Formulae List

7/30/2019 Formulae List

1/22

Contents

1 Algebra 1

1.1 Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Partial fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Law of indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.5 Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.6 Arithmetic Progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.7 Geometric Progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.8 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Geometry 5

3 Trigonometry 6

3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Signs and variations of trigonometric functions . . . . . . . . . . . . . . . . 6

3.3 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.4 Compound angle addition and subtraction formula . . . . . . . . . . . . . 7

3.5 Double angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.6 Products of sines or cosines into sums or differences . . . . . . . . . . . . . 8

3.7 Sums or differences of sines or cosines into products . . . . . . . . . . . . . 8

3.8 Polar Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.9 Complex exponent forms : imaginary form . . . . . . . . . . . . . . . . . . 9

4 Hyperbolic Function 10

4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Complex Numbers 11

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Contents

6 Differentiation 12

7 Integration 14

8 Differential Equations 15

8.1 First order differential equations . . . . . . . . . . . . . . . . . . . . . . . . 15

8.2 Second order differential equations . . . . . . . . . . . . . . . . . . . . . . 16

9 Laplace Transform 18

9.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

9.2 Laplace Transform Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

9.3 Laplace Transform rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

10 Fourier Series 20

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1.3. Law of indices

repeated linear factors :f(x)

(x + )n =

A1

x + + . . . +

An

(x + )n

a quadratic factor :f(x)

(1x2 + 1x + 1)(2x2 + 2x + 2)=

Ax + B

(1x2 + 1x + 1)+

Cx + D

(2x2 + 2x + 2)

1.3 Law of indices

am an = am+nam

an= amn

(am)n = amn

an =1

an

a0 = 1

nam = am/nn

ab = n

a n

b

n

a

b=

n

an

b

1.4 Logarithm

Definition : If y = ax, then

x = loga y (log y to base a)

y = aloga y (log and antilog with the same base cancel)

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1.5. Quadratic Formula

Logarithmic rules

log(a

b) = log a + log b

log

a

b

= log a log b

log(xa) = a log x

log(y)1/n =log y

n

logb a =logc a

logc b(Change of base)

log(a + jb) = log

a2 + b2 + j tan1b

a

1.5 Quadratic Formula

If ax2 + bx + c = 0, then

x =b b2 4ac

2a

If b2 4ac > 0, then ax2 + bx + c = 0 yields two real and distinct roots.

If b2 4ac = 0, then ax2 + bx + c = 0 yields two real and equal roots.

If b24ac < 0, then ax2 + bx + c = 0 yields a pair of complex conjugate roots.

1.6 Arithmetic Progression

If a = first term, d = common difference, n = number of terms and l = last term, then

the arithmetic progression is

a, a + d, a + 2d, a + 3d , . . . , a + (n 1)d , . . . , lSum of n terms =

n

2[2a + (n 1)d]

=n

2

(a + l)

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1.7. Geometric Progression

1.7 Geometric Progression

If a = first term, r = common ratio and n = number of terms, then the geometric

progression is :

a, ar, ar2, ar3, . . . , a rn1, . . .

Sum of n terms =

a(rn1)r1 , r > 1

a(1rn)1r , r < 1

Sum to infinity when r < 1 =a

1 r

1.8 Series

Binomial series:

(a + b)n = an + nan1b +n(n 1)

2!an2b2 +

n(n 1)(n 2)3!

an3b3 + . . .

(1 + x)n

= 1 + nx +n(n

1)

2! x2

+n(n

1)(n

2)

3! x3

+ . . . (valid for 1 < x < 1)

Exponential series:

ex = 1 + x +x2

2!+

x3

3!+ . . .

ex = 1 x + x2

2! x

3

3!+ . . .

Taylors expansion:

f(xo + a) = f(xo) + af(xo) +

a2

2!f(xo) +

a3

3!f(x0) + . . .

where x0 is the center and a is the deviation about xo.

Maclaurins form:

f(x) = f(0) + xf(0) +x2

2!f(0) +

x3

3!f(0) + . . .

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Chapter 2

Geometry

Equation of a straight line joining (x1, y1) and (x2, y2) :

y y1x x1 =

y2 y1x2 x1 = m

or y y1 = m(x x1)y = mx + c where m = gradient and c = y-intercept

Equation of a circle, center at (x0, y0), radius r :

(x x0)2

+ (y y0)2

= r2

Equation of an ellipse, center at (x0, y0), semi-axes a and b :

(x x0)2a2

+(y y0)2

b2= 1

Pythagoras Theorem :

a

b

c

c2 = a2 + b2

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9/22

3.3. Identities

3.3 Identities

sin2 A + cos2 A = 1

1 + tan2 A = sec2 A

cot2 A + 1 = cosec2A

cosec2A = 1 + cot2 A

sin(A) = sin Acos(A) = + cos A

tan(A) = tan A

3.4 Compound angle addition and subtraction

formula

sin(A B) = sin A cos B cos A sin Bcos(A B) = cos A cos B sin A sin Btan(A B) = tan A tan B

1 tan A tan B

3.5 Double angles

sin2A = 2 sin A cos A

cos2A = cos2 A sin2 A = 2cos2 A 1 = 1 2sin2 Atan2A =

2tan A

1 tan2 Asin3A = 3 sin A 4sin3 Acos3A = 4 cos3 A 3cos Atan3A =

3tan A tan3 A1

3tan2 A

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3.6. Products of sines or cosines into sums or differences

3.6 Products of sines or cosines into sums or

differences

2sin A cos B = sin(A + B) + sin(A B)2cos A sin B = sin(A + B) sin(A B)2cos A cos B = cos(A + B) + cos(A B)2sin A sin B = cos(A B) cos(A + B)

3.7 Sums or differences of sines or cosines into

products

sin A + sin B = 2 sinA + B

2cos

A B2

sin A sin B = 2 cos A + B2

sinA B

2

cos A + cos B = 2 cos A + B2

cos A B2

cos A cos B = 2sin A + B2

sinA B

2

3.8 Polar Form

a sin x + b cos x = R sin(x + )

where R =

a2 + b2

= tan1b

a

sin =b

R

cos =a

R

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3.9. Complex exponent forms : imaginary form

3.9 Complex exponent forms : imaginary form

cos x = ejx+ejx

2ejx = cos x + j sin x

sin x = ejxejx

2jejx = cos x j sin x j = 1

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Chapter 4

Hyperbolic Function

4.1 Definitions

sinh A =eA eA

2cosech A =

1

sinh A=

2

eA eAcosh A =

eA + eA

2sech A =

1

cosh A=

2

eA + eA

tanh A =eA eAeA + eA

coth A =1

tanh A=

eA + eA

eA eA

4.2 Identities

cosh2 A sinh2 A = 11 tanh2 A = sech2Acoth2 A 1 = cosech2A

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Chapter 5

Complex Numbers

Re

Im

x

y

r

z = x +jy

1 = (2)

z = x + jy where j =

1

= r(cos + j sin )

= r

= rej

r = |z| =

x2 + y2

= arg z = 1

= tan1b

a= sin1

y

|z|= cos1

x

|z|

Addition : (a + bj) + (c + dj) = (a + c) + (b + d)j

Subtraction : (a + bj) (c + dj) = (a c) + (b d)jMultiplication : z1z2 = r1r2(1 + 2)

Division :z1

z2=

r1

r2(1 2)

De Moivres theorem : [r]n = rnn = rn(cos n + j sin n)

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Chapter 6

Differentiation

Function Derivative Function Derivative

x

n

nx

n

1 1

x 1

x2

loge n1x

loga x1

x loge a

ecx cecx ax (a > 0) ax loge a

sin x cos x cos x sin xtan x sec2 x cosec x cosec x cot xsec x sec x tan x cot x cosec2 x

sin1

x1

1x2 cos1

x 1

1x2

tan1 x 11+x2

Product rule :

When y = uv and u and v are functions of x, then

dy

dx= v

du

dx+ u

dv

dx

Quotient rule :

When y = uv

and u and v are functions of x, then

dy

dx=

v dudx u dv

dx

v2

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Chain rule :

If u is a function of x, then

dydx

= dydu

dudx

Implicit differentiation :

d

dx[f(y)] =

d

dy[f(y)] dy

dx

Maximum and minimum values :

If y = f(x), then the stationary points are found by solvingdy

dx= 0.

Let a solution ofdy

dx= 0 be x = a. If the value of

d2y

dx2when x = a is :

positive, the point is a minimum point.

negative, the point is a maximum point.

zero, the point is a point of inflexion (saddle point).

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Chapter 8

Differential Equations

8.1 First order differential equations

1.dy

dx= f(x) or P

dy

dx+ Q = 0, P and Q being functions of x only.

Pdy

dx+ Q = 0 can be written as

dy

dx= Q

P= f(x)

Solution : y =

f(x) dx

2. If dy

dx= f(y) or P

dy

dx+ Q = 0, P and Q being functions of y only.

Pdydx

+ Q = 0 can be written as dydx

= QP

= f(y)

Solution :

dx =

dy

f(y)

3. Separation of variables :dy

dx= f(x) f(y)

Solution :

dy

f(y)=

f(x) dx

4. Homogeneous first order differential equation :dy

dx

= fy

x

Solution :

Introduce the new independent variable v = yx

.

Then, y = vx and dydx

= v(1) + xdv

dxby the product rule.

Substitute y = vx and dydx = v(1) + x dvdx

indy

dx= f

yx

to obtain a separable

differential equation. Solve the separable differential equation.

Solution is obtained by replacing v by yx

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8.2. Second order differential equations

5. Linear first order differential equation :dy

dx+ P(x)y = Q(x)

Solution :

Determine the integrating factor (I.F.) : eP(x)dx

Substitute the I.F. into the equation : yeP(x)dx =

Q(x)e

P(x)dx dx,

yeP(x)dx will be an exact differential and

Q(x)e

P(x)dx dx can be integrated

8.2 Second order differential equations

Linear homogeneous second order differential equations

ad2y

dx2+ b

dy

dx+ cy = 0

Form the characteristic equation : am2 + bm + c = 0

If the roots of the complementary equation are

real and distinct i.e. m = and m = , then the general solution is

y = Aex + Bex

real and equal i.e. m = twice, then the general solution is

y = ex(A + Bx)

complex i.e. m = j, then the general solution is

y = ex(A sin x + B cos x)

Constants A and B are determined from initial conditions.

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8.2. Second order differential equations

Linear non-homogeneous second order differential equations

a

d2y

dx2 + b

dy

dx + cy = f(x)

The total solution of this type of differential equation is made up of :

The general solution of the homogeneous second order differential equation

ad2y

dx2+ b

dy

dx+ cy = 0

The particular integral, which depends on f(x). The following table lists thetrial particular integral to use for various f(x).

f(x) Trial particular integral

f(x) = constant y = k y = kx when general solution contains a

constant

f(x) = L + M x + N x2 + . . . y = a + bx + cx2 + . . .

f(x) = Aex

y = ke

x y = kxex when general solution contains ex

y = kx2ex when general solution containsxex, and so on

f(x) = sinpx + cos x y = a sinpx + b cospx or may be zero y = x(a sinpx + b cospx) when general solution

contains sinpx and/or cospx

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20/22

Chapter 9

Laplace Transform

9.1 Definition

L{f(t)} =0

est f(t) dt

9.2 Laplace Transform Pairs

f(t) F(s)

(t) 1

U(t) 1s

tU(t) 1s2

eatU(t) 1s + a

[sin at] U(t)

a

s2

+ a2

[cos at] U(t) ss2 + a2

[eat sin bt] U(t) b(s + a)2 + b2

[eat cos bt] U(t) s + a(s + a)2 + b2

teatU(t) 1(s + a)2

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9.3. Laplace Transform rule

9.3 Laplace Transform rule

Transform of Derivatives :

L{f(t)} = sF(s) f(0)

Transform of Derivative of order n :

L{fn(t)} = snF(s) n1k=0

sn1kfk(0)

Transform of an Integral :Lt

0

f()d

=

F(s)

s

Derivative of Transforms

F(s) = L{tf(t)}and

dn

dsnF(s) = (1)nL{tnf(t)}

Shift in the time-domain function :

L{f(t t0)U(t t0)} = est0F(s)

Shift in the s-domain function :

L{etf(t)} = F(s + )

Initial Value Theorem :lims

sF(s) = f(0+) = limt0+

f(t)

Final Value Theorem (FVT) :

lims0

sF(s) = limt

f(t)

FVT is applicable only if the signal is a finite constant value, i.e. limt

f(t) =

constant

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Chapter 10

Fourier Series

The Fourier series corresponding to a periodic function f(x) of period 2 is

f(x) = a0 +n=1

(an cos nx + bn sin nx)

where for the range to a0 =

1

2

f(x) dx

an =1

f(x)cos nx dx (n = 1, 2, 3, . . .)

bn =1

f(x)sin nx dx (n = 1, 2, 3, . . .)

For functions of any period 2L, the Fourier series is

f(x) = a0 +n=1

an cos

n

Lx + bn sin

n

Lx

a0 =1

2L

LL

f(x) dx

an =

1

LLL f(x)cos

nx

L dx (n = 1, 2, 3, . . .)

bn =1

L

LL

f(x)sinnx

Ldx (n = 1, 2, 3, . . .)

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