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CHAPTER 12 APPENDIX A
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The following are some sums of series that arise in practice. By definition, 0! the range of convergence is indicated. m m(m 1) 1. a j 1 2 3 c m 2 j 1
l. Where the series is infinite,
2. a j 2
12 1 x 1 1 x) x
22 x2 2! x3 3! x2 2! x x
32 x3 3! x5 5! x4 4! x2 x2 2
c c x7 7! x6 6! x3 x3 3
1)(2m 6 all x
3. e x 4. sin x 5. cos x 6. 1 1
xj a j! j 0 c c c x4 4
` ( 1) jx2 j 1 a (2j 1)! j 0
all x all x 1 1 x 1
( 1) jx 2j a (2j)! j 0
uxu xj aj
7. ln (1
Euler s Formulas
8. eiu 9. cos u cos u eiu 2 i sin u, e
cos u eiu 2i e
i sin u
sin u
The Gamma Function
The gamma function, denoted by (n), is defined by
(n) A recurrence formula is given by
30 t (n
n 1e t dt
n (n) 0 can be obtained by the use of (2). n!
where (l) 1. An extension of the gamma function to n If n is a positive integer, then (n 1)
For this reason (n) is sometimes called the factorial function. An important property of the gamma function is that p ( p) (1 p) (4) sin pp
APPENDIX A
For p
1 , 2
(4) gives
For large values of n we have Stirling s asymptotic formula: (n 1) , !2pn nn e
where the sign , indicates that the ratio of the two sides approaches 1 as n S ` . In particular, if n is a large positive integer, a good approximation for n! is given by n! , !2pn nn e
1 2
The Beta Function
The beta function, denoted by B(m, n), is defined as
B(m, n) It is related to the gamma function by
30 u
m 1(1
u)n 1 du
0, n
B(m, n)
(m) (n) (m n)
Special Integrals
The following are some integrals which arise in probability and statistics.
10. 3 e 0
ax2 dx
1 p 2Aa m 2
1)>2
11. 3 xme 0
ax2 dx
2a(m
12. 3 e 0
ax2 cos bx dx
1 p e 2Aa a a2 b a2 b2
b2>4a
0, m a 0
13. 3 e 0
ax cos bx dx
a a 0, p
4ac)>4a
0 0 0 a 0
14. 3 e 0
ax sin bx dx
15. 3 x p 1e 0
ax dx
(p) ap
16. 3 e `
(ax2 bx c) dx
p (b2 e Aa 1 p (b2 e 2Aa
17. 3 e 0
(ax2 bx c) dx
4ac)>4a erfc
where erfc(u) 1
erf(u)
b 2!a
2 u e !p 30
x2 dx
2 ` e !p 3u
x2 dx
is called the complementary error function.
` cos vx 18. 3 2 dx a2 0x p>2
p e 2a
0, v
0 m 0, n 0
19. 3 sin2m 1u cos 2n 1u du 0
(m) (n) 2 (m n)
CHAPTER 12 CHAPTER B APPENDIX12
Ordinates y of the Standard Normal Curve at z
z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 0 .3989 .3970 .3910 .3814 .3683 .3521 .3332 .3123 .2897 .2661 .2420 .2179 .1942 .1714 .1497 .1295 .1109 .0940 .0790 .0656 .0540 .0440 .0355 .0283 .0224 .0175 .0136 .0104 .0079 .0060 .0044 .0033 .0024 .0017 .0012 .0009 .0006 .0004 .0003 .0002 1 .3989 .3965 .3902 .3802 .3668 .3503 .3312 .3101 .2874 .2637 .2396 .2155 .1919 .1691 .1476 .1276 .1092 .0925 .0775 .0644 .0529 .0431 .0347 .0277 .0219 .0171 .0132 .0101 .0077 .0058 .0043 .0032 .0023 .0017 .0012 .0008 .0006 .0004 .0003 .0002 2 .3989 .3961 .3894 .3790 .3653 .3485 .3292 .3079 .2850 .2613 .2371 .2131 .1895 .1669 .1456 .1257 .1074 .0909 .0761 .0632 .0519 .0422 .0339 .0270 .0213 .0167 .0129 .0099 .0075 .0056 .0042 .0031 .0022 .0016 .0012 .0008 .0006 .0004 .0003 .0002 3 .3988 .3956 .3885 .3778 .3637 .3467 .3271 .3056 .2827 .2589 .2347 .2107 .1872 .1647 .1435 .1238 .1057 .0893 .0748 .0620 .0508 .0413 .0332 .0264 .0208 .0163 .0126 .0096 .0073 .0055 .0040 .0030 .0022 .0016 .0011 .0008 .0005 .0004 .0003 .0002 4 .3986 .3951 .3876 .3765 .3621 .3448 .3251 .3034 .2803 .2565 .2323 .2083 .1849 .1626 .1415 .1219 .1040 .0878 .0734 .0608 .0498 .0404 .0325 .0258 .0203 .0158 .0122 .0093 .0071 .0053 .0039 .0029 .0021 .0015 .0011 .0008 .0005 .0004 .0003 .0002 5 .3984 .3945 .3867 .3752 .3605 .3429 .3230 .3011 .2780 .2541 .2299 .2059 .1826 .1604 .1394 .1200 .1023 .0863 .0721 .0596 .0488 .0396 .0317 .0252 .0198 .0154 .0119 .0091 .0069 .0051 .0038 .0028 .0020 .0015 .0010 .0007 .0005 .0004 .0002 .0002 6 .3982 .3939 .3857 .3739 .3589 .3410 .3209 .2989 .2756 .2516 .2275 .2036 .1804 .1582 .1374 .1182 .1006 .0848 .0707 .0584 .0478 .0387 .0310 .0246 .0194 .0151 .0116 .0088 .0067 .0050 .0037 .0027 .0020 .0014 .0010 .0007 .0005 .0003 .0002 .0002 7 .3980 .3932 .3847 .3725 .3572 .3391 .3187 .2966 .2732 .2492 .2251 .2012 .1781 .1561 .1354 .1163 .0989 .0833 .0694 .0573 .0468 .0379 .0303 .0241 .0189 .0147 .0113 .0086 .0065 .0048 .0036 .0026 .0019 .0014 .0010 .0007 .0005 .0003 .0002 .0002 8 .3977 .3925 .3836 .3712 .3555 .3372 .3166 .2943 .2709 .2468 .2227 .1989 .1758 .1539 .1334 .1145 .0973 .0818 .0681 .0562 .0459 .0371 .0297 .0235 .0184 .0143 .0110 .0084 .0063 .0047 .0035 .0025 .0018 .0013 .0009 .0007 .0005 .0003 .0002 .0001 9 .3973 .3918 .3825 .3697 .3538 .3352 .3144 .2920 .2685 .2444 .2203 .1965 .1736 .1518 .1315 .1127 .0957 .0804 .0669 .0551 .0449 .0363 .0290 .0229 .0180 .0139 .0107 .0081 .0061 .0046 .0034 .0025 .0018 .0013 .0009 .0006 .0004 .0003 .0002 .0001
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