Concrete mathematics : a foundation for computer science

More documents

Recommendations

Info

512 ANSWERS TO EXERCISES 5.7 Yes, because rs = (--1 ) "/( -r - 1)". We also have rqr + i)" = (2r)9;!% 5.8 f(k) = (k/n - 1)” is a polynomial of degree n whose leading coefficient is nn. By (5.40)~ the sum is n!/nn. When n is large, Stirling’s approximation says that this is approximately &/en. (This is quite different from (1 - l/e), which is what we get if we use the approximation (1 -k/n)” N eek, valid for fixed k as n + oo.) 5 . 9 E,(z)t = tksO t(tk + t)k-‘zk/k! = tk.Jk + l)k '(tz)k/k! = 1, (tz), by (5.60). 5’1o tk>O 2zk/(k + 2) = F(2,l; 3; z), since tk+l/tk = (k + 2)z/(k + 3). 5.11 The first is Besselian and the second is Gaussian: But not Imbesselian. z-~‘sinz = tka,(-l)kz2k/(2k+1)! = F(l;l,i;-z2/4); z- ’ arcsin 2 = tkZo z2k(;)k/(2k+ l)k! = F(;, ;; 5;~~). 5.12 (a) Yes, the term ratio is n. (b) No, the value should be 1 when k = 0; but (k + 1)" works, if n is an integer. (c) Yes, the term ratio is (k+l)(k+3)/(k+2). (d) No, the term ratio is 1 +l/(k+l)Hk; and Hk N Ink isn’t a rational function. (e) Yes, the term ratio is t(k+ 1) T(n - k) t(k) I T(n - k - .I) ’ (f) Not always; e.g., not when t(k) = 2k and T(k) = 1. (g) Yes, the term ratio can be written at(k+l)/t(k) + bt(k-t2)/t(k) + ct(k+3)/t(k) a+bt(k+l)/t(k) +ct(k+2)/t(k) ’ and t(k+m)/t(k) = (t(k+m)/t(k+m-1)) . . . (t(k+ 1)/t(k)) is arational function of k. 5.13 R, = n!n+‘/Pi = Qn/P,, = Qi/n!“+‘. 5.14 The first factor in (5.25) is (,‘i k,) when k < 1, and this is (-1 )Lpkpm x (r-r::). The sum for k 6 1 is the sum over all k, since m 3 0. (The condition n 3 0 isn’t really needed, although k must assume negative values if n < 0.) To go from (5.25) to (5.26), first replace s by -1 -n - q. 5.15 If n is odd, the sum is zero, since we can replace k by n-k. If n = 2m, the sum is (-1)“(3m)!/m!3, by (5.29) with a = b = c = m.
A ANSWERS TO EXERCISES 513 5.16 This is just (;!a)! (2b)! (2c)!/(a+ b)! (b+c)! (c+ a)! times (5.2g), if we write the summands in terms of factorials. 27/2) 5.17 ( = (;;)/22”; (2-/2) = (;;)/24”; so (2nn’/2) = 22n(2-/2). 5.18 (:;)(,"k",k)i33k. 5.19 Bl .t(-2) ’ := tkzO (kP’,“P’) (-l/(k ~ tk - 1)) (-.z)~, by (5.60), and this is tkaO (tt)(l/(tk- k+ 1))~~ = ‘BH,(z). 5.20 It equals F(-al, . ,-a,; -bl, . . . ,-b,; (-l)mfnz); see exercise 2.17. 5.21 lim,,,(n + m)c/nm = 1. 5.22 Multiplying and dividing instances of (5.83) gives (-l/2)! x!(x-l/2)! by (5.34) and (5.36). Also 1/(2x)! = lim 2n+2x n--c% 2n = &c ("n'") (n+xc1'2)n 2r/(n-i'2) 2n + 2x n--2X = lim n+cc ( 2n > ’ ( ) (2n) -2x . Hence, etc. The Gamma function equivalent, incidentally, is T(x) l-(x + ;, = r(2x) r(;)/22x- ’ 5.23 (-l)"ni, see (5.50). 5.24 This sum is (1:) F( ",~~"ll> = (fz), by (5.35) and (5.93). 5.25 This is equivalent to the easily proved identity a’ (a+llEemb< ( a - b ) - - = oP (b + II)” (b+l)k bk as well as to the operator formula a - b = (4 + a) - (4 + b). Similarly, we have (al - a21 F al,a2,a3, . . . . am bl, . . . , bn = alF al+l,a2,a3,...,am bl, . , b, 13 -wF( al,a2+l,a3,...,am
Page 1 and 2:
CONCRETE MATHEMATICS Dedicated to L
Page 3 and 4:
CONCRETE MATHEMATICS Ronald L. Grah
Page 5 and 6:
“A odience, level, and treatment
Page 7 and 8:
‘I a concrete life preserver thro
Page 9 and 10:
I’m unaccustomed to this face. De
Page 11 and 12:
n [ n-l 1 n {I m Prestressed concre
Page 13:
CONTENTS xiii 5.3 Tricks of the Tra
Page 16 and 17:
2 RECURRENT PROBLEMS It’s not imm
Page 18 and 19:
4 RECURRENT PROBLEMS Of course the
Page 20 and 21:
6 RECURRENT PROBLEMS through any of
Page 22 and 23:
8 RECURRENT PROBLEMS From these sma
Page 24 and 25:
10 RECURRENT PROBLEMS But what abou
Page 26 and 27:
12 RECURRENT PROBLEMS that is, in t
Page 28 and 29:
14 RECURRENT PROBLEMS by separating
Page 30 and 31:
16 RECURRENT PROBLEMS For example,
Page 32 and 33:
18 RECURRENT PROBLEMS Homework exer
Page 34 and 35:
20 RECURRENT PROBLEMS 21 Suppose th
Page 36 and 37:
22 SUMS The three-dots notation has
Page 38 and 39:
24 SUMS A slightly modified form of
Page 40 and 41:
26 SUMS where A(n), B(n), and C(n)
Page 42 and 43:
28 SUMS Let’s apply these ideas t
Page 44 and 45:
30 SUMS 2.3 MANIPULATION OF SUMS No
Page 46 and 47:
32 SUMS Typically we use rule (2.20
Page 48 and 49:
34 SUMS because the derivative of a
Page 50 and 51:
36 SUMS instead of summing over all
Page 52 and 53:
38 SUMS The second sum here is zero
Page 54 and 55:
40 SUMS The normal way to evaluate
Page 56 and 57:
42 SUMS First, as usual, we look at
Page 58 and 59:
44 SUMS order to get an equation fo
Page 60 and 61:
46 SUMS One way to use this fact is
Page 62 and 63:
48 SUMS definition where the factor
Page 64 and 65:
50 SUMS Let’s try to recap this;
Page 66 and 67:
52 SUMS we need an appropriate defi
Page 68 and 69:
54 SUMS This formula indicates why
Page 70 and 71:
56 SUMS u(x) = x and Av(x) = 2’;
Page 72 and 73:
58 SUMS The definition in the previ
Page 74 and 75:
60 SUMS In fact, our definition of
Page 76 and 77:
62 SUMS tCj,kiEF a. J, k < A for al
Page 78 and 79:
64 SUMS 18 Let 9%~ and Jz be the re
Page 80 and 81:
66 SUMS 34 35 36 Prove that if the
Page 82 and 83:
68 INTEGER FUNCTIONS below the line
Page 84 and 85:
70 INTEGER FUNCTIONS same inequalit
Page 86 and 87:
72 INTEGER FUNCTIONS An important s
Page 88 and 89:
74 INTEGER FUNCTIONS By the way, th
Page 90 and 91:
76 INTEGER FUNCTIONS (Previously K
Page 92 and 93:
78 INTEGER FUNCTIONS This derivatio
Page 94 and 95:
80 INTEGER FUNCTIONS We’ve got mo
Page 96 and 97:
82 INTEGER FUNCTIONS division, and
Page 98 and 99:
84 INTEGER FUNCTIONS more than one
Page 100 and 101:
86 INTEGER FUNCTIONS 3.5 FLOOR/CEIL
Page 102 and 103:
88 INTEGER FUNCTIONS For this purpo
Page 104 and 105:
90 INTEGER FUNCTIONS it comes out i
Page 106 and 107:
92 INTEGER FUNCTIONS To summarize,
Page 108 and 109:
94 INTEGER FUNCTIONS Also, as we gu
Page 110 and 111:
96 INTEGER FUNCTIONS Basics 10 Show
Page 112 and 113:
98 INTEGER FUNCTIONS 31 Prove or di
Page 114 and 115:
100 INTEGER FUNCTIONS 43 Find an in
Page 116 and 117:
4 Number Theory INTEGERS ARE CENTRA
Page 118 and 119:
104 NUMBER THEORY The left side can
Page 120 and 121:
106 NUMBER THEORY product of primes
Page 122 and 123:
108 NUMBER THEORY only finitely man
Page 124 and 125:
110 NUMBER THEORY But 2P - 1 isn’
Page 126 and 127:
112 NUMBER THEORY arrangements in a
Page 128 and 129:
114 NUMBER THEORY The binary repres
Page 130 and 131:
116 NUMBER THEORY A fraction m/n is
Page 132 and 133:
118 NUMBER THEORY A mediant fractio
Page 134 and 135:
120 NUMBER THEORY This representati
Page 136 and 137:
122 NUMBER THEORY This means that w
Page 138 and 139:
124 NUMBER THEORY Since x mod m dif
Page 140 and 141:
126 NUMBER THEORY than to say that
Page 142 and 143:
128 NUMBER THEORY require division,
Page 144 and 145:
130 NUMBER THEORY Now we must show
Page 146 and 147:
132 NUMBER THEORY And it’s possib
Page 148 and 149:
134 NUMBER THEORY For example, (~(1
Page 150 and 151:
136 NUMBER THEORY Conversely, if f(
Page 152 and 153:
138 NUMBER THEORY (We must add 1 to
Page 154 and 155:
140 NUMBER THEORY The problem of co
Page 156 and 157:
142 NUMBER THEORY n is odd, n4 and
Page 158 and 159:
144 NUMBER THEORY But the inner sum
Page 160 and 161:
146 NUMBER THEORY 22 The number 111
Page 162 and 163:
148 NUMBER THEORY 40 If the radix p
Page 164 and 165:
150 NUMBER THEORY 57 Let S(m,n) be
Page 166 and 167:
152 NUMBER THEORY 73 If the 0(n) +
Page 168 and 169:
154 BINOMIAL COEFFICIENTS For examp
Page 170 and 171:
156 BINOMIAL COEFFICIENTS And now t
Page 172 and 173:
158 BINOMIAL COEFFICIENTS property
Page 174 and 175:
160 BINOMIAL COEFFICIENTS (3, then
Page 176 and 177:
162 BINOMIAL COEFFICIENTS Binomial
Page 178 and 179:
164 BINOMIAL COEFFICIENTS Table 164
Page 180 and 181:
166 BINOMIAL COEFFICIENTS not rows.
Page 182 and 183:
168 BINOMIAL COEFFICIENTS Equation
Page 184 and 185:
170 BINOMIAL COEFFICIENTS that m =
Page 186 and 187:
172 BINOMIAL COEFFICIENTS ifweuse (
Page 188 and 189:
Page 190 and 191:
176 BINOMIAL COEFFICIENTS sum on th
Page 192 and 193:
178 BINOMIAL COEFFICIENTS some data
Page 194 and 195:
180 BINOMIAL COEFFICIENTS Problem 4
Page 196 and 197:
182 BINOMIAL COEFFICIENTS We’re l
Page 198 and 199:
184 BINOMIAL COEFFICIENTS We can’
Page 200 and 201:
186 BINOMIAL COEFFICIENTS 5.3 TRICK
Page 202 and 203:
188 BINOMIAL COEFFICIENTS if we app
Page 204 and 205:
190 BINOMIAL COEFFICIENTS The Newto
Page 206 and 207:
192 BINOMIAL COEFFICIENTS One examp
Page 208 and 209:
194 BINOMIAL COEFFICIENTS We can de
Page 210 and 211:
196 BINOMIAL COEFFICIENTS This obse
Page 212 and 213:
198 BINOMIAL COEFFICIENTS Generatin
Page 214 and 215:
200 BINOMIAL COEFFICIENTS can be pu
Page 216 and 217:
Page 218 and 219:
204 BINOMIAL COEFFICIENTS This hold
Page 220 and 221:
206 BINOMIAL COEFFICIENTS It’s ou
Page 222 and 223:
208 BINOMIAL COEFFICIENTS into this
Page 224 and 225:
210 BINOMIAL COEFFICIENTS integer.
Page 226 and 227:
212 BINOMIAL COEFFICIENTS into to b
Page 228 and 229:
214 BINOMIAL COEFFICIENTS Now sin(
Page 230 and 231:
216 BINOMIAL COEFFICIENTS But look
Page 232 and 233:
218 BINOMIAL COEFFICIENTS It’s al
Page 234 and 235:
220 BINOMIAL COEFFICIENTS A similar
Page 236 and 237:
222 BINOMIAL COEFFICIENTS One use o
Page 238 and 239:
224 BINOMIAL COEFFICIENTS Therefore
Page 240 and 241:
226 BINOMIAL COEFFICIENTS Now f(-b)
Page 242 and 243:
228 BINOMIAL COEFFICIENTS negative
Page 244 and 245:
230 BINOMIAL COEFFICIENTS valid for
Page 246 and 247:
232 BINOMIAL COEFFICIENTS 18 Find a
Page 248 and 249:
234 BINOMIAL COEFFICIENTS 37 Show t
Page 250 and 251:
236 BINOMIAL COEFFICIENTS 57 Use Go
Page 252 and 253:
238 BINOMIAL COEFFICIENTS 72 Prove
Page 254 and 255:
240 BINOMIAL COEFFICIENTS 86 Let al
Page 256 and 257:
242 BINOMIAL COEFFICIENTS Research
Page 258 and 259:
244 SPECIAL NUMBERS Table 244 Stirl
Page 260 and 261:
246 SPECIAL NUMBERS There are eleve
Page 262 and 263:
248 SPECIAL NUMBERS cycle arrangeme
Page 264 and 265:
250 SPECIAL NUMBERS Table 250 Basic
Page 266 and 267:
252 SPECIAL NUMBERS We can remember
Page 268 and 269:
254 SPECIAL NUMBERS Table 254 Euler
Page 270 and 271:
{'l 'I L\‘, 0 1 / 2 256 SPECIAL N
Page 272 and 273:
258 SPECIAL NUMBERS Table 258 Stirl
Page 274 and 275:
260 SPECIAL NUMBERS ’ ’ ’ (Th
Page 276 and 277:
262 SPECIAL NUMBERS we found that a
Page 278 and 279:
264 SPECIAL NUMBERS Rearranging, we
Page 280 and 281:
266 SPECIAL NUMBERS Thus we have th
Page 282 and 283:
268 SPECIAL NUMBERS (We have used t
Page 284 and 285:
270 SPECIAL NUMBERS Bernoulli numbe
Page 286 and 287:
272 SPECIAL NUMBERS trigonometric f
Page 288 and 289:
274 SPECIAL NUMBERS Recurrence (6.9
Page 290 and 291:
276 SPECIAL NUMBERS Gnethingwecando
Page 292 and 293:
278 SPECIAL NUMBERS cases are: a0 =
Page 294 and 295:
280 SPECIAL NUMBERS The process of
Page 296 and 297:
282 SPECIAL NUMBERS Fk < n < Fk+l;
Page 298 and 299:
284 SPECIAL NUMBERS We have now boi
Page 300 and 301:
286 SPECIAL NUMBERS Before we stop
Page 302 and 303:
288 SPECIAL NUMBERS The continuant
Page 304 and 305:
290 SPECIAL NUMBERS in four steps:
Page 306 and 307:
292 SPECIAL NUMBERS Thus we can con
Page 308 and 309:
294 SPECIAL NUMBERS numbers are 1,
Page 310 and 311:
296 SPECIAL NUMBERS 6 An explorer h
Page 312 and 313:
298 SPECIAL NUMBERS 24 Prove that t
Page 314 and 315:
300 SPECIAL NUMBERS 44 Prove the co
Page 316 and 317:
302 SPECIAL NUMBERS 61 Prove the id
Page 318 and 319:
304 SPECIAL NUMBERS 80 Show that co
Page 320 and 321:
7 Generating Functions THE MOST POW
Page 322 and 323:
308 GENERATING FUNCTIONS are added
Page 324 and 325:
310 GENERATING FUNCTIONS (The last
Page 326 and 327:
312 GENERATING FUNCTIONS Now we hav
Page 328 and 329:
314 GENERATING FUNCTIONS The first
Page 330 and 331:
316 GENERATING FUNCTIONS determine
Page 332 and 333:
318 GENERATING FUNCTIONS operation
Page 334 and 335:
320 GENERATING FUNCTIONS Table 320
Page 336 and 337:
322 GENERATING FUNCTIONS middle of
Page 338 and 339:
324 GENERATING FUNCTIONS Fortunatel
Page 340 and 341:
326 GENERATING FUNCTIONS Once we’
Page 342 and 343:
328 GENERATING FUNCTIONS Step 1 is
Page 344 and 345:
330 GENERATING FUNCTIONS nice prope
Page 346 and 347:
332 GENERATING FUNCTIONS Finally, s
Page 348 and 349:
334 GENERATING FUNCTIONS Step 3 was
Page 350 and 351:
336 GENERATING FUNCTIONS This is a
Page 352 and 353:
338 GENERATING FUNCTIONS Table 337
Page 354 and 355:
340 GENERATING FUNCTIONS F(z)' = i
Page 356 and 357:
342 GENERATING FUNCTIONS We can app
Page 358 and 359:
344 GENERATING FUNCTIONS Step 3 is
Page 360 and 361:
346 GENERATING FUNCTIONS Raney’s
Page 362 and 363:
348 GENERATING FUNCTIONS Notice tha
Page 364 and 365:
350 GENERATING FUNCTIONS Chapter 5
Page 366 and 367:
352 GENERATING FUNCTIONS Now let’
Page 368 and 369:
354 GENERATING FUNCTIONS function f
Page 370 and 371:
356 GENERATING FUNCTIONS 7.7 DIRICH
Page 372 and 373:
358 GENERATING FUNCTIONS 5 Find a g
Page 374 and 375:
360 GENERATING FUNCTIONS 20 A power
Page 376 and 377:
362 GENERATING FUNCTIONS 29 What is
Page 378 and 379:
364 GENERATING FUNCTIONS 43 The New
Page 380 and 381:
366 GENERATING FUNCTIONS 53 The seq
Page 382 and 383:
368 DISCRETE PROBABILITY total of 6
Page 384 and 385:
370 DISCRETE PROBABILITY about R if
Page 386 and 387:
372 DISCRETE PROBABILITY The mean o
Page 388 and 389:
374 DISCRETE PROBABILITY more sprea
Page 390 and 391:
376 DISCRETE PROBABILITY hence VS =
Page 392 and 393:
378 DISCRETE PROBABILITY And we can
Page 394 and 395:
380 DISCRETE PROBABILITY variance b
Page 396 and 397:
382 DISCRETE PROBABILITY (1 +z+...
Page 398 and 399:
384 DISCRETE PROBABILITY ~1 and ~2
Page 400 and 401:
386 DISCRETE PROBABILITY where Ug i
Page 402 and 403:
388 DISCRETE PROBABILITY Repeating
Page 404 and 405:
390 DISCRETE PROBABILITY There’s
Page 406 and 407:
392 DISCRETE PROBABILITY But the co
Page 408 and 409:
394 DISCRETE PROBABILITY In the spe
Page 410 and 411:
396 DISCRETE PROBABILITY Here 1 is
Page 412 and 413:
398 DISCRETE PROBABILITY For exampl
Page 414 and 415:
400 DISCRETE PROBABILITY Thus the m
Page 416 and 417:
402 DISCRETE PROBABILITY Let sj be
Page 418 and 419:
404 DISCRETE PROBABILITY Therefore
Page 420 and 421:
406 DISCRETE PROBABILITY The mean v
Page 422 and 423:
408 DISCRETE PROBABILITY Complicate
Page 424 and 425:
410 DISCRETE PROBABILITY all k and
Page 426 and 427:
412 DISCRETE PROBABILITY to be a pg
Page 428 and 429:
414 DISCRETE PROBABILITY 10 What’
Page 430 and 431:
416 DISCRETE PROBABILITY 28 What is
Page 432 and 433:
418 DISCRETE PROBABILITY 38 What is
Page 434 and 435:
420 DISCRETE PROBABIL1T.Y a Once he
Page 436 and 437:
422 DISCRETE PROBABILITY 50 Conside
Page 438 and 439:
424 DISCRETE PROBABILITY 58 Are the
Page 440 and 441:
426 ASYMPTOTICS The word asymptotic
Page 442 and 443:
428 ASYMPTOTIC3 How about the funct
Page 444 and 445:
430 ASYMPTOTICS N. G. de Bruijn beg
Page 446 and 447:
432 ASYMPTOTICS subtle consideratio
Page 448 and 449:
434 ASYMPTOTICS inefficient “beca
Page 450 and 451:
436 ASYMPTOTICS 9.3 0 MANIPULATION
Page 452 and 453:
138 ASYMPTOTICS Table 438 Asymptoti
Page 454 and 455:
440 ASYMPTOTICS Notice how the 0 te
Page 456 and 457:
442 ASYMPTOTICS Problem 3: The nth
Page 458 and 459:
444 ASYMPTOTICS Problem 4: A sum fr
Page 460 and 461:
446 ASYMPTOTICS Problem 5: An infin
Page 462 and 463:
448 ASYMPTOTICS We might as well do
Page 464 and 465:
450 ASYMPTOTIC3 a rough estimate an
Page 466 and 467:
452 ASYMPTOTICS This last estimate
Page 468 and 469:
454 ASYMPTOTICS Here’s another ex
Page 470 and 471:
456 ASYMPTOTICS On the left is a ty
Page 472 and 473:
458 ASYMPTOTICS (The Bernoulli poly
Page 474 and 475:
460 ASYMPTOTICS The graph of B,(x)
Page 476 and 477: 462 ASYMPTOTICS 9.6 FINAL SUM:MATIO
Page 478 and 479: 464 ASYMPTOTICS The integral JT B4(
Page 480 and 481: 466 ASYMPTOTIC3 where 0 < a,,,, < 1
Page 482 and 483: 468 ASYMPTOTICS In Chapter 5, equat
Page 484 and 485: 470 ASYMPTOTICS For example, we hav
Page 486 and 487: 472 ASYMPTOTICS The final task seem
Page 488 and 489: 474 ASYMPTOTICS This is our approxi
Page 490 and 491: 476 ASYMPTOTIC23 9 Prove (9.22) rig
Page 492 and 493: 478 ASYMPTOTICS 39 Evaluate xOsk
Page 494 and 495: 480 ASYMPTOTICS a What is the asymp
Page 496 and 497: 482 ASYMPTOTIC3 Research problems 6
Page 498 and 499: 484 ANSWERS TO EXERCISES Venn [294]
Page 500 and 501: 486 ANSWERS TO EXERCISES move the b
Page 502 and 503: 488 ANSWERS TO EXERCISES 2.2 This i
Page 504 and 505: 490 ANSWERS TO EXERCISES 2.24 Summi
Page 506 and 507: 492 ANSWERS TO EXERCISES 2.36 (a) B
Page 508 and 509: 494 ANSWERS TO EXERCISES 3.21 If 10
Page 510 and 511: 496 ANSWERS TO EXERCISES the circle
Page 512 and 513: 498 ANSWERS TO EXERCISES 3.40 Let L
Page 514 and 515: 500 ANSWERS TO EXERCISES 3.50 H. S.
Page 516 and 517: 502 ANSWERS TO EXERCISES 4.22 (b,
Page 518 and 519: 504 ANSWERS TO EXERCISES 4.40 Let f
Page 520 and 521: 506 ANSWERS TO EXERCISES 4.50 (a) I
Page 522 and 523: 508 ANSWERS TO EXERCISES and (k- 1)
Page 524 and 525: 510 ANSWERS TO EXERCISES In both ca
Page 528 and 529: 514 ANSWERS TO EXERCISES because al
Page 530 and 531: 516 ANSWERS TO EXERCISES 5.44 Cance
Page 532 and 533: 518 ANSWERS TO EXERCISES and this i
Page 534 and 535: 520 ANSWERS TO EXERCISES The infini
Page 536 and 537: 522 ANSWERS TO EXERCISES 5.82 Let ~
Page 538 and 539: 524 ANSWERS TO EXERCISES Since ~06j
Page 540 and 541: 526 ANSWERS TO EXERCISJES 5.94 This
Page 542 and 543: 528 ANSWERS TO EXERCISES 6.21 The h
Page 544 and 545: 530 ANSWERS TO EXERCISIES 6.40 If 6
Page 546 and 547: 532 ANSWERS TO EXERCISES solutions
Page 548 and 549: 534 ANSWERS TO EXERCISES 6.62 (a) A
Page 550 and 551: 536 ANSWERS TO EXERCISES 6.74 If p(
Page 552 and 553: 538 ANSWERS TO EXERCISIES One of th
Page 554 and 555: 540 ANSWERS TO EXERCISES 6.85 The p
Page 556 and 557: 542 ANSWERS TO EXERCISES Hence gzn+
Page 558 and 559: 544 ANSWERS TO EXERCISES 7.24 ntk,+
Page 560 and 561: 546 ANSWERS TO EXERCISES 7.40 The e
Page 562 and 563: 548 ANSWERS TO EXERCISES 1 + a. Hen
Page 564 and 565: 550 ANSWERS TO EXERCISES applied to
Page 566 and 567: 552 ANSWERS TO EXERCISES 8.13 (Solu
Page 568 and 569: 554 ANSWERS TO EXERCISES 8.24 (a) A
Page 570 and 571: 556 ANSWERS TO EXERCISES 8.32 By sy
Page 572 and 573: 558 ANSWERS TO EXERCISES 8.37 The n
Page 574 and 575: 560 ANSWERS TO EXERCISES probabilit
Page 576 and 577:
562 ANSWERS TO EXERCISES And since
Page 578 and 579:
564 ANSWERS TO EXERCISES 8.56 If m
Page 580 and 581:
566 ANSWERS TO EXERCISES We can now
Page 582 and 583:
568 ANSWERS TO EXERCISES 9.19 Hlo =
Page 584 and 585:
570 ANSWERS TO EXERCIS:ES 9.30 Let
Page 586 and 587:
572 ANSWERS TO EXERCISES 9.42 The h
Page 588 and 589:
574 ANSWERS TO EXERCISIES length of
Page 590 and 591:
576 ANSWERS TO EXERCISES 9.58 Let 0
Page 592 and 593:
B Bibliography HERE ARE THE WORKS c
Page 594 and 595:
580 BIBLIOGRAPHY 24' 25 26 27 28 29
Page 596 and 597:
582 BIBLIOGRAPHY 51 52 53 54 55 56
Page 598 and 599:
584 BIBLIOGRAPHY 78 79 80 81 82 83
Page 600 and 601:
586 BIBLIOGRAPHY 100 R. A. Fisher,
Page 602 and 603:
588 BIBLIOGRAPHY 128 Ronald L. Grah
Page 604 and 605:
590 BIBLIOGRAPHY 160 K. Inkeri, “
Page 606 and 607:
592 BIBLIOGRAPHY 188 E.E. Kummer,
Page 608 and 609:
594 BIBLIOGRAPHY 214 Z.A. Melzak, C
Page 610 and 611:
596 BIBLIOGRAPHY 243 George N. Rane
Page 612 and 613:
598 BIBLIOGRAPHY 271 A. D. Solov’
Page 614 and 615:
600 BIBLIOGRAPHY 299 Louis Weisner,
Page 616 and 617:
602 CREDITS FOR EXERCISES 1.1 1.2 1
Page 618 and 619:
604 CREDITS FOR EXERCISES 6.46 6.47
Page 620 and 621:
Index WHEN AN INDEX ENTRY refers to
Page 622 and 623:
608 INDEX Bois-Reymond, Paul David
Page 624 and 625:
610 INDEX Degenerate hypergeometric
Page 626 and 627:
612 INDEX Feder, Tom&s, 604. Feigen
Page 628 and 629:
614 INDEX Hall, Marshall, Jr., 588.
Page 630 and 631:
616 INDEX Lincoln, Abraham, 387. Li
Page 632 and 633:
618 INDEX Patashnik, Oren, iii, iv,
Page 634 and 635:
620 INDEX Running time, 411-412. Ru
Page 636 and 637:
622 INDEX Sylvester, James Joseph,
Page 638 and 639:
List of Tables Sums and differences
show all

Concrete mathematics : a foundation for computer science

Create successful ePaper yourself

Delete template?

Save as template?