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Tabela De Integrais
tabela de integrais
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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(&) = 79264 2.30258 190.. 12707 6512. 9852.. 00128 1468.. 1.77245 2.67893 85347 07748.. . 1.25 r(i) 3.62560 99082 21908.. . 1-26 1 radian 1.27 1” = ~/180 radians . = = .. . 57.29577 0.01745 .. 7232. . 69139 5245.. .. = Eukr's co%stu~t [see 1.201 . 38509 05516 II’(&) = 180°/7r . 02729 ~ZLYLC~~OTZ [sec pages 1.24 = . 50929 94045 68401 7991.. 01532 86060 F is the gummu = . 99789 6964.. 24179 90197 1.64872 where = .. 8167.. .O 95130 8232.. 32925 . 101-102). 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 = $nb?- cet c 4.10 Perimeter = POLYGON inbz- FORMULAS OF n SIDES EACH CJf 1ENGTH b COS(AL) sin (~4%) = nb 7,’ 0.’ 0 Fig. 4-5 CIRÇLE OF RADIUS 4.11 Area 4.12 Perimeter r = & = 277r Fig. 4-6 SEClOR 4.13 4.14 Area = &r% OF CIRCLE OF RAD+US Y [e in radians] T Arc length s = ~6 A 8 0 T Fig. 4-7 RADIUS 4.15 OF C1RCJ.E INSCRWED r= where &$.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 4.16 R= 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 =. $ab r 2 . ABC r = e -&2dw a n a e r to 4 c +n E5 p u g e ar p m e b F r 4e l i -r o e g a 4 gl 1 a ) tn + h AOC @ T b Fig. 4-14 - f 1i GEOMETRIC 8 RECTANGULAR 4.26 Volume = 4.27 Surface area PARALLELEPIPED FORMULAS OF LENGTH u, HEIGHT r?, WIDTH c ubc Z(ab + CLC + bc) = a Fig. 4-15 PARALLELEPIPED 4.28 Volume = Ah = OF CROSS-SECTIONAL AREA A AND HEIGHT h abcsine Fig. 4-16 SPHERE 4.29 Volume = OF RADIUS ,r + 1 ---x ,------- 4.30 Surface area = 4wz @ Fig. 4-17 RIGHT 4.31 Volume 4.32 Lateral = CIRCULAR CYLINDER OF RADIUS T AND HEIGHT h 77&2 surface area = h 25dz Fig. 4-18 CIRCULAR 4.33 Volume 4.34 Lateral = m2h surface area CYLINDER = OF RADIUS r AND SLANT HEIGHT 2 ~41 sine = 2777-1 = 2wh z = 2wh csc e Fig. 4-19 . GEOMETRIC CYLINDER = OF CROSS-SECTIONAL 4.35 Volume 4.36 Lateral surface area Ah FORMULAS 9 A AND AREA SLANT HEIGHT I Alsine = = pZ = GPh -- ph csc t Note that formulas 4.31 to 4.34 are special cases. Fig. 4-20 RIGHT = CIRCULAR 4.37 Volume 4.38 Lateral surface area CONE OF RADIUS ,r AND HEIGHT h jîw2/z = 77rd77-D = ~-7-1 Fig. 4-21 PYRAMID 4.39 Volume = OF BASE AREA A AND HEIGHT h +Ah Fig. 4-22 SPHERICAL 4.40 Volume (shaded in figure) 4.41 Surface area = CAP = OF RADIUS ,r AND HEIGHT h &rIt2(3v - h) 2wh Fig. 4-23 FRUSTRUM = OF RIGHT 4.42 Volume 4.43 Lateral surface area +h(d CIRCULAR CONE OF RADII u,h AND HEIGHT h + ab + b2) = T(U + b) dF = n(a+b)l + (b - CL)~ Fig. 4-24 10 SPHEMCAt hiiWW 4.44 Area of triangle ABC = GEOMETRIC FORMULAS OF ANG%ES A,&C Ubl SPHERE OF RADIUS (A + B + C - z-)+ Fig. 4-25 TOW$ &F lNN8R 4.45 Volume 4.46 w Surface area = 7r2(b2- u2) 4.47 Volume = RADlU5 a AND OUTER RADIUS b &z-~(u+ b)(b - u)~ = $abc Fig. 4-27 T. 4.4a Volume = PARAWlO~D aF REVOllJTlON &bza Fig. 4-28 Y 5 TRtGOhiOAMTRiC D OE T FF R WNCTIONS F l I FU A R N G T ON Triangle ABC bas a right angle (9Oo) at C and sides of length u, b, c. angle A are defined as follows. sintz . of A 5 5 5 5 5 sin A 1= : = opposite hypotenuse i = adjacent hypotenuse cosine . of A = ~OSA 2= . of A = tanA 3= f = -~ . of A = of A tangent c 5.5 = secant cosecant . of A 4= k = adjacent t opposite = sec A = t = -~ = csc A 6= z = hypotenuse opposite E l O R RC functions G T N I T of B opposite adjacent A o cet The trigonometric I TX A c z A n g hypotenuse adjacent W OT Fig. 5-1 N M 3 HG E G A TE I R N9L Y H C E S0 E A H A I ’ 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. If it is described dockhse from The angle A described cozmtwcZockwLse from OX is considered pos&ve. 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, II, III and IV called the first, second, third and fourth quadrants 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 II 1 II 1 III IV III IV Y’ Y’ Fig. 5-3 Fig. 5-2 11 f TRIGONOMETRIC 12 FUNCTIONS For an angle A in any quadrant the trigonometric functions of A are defined as follows. 5.7 sin A = ylr 5.8 COSA = xl?. 5.9 tan A = ylx 5.10 cet A = xly 5.11 sec A = v-lx 5.12 csc A = riy RELAT!ONSHiP BETWEEN DEGREES AN0 RAnIANS N 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°/~ 5.14 10 = ~/180 radians = 1 r e 0 57.29577 95130 8232. . . o r B = 0.01745 32925 19943 29576 92.. .radians Fig. 5-4 REkATlONSHlPS 5.15 tanA = 5 5.16 &A ~II ~ 1 5.17 sec A = ~ 5.18 cscA = - tan A AMONG COSA sin A zz - 1 COS A TRtGONOMETRK 5.19 sine A + ~OS~A 5.20 sec2A - tane 5.21 csceA - cots A II III IV 1 A = 1 = 1 1 sin A SIaNS AND VARIATIONS 1 = FUNCTItB4S + 0 to 1 + 1 to 0 + 1 to 0 0 to -1 0 to -1 -1 to 0 OF TRl@ONOMETRK + 0 to m -mtoo + 0 to d -1 to 0 + 0 to 1 + CCto 0 oto-m + Ccto 0 - -- too oto-m FUNCTIONS + 1 to uz + m to 1 -cc to -1 + 1 to ca -1to-m + uz to 1 --COto-1 -1 to -- M TRIGONOMETRIC E Angle A in degrees 00 X F Angle A in radians A T A O RL FC R 1 IU O UT O S sec A csc A 0 1 0 w 1 cc ii/6 1 +ti 450 zl4 J-fi $fi 60° VI3 Jti 750 5~112 900 z.12 105O 7~112 *(fi+&) -&(&-Y% -(2+fi) -(2-&) 120° 2~13 *fi -* -fi -$fi 1350 3714 +fi -*fi 150° 5~16 4 -+ti #-fi) 2-fi &(&+fi) fi 1 0 fi) -&(G+ 0 -*fi -fi -(2-fi) -(2+fi) 180° ?r -1 1950 13~112 210° 7716 225O 5z-14 -Jfi 240° 4%J3 -# 255O 17~112 270° 3712 -1 285O 19?rll2 -&(&+fi) 3000 5ïrl3 -*fi 2 315O 7?rl4 -4fi *fi -1 330° 117rl6 *fi -+ti 345O 237112 360° 2r -$(fi-fi) -*(&+fi) 2-fi - 1 4 -*fi -i(fi- 2+fi 0 -(2+6) &(&+ -ti fi) 1 0 see pages 206-211 -(2 - fi) 0 ++ -fi \h -+fi 2 -(fi-fi) f -(&-fi) -2 -(&+?cz) -@-fi) &+fi -(2+6) T-J i -36 -(fi-fi) -1 -(fi-fi) 2 -1 f -fi Tm -*fi -ti -2 g -fi 0 *ca -(&+fi) i - &fi 2-6 Vz+V-c? -1 3 1 km *(&-fi) 6) angles ti -&(&-fi) 1 l 1 -4 -&&+&Q 6 fi-fi -2 2 + ti & 1 -(&+fi) Tm 0 fi-fi km -1 -1 TG ;G &+fi 0 N fi 2 2-& *CU fi) fi .+fi 2+& R 2 $fi 1 C N 3 &+fi fi-fi fi 1 @-fi) $(fi- 2+* *fi r1 i(fi+m other A cet A 300 involving FN A tan A rIIl2 tables GE COSA 0 llrll2 V sin A 15O 165O For V FUNCTIONS fi $fi fi-fi -$fi -fi -2 -(&+fi) 1 ?m and 212-215. f I TRIGONOMETRIC 5.89 y = cet-1% 5.90 y = FUNCTIONS 19 sec-l% 5.91 _--/ y = csc-lx Y I T --- , /A-- /’ / -77 -// , Fig. 5-14 Fig. 5-15 RElAilONSHfPS BETWEEN The following results hold for sides a, b, c and angles A, B, C. 5.92 ANGtGS any plane triangle ABC OY A PkAtM with TRlAF4GlG ’ A Law of Sines a -=Y=sin A 5.93 SIDES AND Fig. 5-16 1 b c sin B sin C C Law of Cosines /A cs = a2 + bz - Zab COS f C with similar relations involving the other sides and angles. 5.94 Law of Tangents tan $(A + B) a+b -a-b = tan i(A -B) with similar relations involving the other sides and angles. 5.95 sinA where s = &a + b + c) = :ds(s is the semiperimeter - a)(s - b)(s - c) of the triangle. B and C cari be obtained. See also formulas 4.5, page 5; 4.15 and 4.16, page 6. Spherieal triangle ABC is on the surface of in Fig. 5-18. Sides a, b, c [which are arcs of measured by their angles subtended at tenter 0 of are the angles opposite sides a, b, c respectively. results hold. 5.96 Law of Sines sin a -z-x_ sin A 5.97 sin b sin B a sphere as shown great circles] are the sphere. A, B, C Then the following sin c sin C Law of Cosines sinbsinccosA cosa = cosbcosc COSA = - COSB COSC + + Fig. 5-1’7 sinB sinccosa with similar results involving other sides and angles. Similar relations involving angles 2 T 0 L 5 o. w T a s 5 f 9 ri s = & S = + f e E g i c S( 8 n & + B a t( t & a= t( 4l op f r x F s ii 1e o mn s e I g U $ ) + n b ) sh a t ai v i r) G e aA ( N ) n a A n O n C T t u n u n h nl o d N s i l d e a g l e t r O ( rl v s s e i u i hr f e. o .s r) A i rh ra 0 FGR RtGHT o C it c na i b -va b i m + ose o sa t a i l i r u n h n l dd ld e g a e t r lr s 0 ( Se RlJlES a wn - B 0 h+ B + C NAPIER’S a t 1 w a et F 9 1+ h . S w 9 w 5 a i i . R 4o f e. o m g . meos 4 eu ANGLED rf , gh p e gor s i c gB e le, , u , . . o sa t a i l i , l SPHERICAL ha e p t lef 9n A l pv Atr r un h n l dd ld e ga e t r lr a TRIANGLES t rwe , d he i Z aet ih ei f t r3s o a i n r h rC r nc a C F S [ c A a t i o p 5 uq ot i h hn o t n p n da t a t - g a pu a fi hce ri a m s ia ea d a e c to a om c r of th y ac e rj n h w r p T s. o a h m i1 fp 5.102 T s o a h m i fp n ee i n S T x C c = 9i o ch a- n ee i n0 t O C s a s ( ba = t n0= m 9A t = o f . oe F 9 ri p aar c s a en io nr Fi a n l p A a npi B e m oc ayd 5 E 1 5 5e s n w s t a ir ndoc g . c p rt ti tsl a ps r p ea Te an cs laN u d f oeh t oa a a l a p y q d eo ht r rc u d fo eht t ooo O w c° h 0p, B A - e e a °l n s , ( a na C o C i C a C ( nO Oo O C ~ uts rl i rt 5 rhe o es p a er fb g e 2p t hd ead wl xi hr pv le r l aao s f p eh B dsp et Cg.l h eoc tn eu t a -ei e p sr . 0r t ri O ee e ni n s rl da sl i n ae ug j s l e ae ve r uip s r frg di ie ce a n co e: i AS =-r SC OaOs 2 ee et f eph dn d l e a l n = rOt b it - w - a ehi ngc t dta l m p t a p y q d eo1 ht r rt -- i O -a B A oa 1. s e n os a n O - mi 99 e Bn S i ) ug a )b SB n n .7 l e e 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 6.1 a+bi = c+di if and only if conjugates 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+ of each other. bc)i Note that the above operations are obtained by using the ordinary rules of algebra and replacing 9 by -1 wherever it occurs. 21 22 COMPLEX GRAPH NUMBERS 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 in Fig. 6-1 P represents the complex number -3 + 4i. A eomplex number cari also be interpreted as a wector p,----. y 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 COS6, y = r sine we have 6.6 x + iy = ~(COS 0+ called the poZar form the mocklus of the complex and t the amplitude i sin 0) number. L We often - X cal1 r = dm of x + iy. Fig. 6-2 tWJLltFltCATt43N [rl(cos 6.7 AND DtVlStON OF CWAPMX el + i sin ei)] [re(cos ez + i sin es)] V-~(COSe1 + i sin el) 6.8 ZZZ 2 rs(cos ee + i sin ez) If p is any real number, De Moivre’s [r(cos rrrs[cos 1bJ POLAR ilj 0” FtMM tel + e2) + i sin tel + e2)] [COS(el - e._J + i sin (el - .9&] DE f#OtVRtt’S 6.9 = NUMBRRS THEORRM theorem states e + i sin e)]p = that rp(cos pe + i sin pe) . RCWTS If p = l/n where k=O,l,2 integer, [r(cos e + i sin e)]l’n 6.10 where n is any positive OF CfMMWtX k is any ,..., integer. n-l. From this the = n nth NUtMB#RS 6.9 cari be written rl’n roots L e + 2k,, ~OSn of a complex + e + 2kH i sin ~ number n cari 1 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/aqE @-Q 7.3 (&y E rp4 7.4 u”=l, 7.5 a-p = l/ap 7.6 (ab)p = &‘bp 7.7 & 7.8 G 7.9 Gb a#0 z aIIn = pin =%Iî/% In ap, p is called the exponent, a is the base and ao is called the pth power of a. The function is called an exponentd function. If a~ = N where a # 0 or 1, then p = loga N is called the loga&hm N = ap is called t,he antdogatithm of p to the base a, written arkilogap. Example: Since The fumAion 3s = 9 we have y = ax of N to the base a. The number log3 9 = 2, antilog3 2 = 9. v = loga x is called a logarithmic jwzction. 7.10 logaMN = loga M + loga N 7.11 log,z ; = logG M - 7.12 loga Mp = p lO& M loga N Common logarithms and antilogarithms [also called Z?rigg.sian] are those in which the base a = 10. The common logarit,hm of N is denoted by logl,, N or briefly log N. For tables of common logarithms and antilogarithms, see pages 202-205. For illuskations using these tables see pages 194-196. 23 EXPONENTIAL 24 AND LOGARITHMIC NATURAL LOGARITHMS FUNCTIONS AND ANTILOGARITHMS Natural logarithms and antilogarithms [also called Napierian] are those in which the base a = e = 2.71828 18. . . [sec page 11. The natural logarithm of N is denoted by loge N or In N. For tables of natural logarithms see pages 224-225. For tables of natural antilogarithms [i.e. tables giving ex for values of z] see pages 226-227. For illustrations using these tables see pages 196 and 200. CHANGE OF BASE OF lO@ARlTHMS The relationship between logarithms of a number N to different bases a and b is given by 7.13 loga N = hb iv hb a - In particular, = ln N 7.14 loge N 7.15 logIO N = logN RElATlONSHlP = 2.30258 50929 94.. . logio N = 0.43429 44819 03.. . h& N BETWEEN EXPONBNTIAL ANO TRl@ONOMETRlC eie = 7.16 These are called Euler’s COS 0 + i sin 8, dent&es. e-iO = COS 13 - sin 6 Here i is the imaginary unit [see page 211. 7.17 sine 7.18 case = = eie- e-ie 2i eie+ e-ie 2 7.19 7.20 2 7.21 sec 0 = &O + e-ie 7.22 csc 6 = eie 7.23 i 2i eiCO+2k~l From this it is seen that @ has period 2G. - e-if3 = eie k = integer FUNCT#ONS ;; E POiAR T p XA FORfvl OF COMPLEX f 7 o h a co o n . 2 6 t o 6 NUMBERS .o hp r 2 (reiO)l/n E LOGARITHM 7.29 COMPLEX a. l OD ym i a tm e ( ffUMBERS e7n ra m 2 t 1r t (q-eio)Pzz q-P&mJ [ 7.2B OF GU EXPRESSE$3 AS AN oxl + i r c u b w m a WITH 7.27 PN or rpe N AN 25 E RC N EXPONENTNAL n re b [if lx 6 pi r e 2 st a ep . a mr 2 et s x o6 g 4 6 + i sin 0) = 9-ei0 x + iy = ~(COS OPERATIONS F fe L [~&O+Zk~~]l/n q f og M t = n u D FORM o 0eh uo ue o h l e e i il g a e vl v h s o NUMBER k e=e i k @n z ) t - ao r rl/neiCO+Zkr)/n OF A COMPLEX = l r n + iT + 2 IN POLAR e i DEIWWOPI OF HYPRRWLK 8.1 Hyperbolic sine of x = sinh x = 8.2 Hyperbolic cosine = coshx = 8.3 Hyperbolic tangent = tanhx = 8.4 Hyperbolic cotangent 8.5 Hyperbolic secant 8.6 Hyperbolic cosecant RELATWNSHIPS of x of x coth x of x = of x AMONG ez + e-= 2 ~~~~~~ 2 ez + eëz HYPERROLIC FUWTIONS = sinh x a coth z = 1 tanh x sech x = 1 cash x 8.10 cschx = 1 sinh x 8.11 coshsx - sinhzx = 1 8.12 sechzx + tanhzx = 1 8.13 cothzx - cschzx = 1 FUNCTIONS 2 = csch x = & tanhx 8.7 # - e-z ex + eCz = es _ e_~ = sech x = of x .:‘.C, FUNCTIONS cash x sinh x = OF NRGA’fWE ARGUMENTS 8.14 sinh (-x) = - sinh x 8.15 cash (-x) = cash x 8.16 tanh (-x) = - tanhx 8.17 csch (-x) = -cschx 8.18 sech(-x) = 8.19 coth (-x) = 26 sechx -~OUIS HYPERBOLIC AWMWM FUNCTIONS 27 FORMWAS 0.2Q sinh (x * y) = sinh x coshg 8.21 cash (x 2 g) = cash z cash y * sinh x sinh y 8.22 tanh(x*v) = tanhx f tanhg 12 tanhx tanhg 8.23 coth (x * y) = coth z coth y 2 1 coth y * coth x 8.24 sinh 2x = 2 ainh x cash x 8.25 cash 2x = coshz x + sinht x 8.26 tanh2x = 2 tanh x 1 + tanh2 x = * cash x sinh y 2 cosh2 x - 1 = 1 + 2 sinh2 z HAkF ABJGLR FORMULAS 8.27 sinht = 8.28 CoshE 2 = 8.29 tanh; = k Z sinh x cash x + 1 .4 [+ if x > 0, - if x < O] cash x + 1 -~ 2 cash x - 1 cash x + 1 ’ MUlTWlE [+ if x > 0, - if x < 0] ZZ cash x - 1 sinh x A!Wlfi WRMULAS 8.30 sinh 3x = 3 sinh x + 4 sinh3 x 8.31 cosh3x = 4 cosh3 x - 8.32 tanh3x = 3 tanh x + tanh3 x 1 + 3 tanhzx 8.33 sinh 4x = 8 sinh3 x cash x + 4 sinh x cash x 8.34 cash 4x = 8 coshd x - 8.35 tanh4x = 4 tanh x + 4 tanh3 x 1 + 6 tanh2 x + tanh4 x 3 cash x 8 cosh2 x -t- 1 2 H 8 YF P O PU HO E N FY& W P J R C E E B T f R R 8 . 3 s 6= &i c 2 - 4 na 8 . 3 c 7= 4 oc 2 + $ sa 8 . 3 s x 8= &i s 3 - 8 . 3 c x 9= &o c + 8 . 4 s 0= 8i - 4 c 2 n+ 4 ca 4x h as % 4 sh x h 8 . 4 c 1= #o + + c 2 s+ & ca 4x h as x 4 sh x h S D 8 U . AI F A hs zh x x hs zh x 2 sn i xx ihn nsh 2 cs o x ahs ssh K NFO x W R & DFF F O P Sl h h3 E x UR D R s 4+ s i = 2 si2 & n + y cn i $ hx - y) anh (x ) s hy x h x h kR U 8 . 4s - s 3i = 2 ci n& + y s an $ hx - Y) i sh (x ) n hy 8 . 4c + c 4o = 2 co is + y c as #(h - Y) a sh xxx ) s hy 8 . 4c - c 5o = 2 so $s + y s is $ (h - Y) i nh ( xx ) n hy 8 . 4s x s y 6i= * i n {- n c h c ho o s s h h ( 8 . 4c x c y 7 a= + a s {+ s c h c ho o s s h h ( s x 4c y i= + a n+ y {- s s x @ h- ) Y sl h i ) -i n } n h h 8 . E I t t OX H f n hw .o 8 s FP FY x e>e 0 ls I oa 1 x = u i c 8( = u !R UPT x < 0 u. l s t f a 9 . n o t t s x i n h c x a s h t x a n h c x o t h s x e c h c x s c h = uh s a c s ou h s p O N ‘ E NEE a e i wme x = 1h n o s i p b s fn i e x =1 xu h t e c h x h F OSC RR g r 8y o dn x = xwh c s T SB n o . rig h c HYPERBOLIC GRAPHS 8.49 y = sinh x OF HYPERBOkfC 8.50 29 FUNCltONS 8.51 y = coshx Fig. S-l 8.52 FUNCTIONS Fig. 8-2 y = coth x 8.53 /i y y = tanh x Fig. 8-3 8.54 y = sech x y = csch x Y \ X 1 7 10 X 0 -1 iNVERSE HYPERROLIC L X Fig. 8-6 Fig. 8-5 Fig. 8-4 0 FUNCTIONS If x = sinh g, then y = sinh-1 x is called the inverse hyperbolic sine of x. Similarly we define the The inverse hyperbolic functions are multiple-valued and. as in the other inverse hyperbolic functions. case of inverse trigonometric functions [sec page 171 we restrict ourselves to principal values for which they ean be considered as single-valued. The following list shows the principal values [unless otherwise indicated] of the inverse hyperbolic functions expressed in terms of logarithmic functions which are taken as real valued. 8.55 sinh-1 x = ln (x + m 8.56 cash-lx = ln(x+&Z-ï) 8.57 tanh-ix = 8.58 coth-ix = 8.59 sech-1 x 8.60 csch-1 x ) -m
(
0
For even n this can be summed 15.81
15.82
15.83
= -12
S-
1
m xdx
ez + 1
0
m
S o
xn-l
dx
eZ+l
=
$+$-$+
r(n)
For some positive
integer
S
=
“cdl:
$ -&+
S
co e-z2-e-*dx
0
15.86
&-
values
+coth;
X
=
&
of Bernoulli ..*
(
0
15.85
in terms
***
=
numbers
9 12
>
of n the series can be summed -
&
[see pages 108-109 and 114-1151.
[see pages 108-109 and 114-1151.
DEFINITE
m e-az
15.87
15.88
_
m e-~x
s
@-bs
_
e-bz
dx
x csc px
0
m e-“‘(lx;
s
‘OS
‘)
,jx
1
xm(ln x)” dx
i -
cot-l
=
tan-l%
a
(--l)%! (m + l)n+l
=
s 0
If
n#0,1,2,...
replace
S o
l - lnx 1+x
dx
=
-$
&
=
-$
’ In (1 + x) dx
S S S S S
2
0
15.94
tan-1
=
-
;
’ ln(l-x) x
0
(a2
+
dx
=
$
=
-?
m > -1,
1)
n = 0, 1,2, . . .
n! by r(n. + 1).
6
1
15.95
In
0
15.90
15.93
99
x set px
15.89
15.91
INTEGRALS
In x In (1 + x) dx
=
2-2ln2-12
In x In (l-x)
=
2 -
572
0
1
15.96
dx
c
0
15.97
- 772WC pn cot pa
O