chapter 3 elementary analytical methods

7/29/2019 Chapter 3 Elementary Analytical Methods

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3 Elementary A nalytical M ethodsMILTON BRAMOWITZ

ContentsE lementary A nalytical M ethods . . . . . . . . . . . . . . . . .3.1. Binomial Theorem and Binomial Coefficients; Arithmetic andGeometric Progressions; Arithmetic. Geometric. Harmonicand Generalized Means . . . . . . . . . . . . . . .3.2. Inequalities . . . . . . . . . . . . . . . . . . . . .3.3. Rules for Differentiation and Integration . . . . . . . . .3.4. Limits.Maxima and Minima . . . . . . . . . . . . . .3.5. Absolute and Relative Errors . . . . . . . . . . . . . .3.6. Infinite Series . . . . . . . . . . . . . . . . . . . . .3.7. Complex Numbers and Functions . . . . . . . . . . . .

3.8. Algebraic Equations . . . . . . . . . . . . . . . . . .3.9. Successive Approximation Methods . . . . . . . . . . .3.10. Theoremson Continued Fractions . . . . . . . . . . . .

PBge10

0101113141416171819

N umerical M ethods . . . . . . . . . . . . . . . . . . . . . . . 193.11. Use and Extension of the Tables . . . . . . . . . . . . 193.12. Computing Techniques. . . . . . . . . . . . . . . . . 19References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23T able 3.1. Powers and Roots . . . . . . . . . . . . . . . . . . 24nk.k=1(1)10. 24. 1/2.1/ 3. 1/4. 1/5

n=2(1)999. Exact or 10s

The author acknowledges the assistanceof Peter J O Hara nd Kermit C Nelson inthe preparation and checkingof the tableof powers and roots

1 National Bureau of Standards (Deceased.)9

1
http://e77c247f7db22f9317af8a3317a85a5.pdf/http://e77c247f7db22f9317af8a3317a85a5.pdf/


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3. Elementary Analytical Methods3.1. B inomial T heorem and B inomial C oeffi-cients; A rithmetic and G eometr ic P rogres-sions; A rithmetic, G eometric, H armonic andG eneralized M eans

B inomial T heorem3.1.1

+C)a"-ag3+. . . +b"( n positive integer)

B inomial C oefficients ( seehapter 24)3.1.2n( n-1) . . ( n-k+ l) - n!-

k! ( n- k) k!

3.1.5

k-n-13*1*3 ( I E ) =( nnk) =( -l) ' ( )n+ l3.1.4 ( k ) =( I E ) + ( knl)

C) =C) =3.1.6

3.1.7 l-c) + c) -.. + ( -l ) " C ) =Ol + c) + C ) + ..+c) =2"

(3able of B inomial C oefficienta3.1.8

For a more extensive table seechapter 24.*Seepage I I .10

11 12--

112 1

3.1.9Sum of A rit hmetic P rogreslrion to n T erms

aS-(a+d)+(a+2d)+ - . + ( a+ ( n--l) d)=na+- n( n- l ) d=2(a+I),2

last term in series=Z=a+(n-1)dSum of G eometr ic P rogreeeion to n T erms

3.1.10

lim s,=a~( l-r) ( -lO, k=1,2, . . .,n)

G eneralized M ean3.1.143.1.15 M (t)=O(tH,equality if nnd only if a1=a2= . . =a,,3.2.2 min. a


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ELEMENTARY ANALYTICAL METHODS 113.2.3 min. a$ uk) (& k)

H older s nequality for Sums1 1P qf -+ =, p>, q>

equality holds if and only if lbkl=cluklp- l (c=con-stant>O). I f p=q=2 we get

Cauchy s nequality3.2.9c constant).

Holder s nequality for Integrals1 1P Pf -+ =, p>, q>

3.2.10J - m d X ) ldxI[ J f(4 dx] [d4PdX]equality holds if and enly if Ig(x) =cl f(r) 1p-l(c=constant>O).I f p=q=2 we get

Schware s nequality3.2.11

M inkowski s nequality for SumsIf p> and U.k, br>O for all k,

3.2.12

equality holds if and only if bk=cuk (c=con-stan t>O) .M inkowski s nequality for Integrals

I f P>,3.2.13

equality holds if and only if g(x)=cj(x) (c=con-stant>O).3.3. R ules for D ifferentiation and I ntegration

D erivativesd du- cu)=c -9 c constantdx dx.3.1

3.3.2

3.3.3

d du dvdx dx dxu+v)=-+-d dv dudx dx dxuv)=u -+v -

3.3.5d du dv- (v)=- -dx dv dx

3.3.6 d (uo)=uo g+ln u2)Leibniz sTheorem for DilT erentiationof an Integral

3.3.7


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12 ELEMENTARY ANALYTICAL M E T H O D S ,Leibniz'sTheorem for Differentiationof a Product

3.3.8d" n dn-2ud2udz"(uv)=g+c) 2+(2) dT 2@

n dn-'u d'v dnv+ . .+(J &z=+.. +u- x"3.3.9 !%@dy dx3.3.10

Integrationby Parts

3.3.12

3.3.13 J uvdx=(s1..> v-J (sudx) $xIntegralsof Rational Algebraic Functiona

(Integrationconstantsare omitted)( n#-1)ax+b)"+'3.3.14 S(ax+b)"dx= a( n+ l )

3.3.15

3.3.263.3.27

The followingformulas are useful for evaluating-s(a:$y'c)n where P(x) is a polynomial andn>l isaninteger.3.3.16

2ax+b(4ac- b2)tarctanx(b2-4~O)(bd) 'la=-.3.283.3.293.3.30


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ELEMENTARY ANALY'MCAL M E T H O D S 13

ax- dG:)2 S[(a+bx)(c+dx)]ll23.3.32

ax2b S[(a+bx) ( c+ d~)]'/ ~-3.3.33

s aZ"+ bx+c)'I2dx=,-'I2 In !2a112(az2+x+c)1/ 2+2ax+ l (a>O)3.3.34

3.3.353.3.36

3.3.37J ax2-3.3.38

(a< b2>4ac, 12ax+b< (b2--4aC)'/ ?

2ax+ b4az+c)1/2dx=- (ax2+bx+c)'I2ax(az2+bx+c)

1/ 2where t= l/ xx atsx ( a9+ bx+c) lI2- -S(a+bt+ct )3.3.39

XdXs a z"+bx+ )

3.3.44

x-a=arcsin-.3.47 J 2a2E2)' a3.3.443

(2ax-x2)++32arcsin x-a-ax- x2)+dx=-S 23.3.49

3.3.50 - 1 [b(~Z" + d) ]+ + ~(b~-d) +2[b(bc--ad)]+ In 1[b(cx2+d)]+- ~( bc- d ) '(bc>d)

3.4. Limits, MaximaandMinimaIndeterminate Forme (L'Hospital's Rule)

3.4.1interval a


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14 ELEMENTARY ANALYTJ CAL METHODBM axima and M inima

3.4.2The function y=f(x) has a maximum at x =qif f ( q) =O and f(q)O. Points 4 forwhich f ( q) =O are called stationary points.3.4.3 (2) Functionso Two Variablesfor those values of (q,,yo) or which

(1) Functionsof One Variable

The function ( x, ) has a maximumor minimum

(a) f(x,y) has a maximum

(b) j ( x,) has a minimumif ->0 and ->0 at (xo,yo).ax2 by2

3.5. Absoluteand RelativeErrors(I) If q is an approximation to the true value

(a) the absold error of q is Ax=xo-x,of x, then3.5.1x--% is the correction to x.

Ax Ax3.5.2 (b) the relative error of x, isax= =-2 xo3.5.3 (c) the percentage error is 100 times therelative error.3.5.4 (2) The absolute error of the s um ordifference of several numbers is t most equal tothe sum of the absolute errors of the individualnumbers.3.5.5 (3) If f(zl, q, . . ., x) is a function ofxl, x2, . . ., xn and the absolute error in xi(i=l,2, . . . n) is Axi, then the absolute errori n f i s

3.5.6 (4) The relative error of the product orquotient of several factors is at most equal to thesum of t.he relative errors of the individual factors.3.5.7(5) If y=f(x), the relative error 8y=-=-Y f ( ) AxY A x)

Approximate ValuesI f e


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ELEMENTARY ANALYT ICAL METHODS 153.6.9(1 +x)==I ax+- x + 2+ . .,(a-1) a(a-I)(a-2)3!!3.6.10(1+x)-1=1-x+x2-2+x4-- . . .


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16 ELEMENTARY ANALYTICAL METHODSRevenion of Serieo

3.6.25 Giveny=a;~+b2?+cza+d;c'+ezs+fx~+~+ . . .

thenx=A y+Byg+C y8+D y'+E $+F ~+C 4y7+

whereaA=1a*B=-ba6C=2b2--aca7D=5abc- 2d- baaOE=a2bd+3a28+14b4- ae 2 lab2ca"F= 7aabe+7aacd+84ab3c-a4falaf=8abf +8a%e+4aW +12 Oa2bad

-28a2b8-42 b6- 28a2bgd+180a2b28+ 32b*-a6g-36a3b2e- 2aabcd- 2a38- 330ab'cKummer's Transformationof Series

m-03.6.26 L et Fak=sbe agiven convergentseriesandgok=c be a given convergent series with known

sum c such that lim %=AZO.Then

0)

k -m cks=xc+& (1-A 2) k.Euler'o Tramformationof Serieo

0)3.6.27 I f & (-l)kak=%-@+az- . . . is a con-vergent seriea with sum s then

-(- )*Akao k k2k+l 'A k%ao-x-0 (-I)* (m ak-mEuler-Maclaurin Summation Formula

3.6.28

3.7. ComplexNumbersandF U ~C ~~O MCartesian Form

3.7.1 z=x+ig/

Polar Form3.7.2 z=refe=r(cose+isin e)3.7.3 Modulus: lzl=(2+y2)r=r3.7.4 Argument: arg z=arctan (&)=e (othernotations for arg z are am z and ph z).3.7.5 Real Part: X = ~ Z = T COSe3.7.6 Imaginary Part: y=Y z=r sin t9

Complex Conjugate of I3.7.7 z=x-iy3.7.8 14=1~13.7.9 arg z=-arg 2-

Multiplicationand DivisionIf zl=z1+ yl z2=z2+iy2 then

3-7-10 Z!Z~=XIZ~Y IV~i( ~ly2 ~ l y d3.7.11 IZlzZl =121112213.7.12 arg (z1zz)=argZ l + q zz

3.7.143.7.15

Powers3.7.16 zn=rnein03.7.17 = rnCOS ne+irn sin ne(n=O,fl,f2,. . .)3.7.18 9=2?-yZ+i(2xy)3i7.19 23=3?-32$+i(3*-$)3.7.20 z'= - w ' + i 42Y -44)3.7.21 9~2-029+52yli-i (5x'y- lox2$ $13.7.22

(n=1:2,. . .)


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EL EM EN T A RY A N A L Y T IC A L METHODS 17If zn=un+ivn, then ~~+1=u,,+~+iv~+lhere3.7.23 ~n + l = U ~- Y V ~ Vn+1= zt)n+yUn9 2 " and Yznare called harmonic polynomials.3.7.243.7.25

ROO--3.7.26 z*=J z=r*e*'@=r*OS @+idin l e

If --r+l*

21=(81+82)-~a2

If zl,a, 3 are the rootsof the cubic equation21+22+a=a2z1z2+z1z3+z2z3=alZlZ2Z3= -UC,

Solution of Quartic Equations3.8.3 Given z'+a3~+a&+~z+a,,=0, ind thereal root ul of the cubic equation

u3&u2+ (a1~-4a,,)u- (a:+& - 4 a ~ )=0and determine the four roots of the quartic aasolutions of the two quadratic equations


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18 ELEMENTARY AN AL Y T I C AL M E T H O D SI f all roots of the cubic equation are real, usethe value of it 1 which gives real coefficients in tlwquadratic equation and select signs so that if

3.9. SuccessiveApproximath Metha&General Comments

3.9.1 Let z=zl be an approximation to z=wheref()=O and both xl and [ re in the intervalasxsb. We definex.+l=z.+cJ (x.) (n=1, 2, . . .I.

Then, if f (x)>O and the constants c,, arenegative and bounded, the sequence x convergesmonotonically to the mot f.I c,,=c=constantO, then theprocess converges but not necessarily monotoni-cally.Degree of Convergence of an Approxirnntion Procem3.9.2 Let xl , x2, x3, . . . be an infinite sequenceof approximations to a number 4. T hen, if

I G +~- I


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3.10. Theoremson Continued FractionsDefinitions3.10.1

(1) Let J =b0++,1b2+%

a1 az %ba+ ' * '

- ---bo+bl+ bz+ b3+ * a *If the number of ternis is finite, .f is cidled (Iteriiiinating continued fraction. If the iiuriiberof ternis is infinite,j sctilledminfinite continuedfraction and the terminating fraction-?=bo+- a1 -a 5j"-Bn h+ b2+ ' ' ' ba

iscalled the nth convergent ofj.(2)tionf is said to be convergent.br are integers there is always convergence.

Theorems(1) If at and br we positive then jZnin+l*A.(2) I fjn=X

An= bnAn- 1+an& -Bn=b&-1 +anBn-zwhere A -l=l , &=bo, B-l=O, Bo=l.

(3) [:;]=[Bn-, Bn-J [:]An-1 4 - 2

I1 *(4) AnBn-i-An-i13n=(-l)n-' LI ~ f iP -1(5) For every n>O,cia1 Ci&CGL C&Qs Cn-lCnan.jn=o+cxTx+ * * * cnbn

(6) l+bl+btba+ . . - +b2b3 - - . n1 b2 b3 bn. . .-- -- bz+l- b3+1- - bn+l1%-1. . .1 1 1 21:-+-+ . . . +-=-u1 uz Un ~ 1 - 1 + ~ 2 - -21,- 1+Un1 G xz X0a~-=~ +-oalaa * - a + (-1)" aoala2. . .a,1 aoa: alx an-15=- ~ -+ al-x+ %-x+ ' ' ' +an-%

FIGURE.1. y=xn.1 1fn-0, -9 -1 1, 2, 5.5 2

3.11. Use and Extension of the TablesExamPlel* Compute X1* and x4' for X=29using Table 3.1.

x 1 L O.Xl0=(1.45071 4598. 101s)(4.207072333. 1014)=6.10326 1248 lon

a?= (z")'/x~(1.25184 008*1W)*/29=5.40388 2547.low

Example2. Computea1 for x=9.19826.(9.19826)"'= (919.826/100)1'4=919.826)1'4/108

Linear interpolation in T able 3.1 gives(919.826)'"m5.507144.N=9 19.826,By Newton's method for fourth rootg with

I [5.507144)a19'826 +a(5.507144)]=5.50714 3845Repetition yields the same result.

~" =5.507143845/101= 1.74151 1796,Thus,

~-~" =~* / ~=.189335683.3.12. Computing Techniques

Example 3. Solve the quadratic equation22- 18.224- 056 given the coefficientsaa 18.2f 1,


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20 ELEMENTARY ANALYTICAL METHODS

~ ~~

QI Qz P P22.003 4 520 -2.550 . 172--

.056f 001. From 3.8.1 the solution isx= + ( l S. 2f 18.~)2-4(.058)]+)=$( 8 . 2 f [331.016]*)=$(18.2f 18.1939)

=18.g&9, .003T he smaller root may be obtained more accuratelyfrom* .05fi/ 18.=9= .0031f OOO .

Example4. Compute (-3 + 0076i)l.From 3.7.26, (-3+.0076i)*=u+iv where

Y(Q1)--011

y T - -2 3u=--,v v+) 9 T = (x2+yZ)t

91 Q2 P P22.00420 2152 4. 51683 7410 -2. 55283 358 . 17530 8659

ThusT = [ -a)*+ (.0076) ]4= (9 .OOOO5776) = 3 OOOOO9627

!/(PI)

A .00000 0011

v = r . O O O O O - ,1.73205 21962 1u=Y= .0076 -.00219 3929262v 2(1.73205 2196)-

We note that the principal square root has beencomputed.

Example 5. Solve the cubic equation2- 18.12To use Newton s method we first form the- 4.8=0.table of f( z) = 2- 18.12-34.8

2 f (44 -43.25 - .36 72.67 181.5We obtain by linear inverse interpolation:

0- (- .3) =5.004.x0=5+72.6- (- .3)Using Newton smethod, (x)= 3x2- 18.1 we get

21=s-f (XO> f (20)9936) =5.00526,57.0200485.004-

Repetition yields x1=5.00526 5097. Dividingf(x) by 2-5.00526 5097 gives xa+5.00526 509724-6.95267 869 the zerosof which are -2.50263 2549f 33036 800i.

Ixample 6. Solve the quartic equation2 -2.37752 4922~ + 6.07350?41x2- 1.17938 023~+ 9.05265 259=0.(z~+pl z+pl ) z~+pzZ+pz)

Resolution Into Quadratic Factorsby Inverse InterpolationStarting with t,hetriiil vdue ql= 1 we computesuccessively

1 5.383 032-2.023

We seek that value of p1for which y(pl)=O.I nverse interpolation in y(pl) gives y(pl)=O forpI ~2.003. T hen,

Inverse interpolation between pl=2.2 Ltnd pl=2.003 gives ql=2.0041, nnd thus,

Inverse interpolution gives q,= 2.00420 2152, nnd we get finally,

*Seepage 11.


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ELEMENTARY ANALYTICAL METHODS 21Double Precision Multiplication and Division on aDesk CalculatorExample7. M ultiplyM =20243 97459 71664 32102by m=69732 82428 43662 95023 on a 10X lOX 20desk calculating machine.Let M 0=20243 97459, & f1=71664 32102, mo=69732 82428, m1=43662 95023. Then Mm=M omlOzo(Moml+MI%)10" +M m .(1) M ultiply M 1m1=31290 75681 96300 28346and record the digits 96300 28346 appearing inpositions 1 to 10 of the product dial.(2) T ransfer the digits 31290 75681 from posi-tions 11 to 20 of the product dial to positions 1 to10 of the product dial.(3) M ultiply cumulativelyM l%+M oml +3129075681=58812 67160 12663 25894 and record thedigits 12663 25894 in positions 1 to 10.(4) T ransfer the digits 58812 67160 from posi-tions 11 to 20 to positions 1 to 10.(5) M ultiply cumulatively M 0m+ 58812 67160=14116 69523 40138 17612. T he results as ob-tained are shown below, 963002834612663 258941411669523401381761214116695234013817612 12663258949630028346

If the product M m is wanted to 20 digits, onlythe result obtained in step 5 need be recorded.Further, if the allowable error in the 20th place isa unit, the operation M lml may be omitted.When either of the factors M or m contains lessthan 20 digits it is convenient to position thenumbers as if they both had 20 digits. Thismultiplication process may be extended to anyhigher accuracy desired.Example 8. Divide N = 14116 69523 40138 17612by d= 20243 97459 71664 32102.M ethod (1 )--linear interpolation.

N / 20243 97459-10" = .69732 82430'90519 39054N / 20243 97460.1010=69732 82427 46057 26941Dif erence=314462 12113.Difference X .71664 32102=24685 644028.10-*Quotient=

(note this is an 11x10 multiplication).(69732 82430 90519 39054 -246856 44028).

= 69732 82428 43662 95028There is an error of 3 uriita in the 20th place dueto neglect of the contribution from second mer -enwe.

M ethod @)-If N and d axe numbers each notmore than 19 digits let N=Nl+No lOg ,d=dl+dolo9where Noand do contain 10 digits and Nland d, not more than 9 digits. T hen

HereN =14116 69523 40138 1761,d=20243 97459 71664 3210No=141 16 69523, & =20243 97459,d1=71664 3210(1) N & =10116 63378 42188 8830 (productdid).(2) (N & )& ,=49973 55504 (quotient dial).(3) N - (N&/&=14116 69522 90164 62106(4) [N - (N dI)/ do]/ dJ 09=69732 8242 8= h t 10(product dial).

digits of quotient in quotient dial. Remainder= ~= 08839 1654, in positions 1 to 10 of productdial.(5) r/ (& 1O9)=.43662 502.10-10=next9digitsofquotient. N / d= .69732 82428 43662 9502. Thismethod may be modified t~give the quotient of20 digit numbers. M ethod (1) may be extendedto quotients of numbers containing more than 20digits by employing higher order interpolation.Example 9. Sum the seriea S = l - + + + - t

+ . . . to 5D using the Euler transform.T he s umof the h t terms is .634524 to 6D .If u,,=l/ n we getn U n A un A%,, Aaun A%,,9 . l l l l l l -1111110 .100000 202011 .090909 1515 15612 .083333 116613 .076923

- 091 - 05-7576 -349-6410

From 3.6.27 we then obtain-111111 (- .011111) .00202022 +28

- 24 +S=.634524+-- 2 (- .000505) .000156= 634524+.055556+.002778+ 000253= 693148(S=h =.6931472 to7D).

+oooO32+.000005


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22 ELEMENTARY ANALYT ICAL METHOD8Example 10. Evaluate the integrallmyxr-- to 4D using the Euler transform.-2

Evaluating t,he integrals in the last sum bynumerical integration we getk

1.85194.43379.25661. 18260 A A2 A s A 4. 14180 -2587. 11593 799- 788 -321.09805 478 153- 310 - 68.08495 310- 000.07495

T he sum to k=3 is 1.49216. Applying theEuler transform to the remainder we obtain1 1 1- (.14180)--zy ( -.02587)+~ (.00799)2

1 1-- - .00321)+5 .00153)24=.07090+.00647+.00100+.00020= 07862 + OW05

We obtain the value of the integral aa 1.57018 ascompared with 1.57080.Example 11. Sum the serieszl -'=x using

the Euler-M aclaurin summation formula.From 3.6.28we have for n= = ,fD

1720- jr- . . .

where j(k)=(k+lO)-'. T hus,k-'=1.54976 7731+.1

-.005+.00016 6667-.OOOOO 0333=1.644934085,

faa compared with x=1.64493 4067.Example 12. Compute

x 2 4 2 9 2arctanx-- ---1+ 34- 54- 74- * * '

0 1K l= l 1 0 111 / = I *L: l=I 1 lIJ=l::4.2 0A * .6 .2 1 1 / = I 3.032 14=.1973963.04 1 .16 15.36 BS[,]=I 3 032 .6 1 1 7 I=/ 21.440 I ~= . I w ~MB, 15.36 3.04 .36 108.6144 B4

Nota that in carryingout the recurrence methodfor computing continued fractions the numeratorsA,, and the denominators B,, must be used asoriginally computed. T he numerators and de-nominators obtained by reducing A,,/B. to lowerterms must not beuaed.


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EL EM EN T A RY A N A L Y T IC A L M E T H O D SReferences

23

Testa[3.1] R . A. Buckingham, Numerical methods (P itmanP ublishing Corp., New Y ork, N .Y ., 1957).[3.2] T . Fort, F inite differences (Clarendon Press, Oxford,England, 1948).[3.3] L . Fox, T he use and construction of mathematicaltables, M athematical T ables, vol. 1, NationalPhysical L aboratory (Her M ajesty s StationeryOffice, London, England, 1956).[3.4] G. H . H ardy, A course of pure mathematics, 9thed. (Cambridge Univ. Press, Cambridge, England,and T he M acmillan Co., New Y ork, N.Y ., 1947).[3.5] D. R . Hartree, Numerical analysis (ClarendonPress, Oxford, England, 1952).13.61 F . B. H ildebrand, I ntroduction to numerical analysis(M cGraw-Hill Book Co., I nc., New York, N.Y .,1956).[3.7] A. S. Householder, P rinciples of numerical analysis

(M cGraw-Hill Book Co., I nc., New York, N.Y .,1953).[3.8] L. V. K antorowitach and V. I . K rylow, Naherungs-methoden der H oheren A nalysis (V EB DeutscherVerlag der Wissenschaften, Berlin, Germany,1956; translated from Russian, M oscow, U.S.S.R .,1952).[3.9] K. K nopp, Theory and application of infinite series(Blackie and Son, Ltd., London, England, 1951).[3.10] Z. K opal, Numerical analysis (J ohn Wiley & Sons,I nc., New York, N .Y ., 1955).[3.11 G. K owalewski, I nterpolation und geniiherte Quad-ratur (B. G. Teubner, Leipaig, Germany, 1932).[3.12] K. S. K unz, Numerical analysis (M cGraw-HillBook Co., nc., N ew Y ork, N .Y ., 1957).[3.13] C. Lancaos, Applied analysis (P rentice-Hall, Inc.,Englewood Cliffs, N.J ., 1956).[3.14] I . M . Longman, Note on a method for computinginfinite integrals of oscillatory functions, P roc.Cambridge P hilos. SOC.62, 764 (1958).[3.15] S. E. M ikeladae, Numerical method8 of mathe-matical analysis (Russian) (Cos. I zdat. T ehn.-Teor. L it., M oscow, US.S.R, 1953).[3.16] W. E. M ilne, Numerical calculus (P rinceton Univ.Press, P rinceton, N.J ., 1949).[3.17] L. M . M ilne-Thomson, The calculus of finite difler-ences (M acmillan and Co., Ltd., London, England,1951).

[3.18] H . M ineur, Techniques de calcul numhrique(L ibrairie P olytechnique Ch. Beranger, Paris,France, 1952).[3.19] National Physical Laboratory, M odern computingmethods, Notea on A pplied Science No. 16 (H erM ajesty s Stationery Office, London, England,1957).13.201 J . B. Rosser, T ransformations to speed the con+vergence of series, J . R esearch NBS 46, 58-84(1951).[3.21] J . B. Scarborough, Numerical mathematical anal-ysis, 3d ed. (The J ohns Hopkins Press, Baltimore,M d.; Oxford Univ. P ress, London, England,1955).[3.22] J . F . Steffensen, Interpolation (Chelsea P ublishingCo., New Y ork, N .Y ., 1950).[3.23] H . 6. Wall, Analytic theory of continued fraction8(D. Van Nostrand Co., I nc., New Y ork, N .Y .,1948).t3.241 E. T . Whittaker and G. R obinson, The calculus ofobservations, 4th ed. (B lackie and Son, L td.,London, England, 1944).[3.25] R . Zurmiihl, P raktische M athematik (Springer-Verlag, Berlin, Germany, 1953).

M athematical T ables and Collectiono of Formulam[3.28] E. P. A dam, Smithsonian mathematical f o r m uand tables of elliptic functions, 3d reprint (TheSmithaonian Institution, Washington, D.C ., 1957).[3.27] L. J . Comrie, Barlow s tables of squares, cubes,

square roota, cube roota and reciprocals of allintegers up to 12,600 (Chemical Publishing Co.,I nc., N ew York, N.Y., 1954).[3.28] H . B. Dwight, T ablesof integrals and other mathe-matical data, 3d ed. (T he M acmillan CQ., NewY ork, N.Y ., 1957).(3.291 Gt. Britain H .M . Nautical A lmanac Office, .I nter-polation and allied tables (Her Majesty sSta-tionery Office, London, England, 1950).(3.301 B. 0. Peirce, A short table of integrals, 4th ed.(G inn and Co., Boston, M ess., 1950).[3.311 G.Schulz, Formelsarnmlung xur praktischen M athe-matik (de Gruyter and Co., Berlin, Germany,1945).

chapter 3 elementary analytical methods

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