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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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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.