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Advanced topics include those from differentia] equations, vector analysis, Fourier series, gamma and beta functions, Bessel and Legendre functions, Fourier and Laplace transforms, ellip

ng

The purpose 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, care 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,

Preface

Topics covered range from elementary to advanced Elementary topics include those

from algebra, geometry, trigonometry, analytie geometry and caleulus Advanced topics include those from differentia] 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 I 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, ete.) as well as

advanced functions (Bessel, Legendre, elliptic, ete.) 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 error due to looking in the wrong column or row

I 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 I 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 Statistical Tables for Biological, Agricultural and Medical Research

I also wish to express my gratitude to Nicola Monti, Henry Hayden and Jack Margolin for their excellent editorial cooperation

M R SPIEGEL

Rensselaer Polytechnic Institute

September, 1968

CONTENTS

pat FORMULAS |

Special Products and Factors

The Binomial Formula and Binomial Coeffieients -: 3 Geometrie Formulas

Trigonometrie Functions Complex Numbers

Exponential and Logarithmie Functions

Hyperbolic Functions

Solutions of Algebraic Equations

Formulas from Plane Analytic Geometry

Special Plane Curves

Formulas from Solid Analytic Geometry

Derivativyes c.c cà {ào

Indefinite Integrals Definite Integrals

The Beta Function

Basic Differential Equations and Solutions

Series of Constants

‘Taylor Series " Bernoulli and Euler Numbers 14 Formulas from Vector Analysis 116

18a Hyperbolic functions sinh

18b Hyperbolic functions cosh x

16c Hyperbolic functions tanh «

Four Place Common Logarithm:

Four Place Common Antilogarithms

Sin x (x in degrees and minutes)

Cos (x in degrees and minutes)

‘Tan a (x in degrees and minutes)

Cot 2 (2 in degrees and minutes)

Sec (x in degrees and minutes) - Csex (x in degrees and minutes) Natural Trigonometric Functions (in radians)

log sinx (x in degrees and minutes) log cosx (x in degrees and minutes) log tan (x in degrees and minutes)

Conversion of radians to degrees, minutes and seconds

Bessel functions Io(z)

Bessel functions J; (:) Bessel functions Ko(x)

Bessel functions Ki(z) Bessel functions Ber (x)

Bessel functions Bei (7) Bessel functions Ker (x)

Bessel functions Kei (x) 249

Values for Approximate Zeros of Bessel Functions Exponential, Sine and Cosine Integrals

Legendre Polynomials P,(z) Legendre Polynomials P, (cos ¢)

Complete Elliptic Integrals of First and Seeond

Incomplete Elliptic Integral of the First Kind Incomplete Elliptic Integral of the Second Kind

Ordinates of the Standard Normal Curve

Areas under the Standard Normal Curve

Percentile Values for the Chi Square Distribution 95th Percentile Values for the F' Distribution 99th Percentile Values for the F Distribution Random Numbers

Part I

FORMULAS

1.27 1° = /180 radians = 0.01745 32925 19943 29576 92 radians

2 ety = etary ty

22 (—y)? = 2-day ty?

23 (ety = 2+ 3ety t+ Sey + `

24 (xy) = z9— 3z9y + 8w? — tŸ

2.5 (ety = att Arty + Gaty? + doy? + yt

260 (x—v)t = z4 — dzêy + 6xty® — dey? +

27 (xt y)8 = x8 + 5zty + 10232 + 102%5 + õzyt + vŠ

2.8 (wy) = 2b — Baty + 10299 — 10r3/9 + Bzy# — 5

2.9 (z+u)® = z9 + 6z5y + 167%/2 + 20z3y3 + 16z2y! + 6zy5 + VẺ

2/10 (@œ—y)* = z9— 6z5y | 16z9y3— 90x33 + 15z3/! = Gzy5 + y®

‘The results 2.1 to 210 above are special cases of the binomial formula [see page 3),

216 (et Wet —2ty + att — 8+)

2.17 z®—w® = (x—-wWlet wet tay t yet ey + vt)

218 tatty = tt ayt yet)

2A zt+ 4y' = (22+ 2zw+9g)(z2— 2zw + 2w)

Some generalizations of the above are given by the following results where n is a positive integer

2.20 inti — yontt = (x — yet + at ty + aye to ys)

4s

= œ-u( - Ray 0085 TT + v= — #zụ cosg Trị + #)

oe (# = Bay coset + ) BBL ened gf yt (aH ylla™ — s Ty | s29 — os +)

3 “The BINOMIAL FORMULA

and BINOMIAL COEFFICIENTS

3.3 + = z® + nớny + aay? + ) mays be tym

This is called the binomial formula It can be extended to other values of n and then is an infinite series

[sce Binomial Seriee, page 110)

(2) bag tees +2) Ê nai ại 1 2, „

where the sum, denoted by =, is taken over all nonnegative integers ?t4,?z„.- -;?t„ for which

Area = bh

Perimeter = 20 +26

TRIANGLE OF ALTITUDE h AND BASE b

Area = Joh = Jabsinz «

GEOMETRIC FORMULAS 1 REGULAR POLYGON OF n SIDES INSCI

428 Volume = Ak = abcsing

SPHERE OF RADIUS + 4.29 Volume = fer

4.30 Surface area = der

Fie 4-17 RIGHT CIRCULAR CYLINDER OF RADIUS 7 AND HEIGHT h

431 Volume = =rth 4.32 Lateral surface aren = 2erh

ig 4-18 CIRCULAR CYLINDER OF RADIUS 7 AND SLANT HEIGHT J

433 Volume = eth = crHsine

434 — Lutoral surface area = Zerl = = Yerh ese

Fig 4-19

GEOMETRIC FORMULAS 9 CYLINDER OF CROSS-SECTIONAL AREA A AND SLANT HEIGHT |

P

4.36 Lateral surface area ol hp = pheacø y

Note that formulas 4.31 to 4.34 are special eases

Fig 4-20 RIGHT CIRCULAR CONE OF RADIUS 7 AND HEIGHT h

438° Lateral surface aren = erVFER = eri nh

i Fig 421

PYRAMID OF BASE AREA A AND HEIGHT h

439 Volume = 4ah JAN

SPHERICAL CAP OF RADIUS r AND HEIGHT h

i

440 Volume (shaded in figure) = }rh%(3r—h)

4A1— Surface area = 2mrh C

Fig 28 FRUSTRUM OF RIGHT CIRCULAR CONE OF RADII a,b AND HEIGHT h

442 Volume = jrh(a? + ab+ 0) 1

4.43 Lateral surface area a+b) ViF> bai

Fig 4-25 TORUS OF INNER RADIUS a AND OUTER RADIUS b Volume = ‡z*(a + ð)(ð—g)*

Surface area = =%(6

Fig 4-25 ELLIPSOID OF SEMI-AXES a,b,c

5 TRIGONOMETRIC FUNCTIONS

DEFINITION OF TRIGONOMETRIC FUNCTIONS FOR A RIGHT TRIANGLE

Triangle ABC has a right angle (90°) at C and sides of length a,b,c The trigonometric functions of angle A are defined as follows

‘ - _ & _ _opposite_

5.1 sine of A = sinA = ¢ = ypotenuse

52 cosine of A = cosA = 2 = diac

5.6 cosecant of A = escA = © = hypotenuse a opposite ie

EXTENSIONS TO ANGLES WHICH MAY BE GREATER THAN 90°

Consider an xy coordinate system [see Fig 5-2 and 5-3 below] A point P in the xy plane has coordinates (x,y) where # is considered as positive along OX and negative along OX’ while y is positive along OY and

negative along OY’ The distance from origin O to point P is positive and denoted by r = Vx? + w°

The angle A described counterclockwise from OX is considered positive If it is described clockwise from

OX it is considered negative We call X’OX and Y’OY the x and y axis respectively

‘The various quadrants are denoted by I, 11,111 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

Fig 5-2

11

5.16 cot ima > 5.20 sec? A — tan? A 1

BIT secA = Shp 5.21 csc? —cot?A = 1

1 5.18 cea = 3t5

SIGNS AND VARIATIONS OF TRIGONOMETRIC FUNCTIONS

Quadrant sind cosa | tan | ese

1 Oot _ 1to0 = 016 ° “tồi Ặ

EXACT VALUES FOR TRIGONOMETRIC FUNCTIONS OF VARIOUS ANGLES

ated et] mà | mee [na | me mt

FUNCTIONS OF NEGATIVE ANGLES

5.28 sin(—A) = —sinA 5.29 cos(—A) = cosA 5.30 tan(—A) = —tanA

5.31 cse(-A) = —cseA 5.32 sec(-A) = secA 5.33 cot(—A) = —cotA

e | —e<4 see A ~escA +eacA see set + se 4 —se 4 +ese A sec A cot | —eotA tana = cot =tanA +eotA

smA=w | cosA=u | tana=u | cotA=w | secA=u | eA=w

sin A “ vi-e 1/V1 + vẽ wire Vit Tlu Vu

cos A “ 1/V1 + tê tựV1 + tế Ve Vưề= 1,

cota vĩ “xã, Vie Vw “ wea ve=1

see A wie tw vite VIF wu u x/Vw2=1

ese A tin 1/VT=tẽ Vite vite wei vive

5.52

TRIGONOMETRIC FUNCTIONS

DOUBLE ANGLE FORMULAS

sin24 = 2sinA cosa cos2d = cost —sintA = 1-2sintA = #coœ24 =1

tana

tan 24

HALF ANGLE FORMULAS

A SeeA [if A/2 is in quadrant I or 1T

sng V3 ~ if A/2 is in quadrant III or IV

A Tied [+ if A/2 isin quadrant I or IV |

we 2 [+ if Av is in quadrant 11 or 111 | A_ wed [+ Mf A/2 ts im quadrant I or HI fang = FGF — if A/2 is in quadrant II or IV

MULTIPLE ANGLE FORMULAS

sin3A = BsinA — Asin contd = AcossA — 3 cond

_ Stand ~ tant tanda = STS tant A sin44 = 4sinA cosA — 8 sin?A cosA

044 = BeostA — BeostA +1

Atan A ~ 4 tant A

wanda = TT p tanta + tanta

sinSA = 5sinA ~ 20sintA + 16 sind A cos5A = 16cos54 — 20 cost + 5 cos A

POWERS OF TRIGONOMETRIC FUNCTIONS

{— Jeos2A 5.57 sintA = § — {co9A + {cos4A

b+ dcos2a 5.58 cost = § + Ðc0S2A + $ cos44, ậsinA — 4 sin3A 5.59 sinðẳA = §sinA — yy sin3A + yy sindA

feos + } cosa 5.60 cos*A = §coSA + ly coS8A + 3h cosa

See also formulas 5.70 through 5.73

TRIGONOMETRIC FUNCTIONS 17

SUM, DIFFERENCE AND PRODUCT OF TRIGONOMETRIC FUNCTIONS

5.61 sinA +sinB = #sinj(A + Bì eo 1(A = Bị

5.62 sinA ~ sin B 2 cus [(A + B) sin 1A — B)

5.63 cosA + cosB = 2cos4(A 1 BỊ cosj(A- ØỊ

5.64 cosA — cosB = 2sin}(A +B) sin }(B—A)

5.65 sinB = jeos(A—B) — cos (A +B)

5.66 cos cos = feos (AB) + cos (A +B))

5.67 sin A cosB = 4{sin(A—B) + sin (A +B))

S70 sineta = CD®”Ígngm-qyg — (Phí T) man sa + C1 l 1) m4)

leercnA +( >) cositn BA + có + (-7)««a}

S1 con 1A, Bt

572 sin™A = a( là) + (ay J jemand — () cos (2 —2)A hoe co an la)

1 (9 1 (® 2n

573 coma = (M+ phi feasena + (2) cos uma + oo + (2, Jeane

INVERSE TRIGONOMETRIC FUNCTIONS

If z=siny then y =sin~!z, ie the angle whose sine is x or inverse sine of x, is a many-valued

function of x which is a collection of single-valued functions called branches Similarly the other inverse

trigonometric functions are multiple-valued

For many purposes a particular branch

for this branch are called principal values required, This is called the principal branch and the values

0 5 see~1z < z/2 | < sec Tz £ 0< se 1z Z z/2 các 1z < 0

RELATIONS BETWEEN INVERSE TRIGONOMETRIC FUNCTIONS

In all cases it is assumed that principal values are used

S74 sin tx + cos tx = x2 5.80 sin *(-2) = —sin tx

5.75 tan lz { cot ly — v2 S81 co=l(Cz) — r = cost

S76 sec lx + các lực — r2 $.82 tan-!(-x) = —tan !z

S77 esete = sin-1 (1/2) ` = — cot-te

S78 se tờ eos~1 (1/3) 5.84 sce“! (—2) 5 — scent

579 cote = tan”! (1/2) 5.85 csc '(-2) = —ese tx

GRAPHS OF INVERSE TRIGONOMETRIC FUNCTIONS

In each graph y is in radians Solid portions of curves correspond to principal val

5.86 w = sin-fz 5.87 w = cos 1x 5.88 y = tan lx

B and C can be obtained

See also formulas 4.5, page 5; 4.15 and 4.16, page 6

RELATIONSHIPS BETWEEN SIDES AND ANGLES OF A SPHERICAL TRIANGLE

Spherical triangle ABC is on the surface of a aphere as shown

in Pig, 548 Sides @,0,¢ [which are arcs of great circles) are

rieasured by their angles subtended at center O of the sphere A,B,C

are the angles opposite sides a,b,c respectively Then the following

5.97 Law of Cosines

cosa = cosbcose + sind sine cos A cosA = —cosBcosC + sinB ain C cosa With cimilar rcoults involving other aides and angles

where S= }(A+B+C) Similar results hold for other sides and angles,

See also formula 4.44, page 10

NAPIER’S RULES FOR RIGHT ANGLED SPHERICAL TRIANGLES

Except for right angle C, there are five parts of spherical triangle ABC which if arranged in the order

as given in Fig 5-19 would be a,b,A,¢,B

Example: Since co-A = 90°—A, e0-B = 90° ~B, we have

sina = tanb tan (co-B) or sing = tanbcotB sin(co-A) = cos@cos(coB) or cosẢ = cosasinE

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

COMPLEX NUMBERS 4

DEFINITIONS INVOLVING COMPLEX NUMBERS

A complex number is generally written as a + bi where a and 6 are real numbers and i, called the

imaginary unit, has the property that # = —1 The real numbers a and 6 are called the real and imaginary

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

EQUALITY OF COMPLEX NUMBERS

a+bi = e+di ifandonlyif a=e and b=d

ADDITION OF COMPLEX NUMBERS

(œ+ bồ + (c+đủ = (a+c) + (b+ đi

SUBTRACTION OF COMI

EX NUMBERS (a+ bi) — (e+ di) = (a~=e) + (b— địi

MULTIPLICATION OF COMPLEX NUMBERS

(a+ bio+di) = (ac—bd) + (ad + be)i

DIVISION OF COMPLEX NUMBERS

et+bt _ at+bi,e~di _ get bd | be — ad

et+di ctdi e~di ” tt+a@ ere

Note that the above operations are obtained by using the ordinary rules of algebra and replacing # by

—1 wherever it occurs

21

22 COMPLEX NUMBERS

GRAPH OF A COMPLEX NUMBER

A complex number œ + bi can be plotted as a point (ø,b) on an

xy plane called an Argand diagram or Gaussian plane For example

in Fig 6-1 P represents the complex number —8 + 4i

A complex number can also be interpreted as a vector from

OP

POLAR FORM OF A COMPLEX NUMBER

In Fig 6-2 point P with ecordinates (z,y) represents the complex number x + iy Point P ean also be represented by polar coordinates (y8) Since z= ros, y=rsing we have

66 atiy = rlcose + ising) _

called the polar form of the complex number We often call r= Vz? + yẺ`

the modulus and 9 the amplitude of z + iy

Fig 6-2 MULTIPLICATION AND DIVISION OF COMPLEX NUMBERS IN POLAR FORM

67 [rs(eos #5 + £ sin @))[ra(eos 8g + i sin eg)] = ryraloos (¢, + #2) + isin (6, + 6)]}

68

DE MOIVRE’S THEOREM

If p is any real number, De Moivre’s theorem states that

69 [ricos ¢ + i sina)]® = r(cos po | i sin pe)

ROOTS OF COMPLEX NUMBERS

If p=1/m where n is any positive integer, 6.9 can be written

mm [=* AES + cain AB]

where is any integer From this the w nth root of a complex number can be obtained by putting

k ¬ 6.10 r(cose + é sin 6)j9/®

TA aPrat=ante 7.2 arlar=arn-a 73 (@)s= a"

z4 aro 78 a-*= tar 7.6 (aby = arb

In af, pis called the exponent, a is the base and av is called the pth power of a The function = 4#

is called an exponential function

LOGARITHMS AND ANTILOGARITHMS

If @=N where a0 or 1, then p= log.N is called the logarithm of N to the base ø The number

N = ar is called the antilogarithm of p to the base a, written antilog, p

Example: Since 32 =$9 we have log; 9 = 2, antilogy2 = 9

‘The function y= loggz is called a logarithmic function

LAWS OF LOGARITHMS

7.10 logs MN = log M + logaN

zm logs = og, M — loge N

7.12 log, MP? = plog, 2

COMMON LOGARITHMS AND ANTILOGARITHMS

‘Common logarithms and antilogarithms [also called Briggsian| are those in which the base The common logarithm of N is denoted by logyy N oF briefly log N For tables af common logarithms and

antilogarithms, see pages 202-205 For illustrations using these tables see pages 194-196

23

24 EXPONENTIAL AND LOGARITHMIC FUNCTIONS

NATURAL LOGARITHMS AND ANTILOGARITHMS

Natural logarithms and antilogarithms [also called Napierian] are those in which the base a =e = 2.71828 18 /see page 1} The natural logarithm of N is denoted by log,N or InN For tables of natural

logarithms see pages 224-225 For tables of natural antilogarithms {ie tables giving e* for values of =]

see pages 226-227 For illustrations using these tables see pages 196 and 200

CHANGE OF BASE OF LOGARITHMS

‘The relationship between logarithms of a number N to different bases a and 6 is given by

logs N 7.13 lore N = Teg

In particular,

7.14 loreN = InN 2.30258 50929 94 loge

7.15 logiyN = logN = 0.43429 44819 03 log N

RELATIONSHIP BETWEEN EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS

EXPONENTIAL AND LOGARITHMIC FUNCTIONS 25

POLAR FORM OF COMPLEX NUMBERS EXPRESSED AS AN EXPONENTIAL

‘The polar form of a complex number x + iy can be written in terms of exponentials (sce 6.6, page 22] as

724 z+ iy = (eos® + isine) = rel

OPERATIONS WITH COMPLEX NUMBERS IN POLAR FORM

Formulas 6.7 through 6.10 on page 22 are equivalent to the following

Hyperbolic tangent of 2 = tanhz = = =

Hyperbolic cotangent of » = cothz = S78 Hyperbolic secant of 2 = seche = >

Hyperbolic cosccant of «= cache = => —

RELATIONSHIPS AMONG HYPERBOLIC FUNCTIONS

_ sinh tanhe = cosh,

— 1 _ coshe

cothe = tanhe ~ sinhz

sehz = ~ cosh co

FUNCTIONS OF NEGATIVE ARGUMENTS

sinh(=z) = ~ sinh 815 ©o8sh (—=#) — coshz 8.16 tanh(=2) = —tanhe

esch(—z) = —cschz 8.18 sech(—z) = sechz 8.19 coth(—xz) = —cothe

26

HYPERBOLIC FUNCTIONS ADDITION FORMULAS

8.20 sinh (x+y) = sinhz coshy + eoshz sinh y

s41 S08h(#=v) = cosh coshy + sinh z sinh y

- sahz—1 sinh

G tank? x + tanhés

In the following we assume z >0

SUM, DIFFERENCE AND PRODUCT OF HYPERBOLIC FUNCTIONS

HYPERBOLIC FUNCTIONS

POWERS OF HYPERBOLIC FUNCTIONS

sinh#z = cosh? x = sink? x = cosh? z=

sinhtz = cosht x =

sinhz + sinhự sinhz — sinhy cosh x + cosh y cosh x — cosh y

= Qsinh f(x +9) cosh }z — y)

= 2eosh Lx +y) sinh {z—y)

2 cosh f(x + y) cosh Hx—y)

= 2sinh (x+y) sinh $Íz — #)

$(eosh (z+ v) — cosh Œ# — y)}

Heosh (x + y) + cosh Œ — w)}

Hoinh (z +y) + sinh (@— 9)

EXPRESSION OF HYPERBOLIC FUNCTIONS IN TERMS OF OTHERS

If x <0 use the appropriate sign as indicated by formulas 8.14

cothz | VWÊ+1⁄e nve=1 Vu “ LVT=ẽ vite sechz | 1/VT cu? Vu vi wt each = Vu 1/VWÊ—1 vi- in VWE~=1 wy

HYPERBOLIC FUNCTIONS 29 GRAPHS OF HYPE!

case of inverse trigonometric functions ‘see page 17] we restrict ourselves to principal values for which

they can be considered as single-valued,

‘The following list shows the principal values {unless otherwise indicated] of the inverse hyperbolic unetions expressed In terms of logarithmic functions which are taken as real valued

RELATIONS BETWEEN INVERSE HYPERBOLIC FUNCTIONS

eseh !z = sinh? (I/x) seeh~1Z = eosh~!(1/z) coth~lz = tanh~!(1/z)

sinh“!(~z) = —sinh~tz tanh~!(—z) = —tanh~!z coth~!(Te) = —eoth~!z cạch 1 z) — —eseh~tr

GRAPHS OF INVERSE HYPERBOLIC FUNCTIONS

868 y= sinh~lz 869 y= cosh B70 y = tanh tự

HYPERBOLIC FUNCTIONS 31

RELATIONSHIP BETWEEN HYPERBOLIC AND TRIGONOMETRIC FUNCTIONS

874 sin(@ø) = isinhz 8.75 cos (iz) = cosh x 8.76 tan (iz) = itanhe

8.77 esc (iz) = —icschz 8.78 sec (iz) = sechx 879 cot (ix) = —fecothe

8.80 sinh (iz) = ising 8.81 cosh (iz) = cosz 8.82 tanh (ix) = itanz

8.83 each (ix) = —iesez 8.84 seh(z) = secz 8.85 coth(z) = —ieotz

PERIODICITY OF HYPERBOLIC FUNCTIONS

In the following & is any integer

8.86 sinh (z + 2kz = sinhz 8.87 cosh (x+2kxi) = coshz 8.88 tanh (z+ kei) = tanhz

8.89 esch (x + 2kvi) = esche 8.90 sech (x + 2kzi) sech x 8.91 coth (2+ kri) = eothz

RELATIONSHIP BETWEEN INVERSE HYPERBOLIC AND INVERSE TRIGONOMETRIC FUNCTIONS

If a,b,c are real and if D = 6%—4uc is the discriminant, then the roots are

real and unequal if D> 0 (ii) real and equal if D=0 i) complex conjugate if D <0

92 It xyz, are the roots, then 21 +2; =—B/e and 222 = e/a

CUBIC EQUATION: 2° +a:2*+a2+a, = 0

aya, — 27a, — 2a , R= ae

If 4,,a,, a3 are real and if D = Q°+R? is the diseriminant, then

cone root is real and two complex conjugate if D > 0 (Gi) all roots are real and at least two are equal if D = 0 i) all roots are real and unequal if D <0

If D <0, computation is simplified by use of trigonometry

Íz, = 2V=@ cos (qo)

94 zy = 2V—O cos (he + 120°) where cose = —R/iV=@

ty = 2V—O cos (he + 240°)

95 By tay tay = Moy ney t age tage = Oy eee = ty

where 2),43,25 are the three roots

32