blue book 27
TRANSCRIPT
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MATHEMATICS TABLES
Department of Applied MathematicsNaval Postgraduate School
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TABLE OF CONTENTS
Derivatives and Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Integrals of Elementary Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Integrals Involving au + b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Integrals Involving u2 a2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Integrals Involving
u2 a2, a > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Integrals Involving
a2 u2, a > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Integrals Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Integrals Involving Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Miscellaneous Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Wallis Formulas . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Gamma Function . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Probability and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Discrete Probability Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12Standard Normal CDF and Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12Continuous Probability Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Bessel Functions . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Trigonometric Functions in a Right Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18Trigonometric Functions of Special Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18Complex Exponential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18Relations among the Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
Sums and Multiples of Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19Miscellaneous Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20Side-Angle Relations in Plane Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22First-Order Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23Linear Second-Order Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . 24Vector Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Vector Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Fundamentals of Signals and Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
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Version 2.7, July 2010
Initial typesetting: Carroll WildeGraphics: David Canright
Editing: Elle Zimmerman, David Canright
Cover: in spherical coordinates (, , ), shell surface is (, ) = ek sin ,
where k = 1 ln G and G =1+
5
2 is the Golden Ratio.
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1
FOREWORD
This booklet provides convenient access to formulas and other data that arefrequently used in mathematics courses. If a more comprehensive referenceis needed, see, for example, the STANDARD MATHEMATICAL TABLESpublished by the Chemical Rubber Company, Cleveland, Ohio.
DIFFERENTIALS AND DERIVATIVES
A. Letters u and v denote independent variables or functions of an indepen-dent variable; letters a and n denote constants.
B. To obtain a derivative, divide both members of the given formula for thedifferential by du or by dx.
1. d(a) = 0 2. d(au) = a du
3. d(u + v) = du + dv 4. d(uv) = u dv + v du
5. du
v
=
v du u dvv2
6. d(un) = n un1 du
7. d(eu) = eudu 8. d(au) = au ln a du
9. d(ln u) =du
u10. d(loga u) =
du
u ln a11. d(sin u) = cos u du 12. d(cos u) = sin u du13. d(tan u) = sec2 u du 14. d(cot u) = csc2 u du
15. d(sec u) = sec u tan u du 16. d(csc u) = csc u cot u du17. d(arcsin u) =
du1 u2 18. d(arccos u) =
du1 u2
19. d(arctan u) =du
1 + u220. d(sinh u) = cosh u du
21. d(cosh u) = sinh u du 22. d(arcsinh u) =du
u2 + 1
23. d(arccosh u) =du
u2 1 , u > 1
24. (Differentiation of Integrals) If f is continuous, then
d(u
a
f(t) dt) = f(u) du.
25. (Chain Rule) If y = f(u) and u = g(x), then
dy
dx=
dy
du
du
dx.
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INTEGRALS
A. Letters u and v denote functions of an independent variable such as x;letters a, b, m, and n denote constants.
B. Generally, each formula gives only one antiderivative. To find an expres-sion for all antiderivatives, add a constant of integration.
ELEMENTARY FORMS
1.
a du = a u
2.
a f(u) du = a
f(u) du
3.
(f(u) + g(u)) du =
f(u) du +
g(u) du
4.
u dv = u v
v du (Integration by parts)
5.
un du =
un+1
n + 1, n = 1
6.
du
u= ln |u|
7.
eudu = eu
8.
audu = auln a
, a > 0
9.
cos u du = sin u
10.
sin u du = cos u
11.
sec2 u du = tan u
12.
csc2 u du = cot u
13. sec u tan u du = sec u14.
csc u cot u du = csc u
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15.
tan u du = ln | cos u|
16.
cot u du = ln | sin u|
17.
sec u du = ln | sec u + tan u|
18.
csc u du = ln | csc u cot u|
19.
du
u2 + a2=
1
aarctan
u
a
20. du
a2
u2
= arcsinu
a
21.
du
u
u2 a2 =1
aarcsec
u
a
INTEGRALS INVOLVING au + b
22.
(au + b)n du =
1
a
(au + b)n+1
n + 1, n = 1
23.
du
au + b=
1
aln |au + b|
24. u du
au + b
=u
a
b
a2
ln
|au + b
|25.
u du
(au + b)2=
b
a2(au + b)+
1
a2ln |au + b|
26.
du
u(au + b)=
1
bln
uau + b
27.
u
au + b du =2(3au 2b)
15a2(au + b)
32
28.
u duau + b
=2(au 2b)
3a2
au + b
29.
u2
au + b du =2(8b2 12abu + 15a2u2)
105a3(au + b)
32
30. u2 duau + b = 2(8b
2
4abu + 3a2u2)
15a3 au + b
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INTEGRALS INVOLVING TRIGONOMETRIC FUNCTIONS
58.
sin2 audu =
u
2 sin2au
4a
59.
cos2 audu =
u
2+
sin2au
4a
60.
sin3 audu =
cos3 au
3a cos au
a
61.
cos3 audu =
sin au
a sin
3 au
3a
62. sinn audu = sin
n1 au cos auna
+n 1
n sinn2 au du,
63.
cosn audu =
cosn1 au sin auna
+n 1
n
cosn2 au du,
64.
sin2 au cos2 audu =
u
8 sin4au
32a
65.
du
sin2 au= 1
acot au
66.
du
cos2 au=
1
atan au
67.
tan2 audu =
1
atan au u
68. cot2 audu = 1
a
cot au
u
69.
sec3 audu =
1
2asec au tan au +
1
2aln | sec au + tan au|
70.
csc3 audu = 1
2acsc au cot au +
1
2aln | csc au cot au|
71.
u sin audu =
1
a2sin au u
acos au
72.
u cos audu =
1
a2cos au +
u
asin au
73.
u2 sin audu =
2u
a2sin au a
2u2 2a3
cos au
74.
u2
cos audu =
2u
a2 cos au +
a2u2
2
a3 sin au
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INTEGRALS INVOLVING EXPONENTIAL FUNCTIONS
75.
ueau du =
eau
a2(au 1)
76.
u2eau du =
eau
a3(a2u2 2au + 2)
77.
uneau du =
uneau
a n
a
un1eau du, n > 1
78.
eau sin budu =
eau
a2 + b2(a sin bu b cos bu)
79.
eau cos budu =
eau
a2 + b2(a cos bu + b sin bu)
80. du
1 + eau = u1
a ln(1 + eau)
MISCELLANEOUS INTEGRALS
81.
arcsin audu = u arcsin au +
1 a2u2
a
82.
arccos audu = u arccos au
1 a2u2
a
83.
arctan audu = u arctan au 1
2aln(1 + a2u2)
84.
arccot audu = u arccot au +1
2a ln(1 + a2u2)
85.
ln audu = u ln au u
86.
un ln audu = un+1
ln au
n + 1 1
(n + 1)2
87.
0
ea2u2 du =
2a
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WALLIS FORMULAS
88.
2
0
sinm u du =
(m1)(m3)(3)(1)(m)(m2)(4)(2)
2 , m even;
(m1)(m3)(4)(2)(m)(m2)(5)(3) , m odd
89.
2
0
cosm u du =
2
0
sinm u du
90.
2
0
sinm u cosn u du =
(m1)(m3)(3)(1)(n1)(n3)(4)(2)(m+n)(m+n2)(3)(1) , m even, n odd;
(m
1)(m
3)
(4)(2)(n
1)(n
3)
(3)(1)
(m+n)(m+n2)(3)(1) , m odd, n even;(m1)(m3)(4)(2)(n1)(n3)(4)(2)
(m+n)(m+n2)(4)(2) , m odd, n odd;
(m1)(m3)(3)(1)(n1)(n3)(3)(1)(m+n)(m+n2)(4)(2)
2
, m even, n even
GAMMA FUNCTION
Definition :
(x) =
0
ettx1 dt, x > 0
Shift Property :
(x + 1) = x(x), x > 0
Definition :
(x) =(x + 1)
x, x < 0, x = 1,2, . . .
Special Values :
(1
2) =
(n) = (n 1)!, n = 1, 2, . . .
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PROBABILITY AND STATISTICS
Formulas for counting permutations and combinations:
(a) nPk =n!
(n k)! (b) nCk =
n
k
=
n!
k!(n k)!Let A and B be subsets (events) of the same sample space.
(a) P(A B) = P(A) + P(B) P(A B).(b) The conditional probability of B given A, P(B|A), is defined by
P(B|A) = P(A B)P(A)
, P(A) = 0.
(c) A and B are statistically independent if and only if
P(A B) = P(A)P(B).Let X be a discrete random variable with probability function f(x).
(a) P(X = x) = f(x), where x S range of X(b) E(X) =
S
xf(x) = = mean of X
(c) var(X) = E((X )2) = E(X2) 2 =S
x2f(x) 2 = 2
= variance of X
(d) =
var(X) = standard deviation of X.
Let X be a continuous random variable with and probability density f(x).
(a) P(a X b) =b
a
f(x) dx.
(b) E(X) =
S
xf(x) dx = = mean of X
(c) var(X) = E((X )2) = E(X2) 2 =S
x2f(x) dx 2 = 2
= variance of X
(d) =
var(X) = standard deviation of X.
Let X1, X2, . . . , X n be n statistically independent random variables, each hav-ing the same expected value, , and the same variance, 2. Let
Y =X1 + X2 + + Xn
n.
Then Y is a random variable for which E(Y) = and var(Y) = 2
n .
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PROBABILITY FUNCTIONS (Continuous Random Variables)
Uniform
f(x; a, b) =1
b a (a x b).
=a + b
2; 2 =
(b a)212
.
Exponential
f(x; b) = bebx (x 0, b > 0).
=
1
b ; 2
=
1
b2 .Normal
f(x; a, b) =1
b
2e
(xa)2
2b2 ( < x < ).
= a; 2 = b2.
FOURIER SERIES
The Fourier series expansion of a function f(t) is defined by:
f(t) = a0 +
n=1
(an cosnt
L + bn sinnt
L ),
where
a0 =1
2L
LL
f(t) dt
an =1
L
LL
f(t)cosnt
Ldt, n 1
bn =1
L L
Lf(t)sin
nt
Ldt, n 1
(Note: for the complex Fourier Series, see page 29)
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FOURIER SERIES (Continued)
Examples of Fourier Series
f(t) =
1, 0 < t < L;1, L < t < 0.
f(t)
t
1
L
4
sin
t
L+
1
3sin
3t
L+
1
5sin
5t
L+ )
f(t) =
0, L2 < |t| < L1, 0 < |t| < L2 .
f(t)
t
1
L
1
2+
2
cos
t
L 1
3cos
3t
L+
1
5cos
5t
L + )
f(t) = t, L < t < Lf(t)
t
L
L
2L
sin
t
L 1
2sin
2t
L+
1
3sin
3t
L + )
f(t) = |t|, L < t < Lf(t)
t
L
L
L2 4L2
cos tL + 132 cos 3tL + 152 cos 5tL + )
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SEPARATION OF VARIABLES
Eigenvalues and Eigenfunctions for the differential equation
+ = 0.
1. (0) = (L) = 0
n = (nL )
2, n = sinnL x, n = 1, 2, . . .
2. (0) = (L) = 0
n =
(n 12)L2
, n = cos(n 12 )Lx, n = 1, 2, . . .3. (0) = (L) = 0
n =
(n 12)L 2, n = sin(n 12)Lx, n = 1, 2, . . .4. (0) = (L) = 0
n = (nL )
2, n = cosnL
x, n = 0, 1, 2, . . .
5. (L) = (L), (L) = (L)0 = 0, 0 = 1,
n = (nL )
2, n = sinnL x and cos
nL x, n = 1, 2, . . .
Solution of inhomogeneous ordinary differential equations
1. un
+ k nun = hn(t)
un(t) = aneknt +t0
hn()ekn(t) d
2. un + 2n un = hn(t)
un(t) = an cos nt + bn sin nt +1n
t0
hn()sin n(t ) d
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BESSEL FUNCTIONS
1. The differential equation
x2y + xy + (2x2 n2)y = 0has solutions
y = yn(x) = c1Jn(x) + c2Yn(x),
where
Jn(t) =m=0
(1)mtn+2m2n+2mm!(n + m + 1)
.
2. General properties:
Jn(t) = (1)nJn(t); J0(0) = 1; Jn(0) = 0, n = 1, 2, 3, . . . ;for fixed n, Jn(t) = 0 has infinitely many solutions.
3. Identities:
(a) ddt [tnJn(t)] = tnJn1(t).
(b) ddt [tnJn(t)] = tnJn+1(t); J0(t) = J1(t)
(c) Jn(t) =12 [Jn1(t) Jn+1(t)]
= Jn1(t) nt Jn(t) = nt Jn(t) Jn+1(t)(d) Jn+1(t) =
2nt Jn(t) Jn1(t) (recurrence).
4. Orthogonality:
Solutions yn(0x), yn(1x), . . . , yn(ix), . . . of the differential eigensystemx2y + xy + (2x2 n2)y = 0A1y(x1) B1y(x1) = 0, A2y(x2) + B2y(x2) = 0
have the propertyx2x1
xyn(ix)yn(jx) dx = 0
for i = j, andx2x1
xy2n(ix) dx =(2ix
2 n2)y2n(ix) + x2ddxyn(ix)
222i
x2
x1
for i = j.
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5. Integration properties:
(a)
tJ1(t) dt = tJ(t) + C
(b)
tJ+1(t) dt = tJ(t) + C
(c)
tJ0(t) dt = tJ1(t) + C
(d)
t3J0(t) dt = (t
3 4t)J1(t) + 2t2J0(t) + C
(e)
t2J1(t) dt = 2tJ1(t) t2J0(t) + C
(f) t4J1(t) dt = (4t3 16t)J1(t) (t4 8t2)J0(t) + CLEGENDRE POLYNOMIALS
The differential equation
(1 x2)y 2xy + n(n + 1)y = 0has solution y = Pn(x), where
Pn(x) =1
2n
[n/2]m=0
(1)m
n
m
2n 2m
n
xn2m
on the interval [
1, 1]. The Legendre polynomials can also be obtained itera-
tively from the first two,
P0(x) = 1 and P1(x) = x,
and the recurrence relation,
(n + 1)Pn+1(x) = (2n + 1)xPn(x) nPn1(x).The Legendre polynomials are also given by Rodrigues formula:
Pn(x) =(1)n2nn!
dn
dxn[(1 x2)n].
Orthogonality:
1
1Pn(x)Pm(x) dx =
0, n = m;2
2n+1, n = m.
Standardization:Pn(1) = 1.
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RELATIONS AMONG THE FUNCTIONS
sin x = 1cscx csc x =1
sin x
cos x = 1secx sec x =1
cos x
tan x = 1cotx =sinxcosx cot x =
1tan x =
cosxsinx
sin2 x + cos2 x = 1 1 + tan2 x = sec2 x1 + cot2 x = csc2 x sin x = 1 cos2 x cos x =
1 sin2 x
tan x = sec2 x 1 sec x =
1 + tan2 x cot x = csc2 x 1 csc x =
1 + cot2 x
sin x = cos(2 x) = sin( x) cos x = sin(2 x) = cos( x)tan x = cot(
2 x) = tan( x) cot x = tan(
2 x) = cot( x)
SUMS AND MULTIPLES OF ANGLES
sin(x y) = sin x cos y cos x sin ycos(x y) = cos x cos y sin x sin ytan(x y) = tan x tan y
1 tan x tan ysin2x = 2 sin x cos xcos2x = cos2 x sin2 x = 2 cos2 x 1 = 1 2sin2 xsin3x = 3 sin x 4sin3 xcos3x = 4 cos3 x 3cos xsin4x = 8 sin x cos3 x
4sin x cos x
cos4x = 8 cos4 x 8cos2 x + 1tan2x =
2tan x
1 tan2 x cot2x =cot2 x 1
2cot x
tan3x =3tan x tan3 x
1 3tan2 x sin
1
2x =
1 cos x
2 cos
1
2x =
1 + cos x
2
tan1
2x =
1 cos x1 + cos x
=1 cos x
sin x=
sin x
1 + cos x
* The choice of the sign in front of the radical depends on the quadrantin which x, regarded as an angle, falls.
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SIDE-ANGLE RELATIONS IN PLANE TRIANGLES
Letters A, B, and Cdenote the angles of a plane triangle with opposite sides a,b, and c, respectively. The letter s denotes the semi-perimeter of the triangle,that is,
s =1
2(a + b + c).
a
sin A=
b
sin B=
c
sin C= diameter of the circumscribed circle
a2 = b2 + c2 2 b c cos Aa = b cos C+ c cos B
tanA B
2
=a b
a + b
cotC
2sin A =
2
bc
s(s a)(s b)(s c)
area =
s(s a)(s b)(s c)
sinA
2=
(s b)(s c)
bc
cosA
2=
s(s a)
bc
tanA
2=
(s b)(s c)
s(s a)a + b
a b =sin A + sin B
sin A sin B =tan 12 (A + B)
tan 12 (A B) =cot 12C
tan 12(A B)
h
d
h = dsin sin
sin( + )=
d
cot + cot
h
d
h = dsin sin
sin( ) =d
cot cot
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HYPERBOLIC FUNCTIONS
sinh x =1
2(ex ex), cosh x = 1
2(ex + ex)
tanh x =ex exex + ex
= sinh xcosh x
=1
coth x
sinh x =
cosh2 x 1 = tanh x1 tanh2 x
= 1coth2 x 1
cosh x =
1 + sinh2 x =1
1 tanh2 x= coth x
coth2 x 1
tanh x =sinh x
1 + sinh
2
x
=
cosh2 x 1
cosh x
=1
coth x
coth x =
1 + sinh2 x
sinh x= cosh x
cosh2 x 1=
1
tanh x
sinh(x y) = sinh x cosh y cosh x sinh ycosh(x y) = cosh x cosh y sinh x sinh ytanh(x y) = tanh x tanh y
1 tanh x tanh y coth(x y) =coth x coth y 1coth y coth x
sinh2x = 2cosh x sinh x cosh2x = cosh2 x + sinh2 x
sinh1 x = ln(x +
x2 + 1) cosh1 x = ln(x +
x2 1)
tanh1 x =1
2
ln1 + x1 x coth
1 x =1
2
lnx + 1x 1
Complex hyperbolic functions involve the Euler relations
eiz = cos z + i sin z; cos z =eiz + eiz
2, sin z =
eiz eiz2i
.
cos z = cosh iz
sin z = i sinh iztan z = i tanh izcot z = i coth iz
cosh z = cos iz
sinh z = i sin iztanh z = i tan izcoth z = i cot iz
sinh(x iy) = sinh x cos y i cosh x sin ycosh(x iy) = cosh x cos y i sinh x sin y
ln z = ln |z|+ i arg z; so ln ix = ln |x| + (2n + 12
)i
and ln(x) = ln |x| + (2n + 1)i (n = 0,1,2, . . .)
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FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS
I. Separable equations : N(y) dy = M(x) dx. To solve, integrate both sides:N(y) dy =
M(x) dx + c.
II. Exact equations : M(x, y) dx + N(x, y) dy = 0, whereM
y=
N
x. To
solve,
integratef
x= M(x, y) and
f
y= N(x, y) :
f(x, y) = M(x, y) dx + (y) = N(x, y) dy + (x) = c.III. Linear equations :
dy
dx+p(x) y = q(x).
The solution is given by
y =1
P(x)
P(x)q(x) dx +
c
P(x), where P(x) = e
p(x) dx.
IV. Homogeneous equations :
dy
dx= f
y
xor
dy
dx= f
x
y .
Substitute y = ux,dy
dx= x
du
dx+ u or x = vy,
dx
dy= y
dx
dy+ v,
integrate the resulting expression, then resubstitute for u or v.
V. Equations in the rational form
dy
dx=
ax + by + c
dx + ey + f,
where a, b, c, d, e, f are constants such that ae = bd. Substitute
x = X h, y = Y k, where h = bf ceae bd , k =
cd afae bd ,
to obtain the homogeneous equation
dY
dX=
aX+ bY
dX+ eY.
This equation may be solved as in item IV above.
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LINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS
WITH CONSTANT COEFFICIENTS
I. Homogeneous equation ay + by + cy = 0. Substitute y = erx to find thecharacteristic equation, and write its roots r1 and r2 :
ar2 + br + c = 0; r1, r2 =b b2 4ac
2a.
The solution, yc, is given by the following table.
Case 1. b2 4ac > 0, yc = c1er1x + c2er2xreal and distinct roots.
Case 2. b2 4ac = 0, yc = (c1 + c2x)erxtwo real equal roots.
Case 3. b2 4ac < 0, yc = ex(c1 cos x + c2 sin x),two complex roots. where
=b2a
, =
4ac b2
2a
II. Undetermined Coefficients : ay + by + cy = F(x).
1. Solve the homogeneous equation ay + by + cy = 0 for yc.
2. Assume a particular solution yp for any term in F(x) that is one of threespecial types, as follows.
A. For a polynomial of degree N, yp is a polynomial of degree N.
B. For an exponential kep(x), yp = Aep(x).C. For j cos x + k sin x, yp = A cos x + B sin x.
3. If any term in yp found above appears in yc, multiply the correspondingtrial term by the power of x just high enough that the resulting productcontains no term of yc.
4. For any term in F(x) that is a product of terms of the three types listedabove, let yp be the product of the corresponding trial solutions.
5. Substitute yp into the given O.D.E. and evaluate the unknown coefficientsby equating like terms.
6. The general solution of the given equation is
y = yc + yp.
Evaluate the two arbitrary constants c1 and c2 in the general solution yby applying boundary/initial conditions, as appropriate.
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III. Variation of Parameters : ay + by + cy = F(x).
1. Write the solution of the homogeneous equation ay + by + cy = 0 in theform
yc = c1y1 + c2y2.
2. Assume a particular solution yp for any term in the form
yp = u1y1 + u2y2,
where u1 and u2 are functions determined by the conditions
u1y1 + u2y2 = 0
u1y1 + u
2y2 = G(x) where G(x) =
F(x)
a.
3. The solution of this system of equations is given by
u1 = y2G
Wand u2 =
y1G
W,
where W denotes the Wronskian determinant:
W = W(x) =
y1 y2y1 y2 = 0.
4. Integrate to find the unknown functions:
u1(x) =
y2(x)G(x)
W(x)dx and u2(x) =
y1(x)G(x)
W(x)dx.
5. Form the particular solution yp = u1y1 + u2y2.
6. The general solution of the given equation is
y = yc + yp.
Evaluate the two arbitrary constants c1 and c2 in the general solution yby applying boundary/initial conditions, as appropriate.
IV. Laplace Transforms : ay + by + cy = F(x), y(0) = y0 and y(0) = y0.
1. Take the Laplace transform (see pages 9-10) of both sides of the givenequation, using the given initial conditions. The result is an algebraicequation for F(s), the Laplace transform of the solution.
2. Solve the algebraic equation obtained in Step 1 to find F(s) explicitly.
3. The inverse Laplace transform of F(s) is the solution of the given initialvalue problem.
*The coefficients may also be functions, i.e., b = b(x) and c = c(x).
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VECTOR ALGEBRA
1. Scalar and vector products. Given vectors
a = 1e1 + 2e2 + 3e3 = axi + ayj + azk
b = 1e1 + 2e2 + 3e3 = bxi + byj + bzk,
c = 1e1 + 2e2 + 3e3 = cxi + cyj + czk,
let be the angle from a to b. The scalar (dot) product of a and b is thescalar
a b = |a||b| cos = axbx + ayby + azbz.The vector (cross) product ofa and b is the vector of magnitude
|a b| = |a||b| sin ,
perpendicular to a and b and in the direction of the axial motion of aright-handed screw turning a into b. In coordinate form,
a b =
e2 e3 1 1e3 e1 2 2e1 e2 3 3
=
i ax bx
j ay byk az bz
=ay azby bz
i +az axbz bx
j + ax aybx by
k= (aybz azby)i + (azbx axbz)j + (axby aybx)k.
The box product ofa, b, and c is the scalar quantity given by
a
b
c
[abc] = [bca] = [cab] =
[bac] =
[cba] =
[acb]
=
1 1 12 2 23 3 3
[e1e2e3] =
ax bx cxay by cyaz bz cz
.The product of two box products is given by
[abc][def] =
a d a e a fb d b e b fc d c e c f
.A special case gives Grams determinant :
[abc]2 =
a a a b a cb a b b b cc a c b c c
= [(a b)(b c)(c a)]= (a a)(b b)(c c) + 2(a b)(b c)(a c) (a a)(b c)2
(b b)(a c)2 (c c)(a b)2.
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VECTOR ALGEBRA (Continued)
The vector triple product of vectors a, b, and c is given by
a (b c) = (a c)b (a b)c = b ca b a c
.Three additional formulas for scalar and vector products:
(a b) (c d) = (a c)(b d) (a d)(b c) = a c b ca d b d
.(a b)2 = (a a)(b b) (a b)2
(a b) (c d) = [acd]b [bcd]a = [abd]c [abc]d.
VECTOR ANALYSIS
Differential operators
(x,y,z) = grad(x,y,z) x
i +
yj +
zk
F(x,y,z) = div F(x,y,z) Fxx
+Fyy
+Fzz
F = curl F(x,y,z) Fz
y Fy
z
i +
Fxz
Fzx
j +
Fyx
Fxy
k
= i j k
x
y
z
Fx Fy Fz
(G )F = Gx F
x+ Gy
F
y+ Gz
F
z
= (G Fx)i + (G Fy)j + (G Fz)k
2 = ( ) 2x2
+2
y2+
2
z2
(Laplace operator)
div grad = () = 2grad div F = ( F) = 2F + ( F)curl curl F = ( F) = ( F) 2F
curl grad =
(
) = 0
div curl F = ( F) = 0() = + 2() = 2 + 2() () + 2
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VECTOR ANALYSIS (Continued)
(F G) = (F )G + (G )F + F (G) + G (F) (F) = F + () F
(F G) = G F F G(G )(F) = F(G ) + (G )F (F) = F + () F
(F G) = (G )F (F )G + F( G) G( F)
Cylindrical : x = r cos , y = r sin , z = z
grad =
=
rr +
1
r
+
zk
div F = F = 1r
(rFr)
r+
1
r
F
+Fzz
curl F = F =
1r r
1r k
r
z
Fr rF Fz
Laplacian = 2 = 1
r
r
r
r
+
1
r22
2+
2
z2
Spherical : x = cos sin , y = sin sin , z = cos
grad = =
+1
+
1
sin
div F = F = 12
(2F)
+ 1 sin
(sin F)
+ 1 sin
F
curl F = F =
1
2 sin 1
sin 1
F F sin F
2 = 1
2
2
+
1
2 sin
sin
+
1
2 sin2
2
2
Integral Theorems
V F(r) dV =
S
F(r) dA (Divergence theorem)V dV + V 2 dV = S() dA (Greens theorem)V(2 2) dV =
S( ) dA (symmetric form)
V2 dV =S dA (Gauss theorem)
S[ F(r)] dA =C
F(r) dr (Stokes theorem)
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FUNDAMENTALS OF SIGNALS AND NOISE
Complex numbers (j2 = 1)
a + bj = Rej , where
R =
a2 + b2,
=
arctanba
, a > 0;
arctanba
, a < 0;2 , a = 0, b > 0;2 , a = 0, b < 0.
Aej Bej = ABej(+)
ej = ej(+2n), n = 1,2, . . .
Unit phasor
j = ej2
ej = 1 1 = ej0 = ej2
j = ej 2 = ej 32
Euler Identity
ejx = cos xj sin x;
cos x = ejx
+ejx
2 ;sin x = e
jxejx2j
.
Fourier series (Periodic signals. For the real number form, see p. 13.)
x(t) =
n=cne
j2nf1t, where
cn =1
T1
T12
T12x(t)ej2nf1t dt,
T1 = period,
f1 = 1T1
= fundamental frequency,
(For real signals, cn = cn (complex conjugate).)
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Fourier transform (aperiodic signals)
X(f) =
x(t)ej2ft dt; x(t) =
X(f)ej2tf df.
The correspondence between x(t) and X(f) is denoted by x(t) X(f).
Transform of Fourier Series
If x(t) =
n=cne
j2nf1t, then X(f) =
n=cn(f nf1).
Convolution (
) and Correlation ()
x y =
x()y(t ) d
y()x(t ) d = y x
x y =
x()y(t + ) d
Properties of the delta function
(t) dt = 1
(t t0)x(t) dt = x(t0)
x(t) (t t0) = x(t t0)x(t)(t t0) = x(t0)(t t0)
Parseval theorem (Average Power in a periodic signal)
P =1
T
T2
T2|x(t)|2 dt =
n=
|cn|2
Rayleigh theorem (Total Energy in an aperiodic signal)
E =
|x(t)|2 dt =
|X(f)|2 df
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Fourier theorems
If x(t) X(f) is a transform pair, thenLINEARITY ax(t) + by(t) aX(f) + bY(f)
SHIFT/MODULATION x(t t0) X(f)ej2t0f
x(t)ej2f0t X(f f0)
PRODUCT/CONVOLUTION x(t)y(t) X Y
x y X(f)Y(f)
SCALE x(at) 1|a|X
fa
DIFFERENTIATIONd
dtx(t) j2f X(f)
INTEGRATION
t
x() d X(f)j2f
.
Fourier pairs
Time Domain Frequency Domain
h(t) = 1 uT02
(|t|) H(f) = sin(T0f)f
1
h(t)
t-T0/2 T0/2
H(f)
f1/T0 2/T0
T0
h(t) =sin(2f0t)
2f0t H(f) = 1
2f0(1 uf0(|f|))
h(t)
t1/2f0 1/f0
11/2f0
H(f)
f-f0 f0
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Fourier pairs Continued
Time Domain Frequency Domain
h(t) = K H(f) = K(f)
K
h(t)
t
H(f)
f
K
h(t) = K(t) H(f) = Kh(t)
t
K K
H(f)
f
h(t) = A cos(2f0t) H(f) = A2
[(f + f0) + (f f0)]h(t)
t1/f0
AH(f)
f
A/2
-f0 f0
h(t) = A sin(2f0t) H(f) = A2
j[(f + f0) (f f0)]h(t)
t
1/f0
AIm[H(f)]
f
A/2
-f0
f0
h(t) =
1 |t|
T0
1 uT0(|t|)
H(f) = sin
2(T0f)
T02f2
1
h(t)
t-T0 T0
H(f)
f1/T0 2/T0
T0
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Fourier pairs Continued
Time Domain Frequency Domain
h(t) = e|t| H(f) = 22 + 42f2
h(t)
t
1H(f)
f
2/
h(t) = exp(t2) H(f) =
exp
2f2
h(t)
t
1H(f)
f
/
h(t) =
et, t > 0
0, t 0 H(f) =1
+ 2jf
h(t)
t
1|H(f)|
f
1/
h(t) = sgn(t) H(f) = jf
1
-1
h(t)
t
Im[H(f)]
f
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Linear time invariant systemsBasic Properties
Ifx(t) y(t)
g(t) q(t)
then
ax(t t0) + bg(t t0) ay(t t0) + bq(t t0)
Transform properties
Input Output
Time Domain (t) h(t)
x(t) y(t) = x(t) h(t)
system
Frequency Domain X(f) Y(f) = X(f) H(f)h(t) H(f) h(t) impulse response; H(f) transfer function
Random signals
x =
xp(x) dx Mean value
x2 =
x2p(x) dx Second moment
< x, x >= x2 = 1T
T2
T2|x(t)|2 dt Norm square
< x > =1
T
T2
T2x(t) dt Time average
For an ergodic process:Second moment = Norm square and Mean value = Time average.
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Autocorrelation
Rxx() = limT
1
T
T2
T2x(t)x(t + ) dt
Power Spectral Density
The power spectral density Gxx is the Fourier transform of the autocor-relation function:
Rxx(t) =
Gxx(f)ej2tf df Gxx(f) =
Rxx(t)ej2ft dt
Noise power (Gxx denotes the power spectral density)
PTotal =
Gxx df = Rxx(0) = x2 = (x)
2
Pac = N = 2x = noise power
Pdc = Rxx() = (x)2
Noise bandwidth (Gxx denotes the power spectral density)
Gyy = Gxx |H|2 (linear system)
Ny =
Gxx |H|2 df = noise power output
N = n0BN = noise power, wheren02
= white noise
BN =
|H|2 df = noise bandwidth
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Ideal filter (low pass)
h(t) =sin(2f0t)
2f0 H(f) = 1 uf0(|f|)
2f0
h(t)
t1/2f0 1/f0
11/2f0
H(f)
f-f0 f0
RC filter (low pass)
VC= h(t)
E(t) h(t) =
1
RC
et/RC
H(f) =
1
1 + 2jfRC
E(t)
VC
CR
h(t)
t
1|H(f)|
f
1
Ideal filter (high pass)
h(t) = (t) sin(2f0t)t
H(f) = uf0(|f|)
h(t)
t
1/2f0 1/f0 1
H(f)
f-f0 f0
RC filter (high pass)
VR = h(t)E(t) h(t) = (t) 1RC
et/RC H(f) = 2jfRC1 + 2jfRC
E(t)
VRCR
h(t)
t
|H(f)|
f
1
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Ideal filter (band pass)
h(t) =sin(2f2t)
t sin(2f1t)
t H(f) = uf2(|f|) uf1(|f|)
h(t)
t
1
H(f)
f-f2 -f1 f1 f2
LRC filter (band pass)
VR = h(t)
E(t) h(t)
H(f) =2jfRC
1 (2f)2LC+ 2jfRC
E(t)
VRC
R L
h(t)
t
|H(f)|
f
1
LC1/(2 )LC1/(2 )
Ideal sampling
Sampled signal
x(t) = x(t) s(t) where s = k=
(t kTs)
Spectrum after sampling
X(f) = X(f) S(f) where S(f) = fs
n=(f nfs)
Sampling theorem
If
Ts 2W (Nyquist rate)
then the signal can be reconstructed with a filter of bandwidth B, givenby:
W < B < fs W.