMETRIC FORMULAS

RECTANGULAR PARALLELEPIPED OF LENGTH u, HEIGHT r?, WIDTH c

4.26 Volume = ubc 4.27 Surface area = Z(ab + CLC + bc)

PARALLELEPIPED OF CROSS-SECTIONAL AREA A AND HEIGHT h 4.28 Volume = Ah = abcsine

4.29 4.30

4.31 4.32

4.3 4.34 a Fig. 4-15

Fig. 4-16 SPHERE OF RADIUS ,r

Volume = +

Surface area = 4wz

1 ,------- ---x .

@ Fig. 4-17

RIGHT CIRCULAR CYLINDER OF RADIUS T AND HEIGHT h

Volume = 7&2

Lateral surface area = 25dz h

Fig. 4-18 CIRCULAR CYLINDER OF RADIUS r AND SLANT HEIGHT 2

Volume = m2h = ~41 sine

2wh Lateral surface area = 2777-1 = z = 2wh csc e

Fig. 4-19

GEOMETRIC FORMULAS 9

4.35 Volume = Ah = Alsine

4.36 Ph - Lateral surface area = pZ = G - ph csc t

Note that formulas 4.31 to 4.34 are special cases.

Fig. 4-20 RIGHT CIRCULAR CONE OF RADIUS ,r AND HEIGHT h

4.37 Volume = jîw2/z 4.38 Lateral surface area = 77rd77-D = ~-7-1

Fig. 4-21 PYRAMID OF BASE AREA A AND HEIGHT h

4.39 Volume = +Ah

Fig. 4-2 SPHERICAL CAP OF RADIUS ,r AND HEIGHT h

4.40 Volume (shaded in figure) = &rIt2(3v - h) 4.41 Surface area = 2wh

Fig. 4-23 FRUSTRUM OF RIGHT CIRCULAR CONE OF RADII u,h AND HEIGHT h

4.42 Volume = +h(d + ab + b2)

4.43 Lateral surface area = T(U + b) dF + (b - CL)~

= n(a+b)l Fig. 4-24

10 GEOMETRIC FORMULAS SPHEMCAt hiiWW OF ANG%ES A,&C Ubl SPHERE OF RADIUS Y

4.4 Area of triangle ABC = (A + B + C - z-)+

4.45 4.46

Volume = &z-~(u + b)(b - u)~ w Surface area = 7r2(b2 - u2)

4.47 Volume = \$abc

Fig. 4-27

PARAWlO~D aF REVOllJTlON T.

4.4a Volume = &bza

Fig. 4-28

5 TRtGOhiOAMTRiC WNCTIONS

DEFlNlTlON OF TRIGONOMETRIC FUNCTIONS FOR A RIGHT TRIANGLE

Triangle ABC bas a right angle (9Oo) at C and sides of length u, b, c. The trigonometric functions of angle A are defined as follows.

5.1 sintz of A = sin A = : = opposite B hypotenuse

5.2 cosine of A = ~OS A = i = adjacent hypotenuse

5.4 5.5 opposite tangent of A = tanA = f = -~ adjacent cotcznged of A = cet A = k = adjacent opposite A hypotenuse secant of A = sec A = t = -~ adjacent

5.6 cosecant of A = csc A = z = hypotenuse opposite Fig. 5-1

EXTENSIONS TO ANGLES WHICH MAY 3E GREATER THAN 90’

Consider an rg coordinate system [see Fig. 5-2 and 5-3 belowl. A point P in the ry plane has coordinates (%,y) where x is eonsidered as positive along OX and negative along OX’ while y is positive along OY and negative along OY’. The distance from origin 0 to point P is positive and denoted by r = dm. The angle A described cozmtwcZockwLse from OX is considered pos&ve. If it is described dockhse from OX it is considered negathe. We cal1 X’OX and Y’OY the x and y axis respectively.

The various quadrants are denoted by 1, I, II and IV called the first, second, third and fourth quad- rants respectively. In Fig. 5-2, for example, angle A is in the second quadrant while in Fig. 5-3 angle A is in the third quadrant.

Y Y I 1 I 1

Y’ Y’ Fig. 5-2 Fig. 5-3

12 TRIGONOMETRIC FUNCTIONS

For an angle A in any quadrant the trigonometric functions of A are defined as follows. 5.7 sin A = ylr

5.8 COS A = xl?.

5.9 tan A = ylx

5.10 cet A = xly 5.1 sec A = v-lx 5.12 csc A = riy

RELAT!ONSHiP BETWEEN DEGREES AN0 RAnIANS

A radian is that angle e subtended at tenter 0 of a eircle by an arc MN equal to the radius r.

Since 2~ radians = 360° we have

 5.13 1 radian = 180°/~ = 57.29577 95130 8232 o 5.14 10 = ~/180 radians = 0.01745 32925 19943 29576 92 radians

1 r e

0 r M

Fig. 5-4

REkATlONSHlPS AMONG TRtGONOMETRK FUNCTItB4S

5.15 tanA = 5 5.19 sine A + ~OS~A = 1

5.16 1 COS A &A ~I ~ z - tan A sin A 5.20 sec2A - tane A = 1

5.17 1 sec A = ~ COS A 5.21 csce A - cots A = 1

5.18 1 cscA = - sin A

SIaNS AND VARIATIONS OF TRl@ONOMETRK FUNCTIONS

1 + + + + + + 0 to 1 1 to 0 0 to m C to 0 1 to uz m to 1

- - I + + 1 to 0 0 to -1 -mtoo oto-m -c to -1 1 to ca

- I + + 0 to -1 -1 to 0 0 to d Cc to 0 -1to-m --CO to-1

-1 to 0 0 to 1 -- too oto-m uz to 1 -1 to --

TRIGONOMETRIC FUNCTIONS 13 EXACT VALUES FOR TRIGONOMETRIC FUNCTIONS OF VARIOUS ANGLES

Angle A Angle A in degrees in radians sin A COS A tan A cet A sec A csc A

0 0 0 1 0 w 1 c

15O rIIl2 #-fi) &(&+fi) 2-fi 2+* fi-fi &+fi

300 i/6 1 +ti *fi fi \$fi 2

450 zl4 J-fi \$fi 1 1 fi fi

60° VI3 Jti r 1 fi .+fi 2 ;G 750 5~112 i(fi+m @-fi) 2+& 2-& &+fi fi-fi 900 z.12 1 0 *CU 0 km 1

105O 7~112 *(fi+&) -&(&-Y% -(2+fi) -(2-&) -(&+fi) fi-fi 120° 2~13 *fi -* -fi -\$fi -2 ++

1350 3714 +fi -*fi -1 -1 -fi \h 150° 5~16 4 -+ti -*fi -fi -+fi 2

165O llrll2 \$(fi- fi) -&(G+ fi) -(2-fi) -(2+fi) -(fi-fi) Vz+V-c? 180° ?r 0 -1 0 Tm -1 *ca

1950 13~112 -\$(fi-fi) -*(&+fi) 2-fi 2 + ti -(&-fi) -(&+fi)

210° 7716 1 -46 &l3 fi -gfi -2

225O 5z-14 -Jfi -*fi 1 1 -fi -fi

240° 4%J3 -# -4 ti &fi -2 -36 255O 17~112 -&&+&Q -&(&-fi) 2+fi 2-6 -(&+?cz) -(fi-fi)

270° 3712 -1 0 km 0 Tm -1 285O 19?rll2 -&(&+fi) *(&-fi) -(2+6) -@-fi &+fi -(fi-fi)

3000 5ïrl3 -*fi 2 -ti -*fi 2 -\$fi

315O 7?rl4 -4fi *fi -1 -1 fi -fi 330° 117rl6 1 *fi -+ti -ti \$fi -2

345O 237112 -i(fi- 6) &(&+ fi) -(2 - fi) -(2+6) fi-fi -(&+fi) 360° 2r 0 1 0 T-J 1 ?m

For tables involving other angles see pages 206-211 and 212-215.

19 5.89 y = cet-1% 5.90 y = sec-l% 5.91 y = csc-lx

Fig. 5-14 Fig. 5-15 Fig. 5-16

RElAilONSHfPS BETWEEN SIDES AND ANGtGS OY A PkAtM TRlAF4GlG ’

The following results hold for any plane triangle ABC with sides a, b, c and angles A, B, C.

5.92 Law of Sines a b c -=Y=- sin A sin B sin C

5.93 Law of Cosines

/A C f

with 5.94 Law with 5.95 cs = a2 + bz - Zab C similar relations involving the other sides and angles.

of Tangents a+b tan \$(A + B)

- = tan i(A -B) a-b similar relations involving the other sides and angles.

sinA = :ds(s - a)(s - b)(s - c)

Fig. 5-1’7 where s = &a + b + c) is the semiperimeter of the triangle. Similar relations involving

B and C cari be obtained. See also formulas 4.5, page 5; 4.15 and 4.16, page 6.

angles

Spherieal triangle ABC is on the surface of a sphere as shown in Fig. 5-18. Sides a, b, c [which are arcs of great circles] are measured by their angles subtended at tenter 0 of the sphere. A, B, C are the angles opposite sides a, b, c respectively. Then the following results hold.

5.96 Law of Sines sin a sin b sin c -z-x_ sin A sin B sin C

5.97 Law of Cosines cosa = cosbcosc + sinbsinccosA COSA = - COSB COSC + sinB sinccosa with similar results involving other sides and angles.

20 TRIGONOMETRIC FUNCTIONS

5.98 Law of Tangents tan &(A + B) tan \$(u + b) tan &(A - B) = tani(a-b) with similar results involving other sides and angles.

5.100 where s = &(u+ 1 + c). Similar results hold for other sides and angles.

where S = +(A + B + C). Similar results hold for other sides and angles. See also formula 4.4, page 10.

NAPIER’S RlJlES FGR RtGHT ANGLED SPHERICAL TRIANGLES

Except for right angle C, there are five parts of spherical triangle AZ3C which if arranged in the order as given in Fig. i-l9 wiuld be a, b,A, c, B.

Fig. 5-19 Fig. 5-20

Suppose these quantities are arranged [indicating complcment] to hypotenuse c and

Any one of the parts of this circle is adjacext parts and the two remaining parts in a circle as in Fig. 5-20 where we attach the prefïx CO angles A and B.

called a middle pav-f, the two neighboring parts are called are called opposite parts. Then Napier’s rules are

5.101 The sine of any middle part equals the product of the tangents of the adjacent parts. 5.102 The sine of any middle part equals the product of the cosines of the opposite parts.

Example: Since CO-A = 90° -A, CO-B = 90° -B, we have sin a = tan b tan (CO-B) or sina = tanbcotB sin (CO-A) = COS a COS (CO-B) or ~OS A = COS a sin B

These cari of course be obtained also from the results 5.97 on page 19.

A complex number is generally written as a + bi where a and b are real numbers and i, called the imaginaru unit, has the property that is = -1. The real numbers a and b are called the real and ima&am parts of a + bi respectively.

The complex numbers a + bi and a - bi are called complex conjugates of each other.

6.1 a+bi = c+di if and only if a=c and b=cZ 6.2 (a + bi) + (c + o!i) = (a + c) + (b + d)i

6.3 (a + bi) - (c + di) = (a - c) + (b - d)i 6.4 (a+ bi)(c+ di) = (ac- bd) + (ad+ bc)i

Note that the above operations are obtained by using the ordinary rules of algebra and replacing 9 by -1 wherever it occurs.

2 COMPLEX NUMBERS

GRAPH OF A COMPLEX NtJtWtER

A complex number a + bi cari be plotted as a point (a, b) on an xy plane called an Argand diagram or Gaussian plane. For example p,----. y in Fig. 6-1 P represents the complex number -3 + 4i.

A eomplex number cari also be interpreted as a wector from 0 to P.

0 -X

Fig. 6-1

POLAR FORM OF A COMPt.EX NUMRER

In Fig. 6-2 point P with coordinates (x, y) represents the complex number x + iy. Point P cari also be represented by polar coordinates (r, e). Since x = r COS 6, y = r sine we have

6.6 x + iy = ~(COS 0 + i sin 0) called the poZar form of the complex number. We often cal1 r = dm the mocklus and t the amplitude of x + iy.

Fig. 6-2

tWJLltFltCATt43N AND DtVlStON OF CWAPMX NUMBRRS 1bJ POLAR FtMM ilj 0”

6.7 [rl(cos el + i sin ei)] [re(cos ez + i sin es)] = rrrs[cos tel + e2) + i sin tel + e2)]

V-~(COS e1 + i sin el) Z rs(cos e + i sin ez) 2 [COS (el - e._J + i sin (el - .9&]

DE f#OtVRtt’S THEORRM

If p is any real number, De Moivre’s theorem states that 6.9 [r(cos e + i sin e)]p = rp(cos pe + i sin pe)

RCWTS OF CfMMWtX NUtMB#RS

If p = l/n where n is any positive integer, 6.9 cari be written

6.10 [r(cos e + i sin e)]l’n = rl’n L e + 2k,, ~OS- + n where k is any integer. From this the n nth roots of a complex k=O,l,2 ,..., n-l.

i sin e + 2kH

~ n 1 number cari be obtained by putting

In the following p, q are real numbers, CL, t are positive numbers and WL,~ are positive integers.

7.1 cp*aq z aP+q 7.2 aP/aq E @-Q 7.3 (&y E rp4

7.4 u”=l, a#0 7.5 a-p = l/ap 7.6 (ab)p = &‘bp

7.7 & z aIIn 7.8 G = pin 7.9 Gb =%Iî/%

(Parte 1 de 3)