GREEK ALPHABET A α alpha N ν nu B β beta Ξ ξ xi Γ γ gamma Oo omicron Δ δ delta Π π pi E , ε epsilon P ρ rho Z ζ zeta Σ σ sigma H η eta T τ tau Θ θ, ϑ theta Υ υ upsilon I ι iota Φ φ, ϕ phi K κ kappa X χ chi Λ λ lambda Ψ ψ psi M μ mu Ω ω omega INDICES AND LOGARITHMS a m × a n = a m+n (a m ) n = a mn log(AB) = log A + log B log(A/B) = log A - log B log(A n ) = n log A log b a = log c a log c b 3
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GREEK ALPHABET
A α alpha N ν nu
B β beta Ξ ξ xi
Γ γ gamma O o omicron
∆ δ delta Π π pi
E ε, ε epsilon P ρ rho
Z ζ zeta Σ σ sigma
H η eta T τ tau
Θ θ, ϑ theta Υ υ upsilon
I ι iota Φ φ, ϕ phi
K κ kappa X χ chi
Λ λ lambda Ψ ψ psi
M µ mu Ω ω omega
INDICES AND LOGARITHMS
am × an = am+n
(am)n = amn
log(AB) = log A + log B
log(A/B) = log A− log B
log(An) = n log A
logb a =logc alogc b
3
Mr.Adil Pangaribuan
Typewriter
Mr. Adil Pangaribuan SMA Neg.4 Medan
Edited by Foxit Reader Copyright(C) by Foxit Software Company,2005-2007 For Evaluation Only.
TRIGONOMETRIC IDENTITIES
tan A = sin A/ cos A
sec A = 1/ cos A
cosec A = 1/ sin A
cot A = cos A/ sin A = 1/ tan A
sin2 A + cos2 A = 1
sec2 A = 1 + tan2 A
cosec 2A = 1 + cot2 A
sin(A±B) = sin A cos B ± cos A sin B
cos(A±B) = cos A cos B ∓ sin A sin B
tan(A±B) = tan A±tan B1∓tan A tan B
sin 2A = 2 sin A cos A
cos 2A = cos2 A− sin2 A
= 2 cos2 A− 1
= 1− 2 sin2 A
tan 2A = 2 tan A1−tan2 A
sin 3A = 3 sin A− 4 sin3 A
cos 3A = 4 cos3 A− 3 cos A
tan 3A = 3 tan A−tan3 A1−3 tan2 A
sin A + sin B = 2 sin A+B2 cos A−B
2
4
sin A− sin B = 2 cos A+B2 sin A−B
2
cos A + cos B = 2 cos A+B2 cos A−B
2
cos A− cos B = −2 sin A+B2 sin A−B
2
2 sin A cos B = sin(A + B) + sin(A−B)
2 cos A sin B = sin(A + B)− sin(A−B)
2 cos A cos B = cos(A + B) + cos(A−B)
−2 sin A sin B = cos(A + B)− cos(A−B)
a sin x + b cos x = R sin(x + φ), where R =√
a2 + b2 and cos φ = a/R, sin φ = b/R.
If t = tan 12x then sin x = 2t
1+t2 , cos x = 1−t21+t2 .
cos x = 12 (eix + e−ix) ; sin x = 1
2i (eix − e−ix)
eix = cos x + i sin x ; e−ix = cos x− i sin x
5
COMPLEX NUMBERS
i =√−1 Note:- ‘j’ often used rather than ‘i’.
Exponential Notation
eiθ = cos θ + i sin θ
De Moivre’s theorem
[r(cos θ + i sin θ)]n = rn(cos nθ + i sin nθ)
nth roots of complex numbers
If z = reiθ = r(cos θ + i sin θ) then
z1/n = n√
rei(θ+2kπ)/n, k = 0,±1,±2, ...
HYPERBOLIC IDENTITIES
cosh x = (ex + e−x) /2 sinhx = (ex − e−x) /2
tanh x = sinh x/ cosh x
sechx = 1/ cosh x cosechx = 1/ sinh x
coth x = cosh x/ sinh x = 1/ tanh x
cosh ix = cos x sinh ix = i sin x
cos ix = cosh x sin ix = i sinh x
cosh2 A− sinh2 A = 1
sech2A = 1− tanh2 A
cosech 2A = coth2 A− 1
6
SERIES
Powers of Natural Numbers
n∑
k=1
k =12
n(n + 1) ;n
∑
k=1
k2 =16
n(n + 1)(2n + 1);n
∑
k=1
k3 =14
n2(n + 1)2
Arithmetic Sn =n−1∑
k=0
(a + kd) =n22a + (n− 1)d
Geometric (convergent for −1 < r < 1)
Sn =n−1∑
k=0
ark =a(1− rn)
1− r, S∞ =
a1− r
Binomial (convergent for |x| < 1)
(1 + x)n = 1 + nx +n!
(n− 2)!2!x2 + ... +
n!(n− r)!r!
xr + ...
wheren!
(n− r)!r!=
n(n− 1)(n− 2)...(n− r + 1)r!
Maclaurin series
f(x) = f(0) + xf ′(0) +x2
2!f ′′(0) + ... +
xk
k!f (k)(0) + Rk+1
where Rk+1 =xk+1
(k + 1)!f (k+1)(θx), 0 < θ < 1
Taylor series
f(a + h) = f(a) + hf ′(a) +h2
2!f ′′(a) + ... +
hk
k!f (k)(a) + Rk+1
where Rk+1 =hk+1
(k + 1)!f (k+1)(a + θh) , 0 < θ < 1.
OR
f(x) = f(x0) + (x− x0)f′(x0) +
(x− x0)2
2!f ′′(x0) + ... +
(x− x0)k
k!f (k)(x0) + Rk+1
where Rk+1 =(x− x0)k+1
(k + 1)!f (k+1)(x0 + (x− x0)θ), 0 < θ < 1
7
Special Power Series
ex = 1 + x +x2
2!+
x3
3!+ ... +
xr
r!+ ... (all x)
sin x = x− x3
3!+
x5
5!− x7
7!+ ... +
(−1)rx2r+1
(2r + 1)!+ ... (all x)
cos x = 1− x2
2!+
x4
4!− x6
6!+ ... +
(−1)rx2r
(2r)!+ ... (all x)
tan x = x +x3
3+
2x5
15+
17x7
315+ ... (|x| < π
2 )
sin−1 x = x +12
x3
3+
1.32.4
x5
5+
1.3.52.4.6
x7
7+
... +1.3.5....(2n− 1)
2.4.6....(2n)x2n+1
2n + 1+ ... (|x| < 1)
tan−1 x = x− x3
3+
x5
5− x7
7+ ... + (−1)n x2n+1
2n + 1+ ... (|x| < 1)
`n(1 + x) = x− x2
2+
x3
3− x4
4+ ... + (−1)n+1xn
n+ ... (−1 < x ≤ 1)
sinh x = x +x3
3!+
x5
5!+
x7
7!+ ... +
x2n+1
(2n + 1)!+ ... (all x)
cosh x = 1 +x2
2!+
x4
4!+
x6
6!+ ... +
x2n
(2n)!+ ... (all x)
tanh x = x− x3
3+
2x5
15− 17x7
315+ ... (|x| < π
2 )
sinh−1 x = x− 12
x3
3+
1.32.4
x5
5− 1.3.5
2.4.6x7
7+
... + (−1)n 1.3.5...(2n− 1)2.4.6...2n
x2n+1
2n + 1+ ... (|x| < 1)
tanh−1 x = x +x3
3+
x5
5+
x7
7+ ...
x2n+1
2n + 1+ ... (|x| < 1)
8
DERIVATIVES
function derivative
xn nxn−1
ex ex
ax(a > 0) ax`na
`nx 1x
loga x 1x`na
sin x cos x
cos x − sin x
tan x sec2 x
cosec x − cosec x cot x
sec x sec x tan x
cot x − cosec 2x
sin−1 x1√
1− x2
cos−1 x − 1√1− x2
tan−1 x1
1 + x2
sinh x cosh x
cosh x sinh x
tanh x sech 2x
cosech x − cosech x coth x
sech x − sech x tanh x
coth x − cosech2x
sinh−1 x1√
1 + x2
cosh−1 x(x > 1)1√
x2 − 1
tanh−1 x(|x| < 1)1
1− x2
coth−1 x(|x| > 1) − 1x2 − 1
9
Product Rule
ddx
(u(x)v(x)) = u(x)dvdx
+ v(x)dudx
Quotient Rule
ddx
(
u(x)v(x)
)
=v(x)du
dx − u(x) dvdx
[v(x)]2
Chain Ruleddx
(f(g(x))) = f ′(g(x))× g′(x)
Leibnitz’s theorem
dn
dxn (f.g) = f (n).g+nf (n−1).g(1)+n(n− 1)
2!f (n−2).g(2)+...+
n!(n− r)!r!
f (n−r).g(r)+...+f.g(n)
10
INTEGRALS
function integral
f(x)dg(x)dx
f(x)g(x)−∫ df(x)
dxg(x)dx
xn(n 6= −1) xn+1
n+11x `n|x| Note:- `n|x|+ K = `n|x/x0|ex ex
sin x − cos x
cos x sin x
tan x `n| sec x|cosec x −`n| cosec x + cot x| or `n
∣
∣
∣tan x2
∣
∣
∣
sec x `n| sec x + tan x| = `n∣
∣
∣tan(
π4 + x
2
)∣
∣
∣
cot x `n| sin x|1
a2 + x2
1a
tan−1 xa
1a2 − x2
12a
`na + xa− x
or1a
tanh−1 xa
(|x| < a)
1x2 − a2
12a
`nx− ax + a
or − 1a
coth−1 xa
(|x| > a)
1√a2 − x2
sin−1 xa
(a > |x|)
1√a2 + x2
sinh−1 xa
or `n(
x +√
x2 + a2)
1√x2 − a2
cosh−1 xa
or `n|x +√
x2 − a2| (|x| > a)
sinh x cosh x
cosh x sinh x
tanh x `n cosh x
cosech x −`n |cosechx + cothx| or `n∣
∣
∣tanh x2
∣
∣
∣
sech x 2 tan−1 ex
coth x `n| sinh x|
11
Double integral∫ ∫
f(x, y)dxdy =∫ ∫
g(r, s)Jdrds
where
J =∂(x, y)∂(r, s)
=
∣
∣
∣
∣
∣
∣
∣
∂x∂r
∂x∂s
∂y∂r
∂y∂s
∣
∣
∣
∣
∣
∣
∣
12
LAPLACE TRANSFORMS
f(s) =∫∞0 e−stf(t)dt
function transform
11s
tnn!
sn+1
eat 1s− a
sin ωtω
s2 + ω2
cos ωts
s2 + ω2
sinh ωtω
s2 − ω2
cosh ωts
s2 − ω2
t sin ωt2ωs
(s2 + ω2)2
t cos ωts2 − ω2
(s2 + ω2)2
Ha(t) = H(t− a)e−as
s
δ(t) 1
eattnn!
(s− a)n+1
eat sin ωtω
(s− a)2 + ω2
eat cos ωts− a
(s− a)2 + ω2
eat sinh ωtω
(s− a)2 − ω2
eat cosh ωts− a
(s− a)2 − ω2
13
Let f(s) = Lf(t) then
L
eatf(t)
= f(s− a),
Ltf(t) = − dds
(f(s)),
L
f(t)t
=∫ ∞
x=sf(x)dx if this exists.
Derivatives and integrals
Let y = y(t) and let y = Ly(t) then
L
dydt
= sy − y0,
L
d2ydt2
= s2y − sy0 − y′0,
L∫ t
τ=0y(τ)dτ
=1sy
where y0 and y′0 are the values of y and dy/dt respectively at t = 0.
Time delay
Let g(t) = Ha(t)f(t− a) =
0 t < a
f(t− a) t > a
then Lg(t) = e−asf(s).
Scale change
Lf(kt) =1kf
( sk
)
.
Periodic functions
Let f(t) be of period T then
Lf(t) =1
1− e−sT
∫ T
t=0e−stf(t)dt.
14
Convolution
Let f(t) ∗ g(t) =∫ tx=0 f(x)g(t− x)dx =
∫ tx=0 f(t− x)g(x)dx
then Lf(t) ∗ g(t) = f(s)g(s).
RLC circuit
For a simple RLC circuit with initial charge q0 and initial current i0,
E =(
r + Ls +1
Cs
)
˜i− Li0 +1
Csq0.
Limiting values
initial value theorem
limt→0+
f(t) = lims→∞
sf(s),
final value theorem
limt→∞
f(t) = lims→0+
sf(s),∫ ∞
0f(t)dt = lim
s→0+f(s)
provided these limits exist.
15
Z TRANSFORMS
Z f(t) = f(z) =∞∑
k=0
f(kT )z−k
function transform
δt,nT z−n(n ≥ 0)
e−at zz − e−aT
te−at Tze−aT
(z − e−aT )2
t2e−at T 2ze−aT (z + e−aT )(z − e−aT )3
sinh atz sinh aT
z2 − 2z cosh aT + 1
cosh atz(z − cosh aT )
z2 − 2z cosh aT + 1
e−at sin ωtze−aT sin ωT
z2 − 2ze−aT cos ωT + e−2aT
e−at cos ωtz(z − e−aT cos ωT )
z2 − 2ze−aT cos ωT + e−2aT
te−at sin ωtTze−aT (z2 − e−2aT ) sin ωT
(z2 − 2ze−aT cos ωT + e−2aT )2
te−at cos ωtTze−aT (z2 cos ωT − 2ze−aT + e−2aT cos ωT )
(z2 − 2ze−aT cos ωT + e−2aT )2
Shift Theorem
Z f(t + nT ) = znf(z)−∑n−1k=0 zn−kf(kT ) (n > 0)
Initial value theorem
f(0) = limz→∞ f(z)
16
Final value theorem
f(∞) = limz→1
[
(z − 1)f(z)]
provided f(∞) exists.
Inverse Formula
f(kT ) =12π
∫ π
−πeikθf(eiθ)dθ
FOURIER SERIES AND TRANSFORMS
Fourier series
f(t) =12a0 +
∞∑
n=1an cos nωt + bn sin nωt (period T = 2π/ω)
where
an =2T
∫ t0+T
t0f(t) cos nωt dt
bn =2T
∫ t0+T
t0f(t) sin nωt dt
17
Half range Fourier series
sine series an = 0, bn =4T
∫ T/2
0f(t) sin nωt dt
cosine series bn = 0, an =4T
∫ T/2
0f(t) cos nωt dt
Finite Fourier transforms
sine transform
fs(n) =4T
∫ T/2
0f(t) sin nωt dt
f(t) =∞∑
n=1fs(n) sin nωt
cosine transform
fc(n) =4T
∫ T/2
0f(t) cos nωt dt
f(t) =12fc(0) +
∞∑
n=1fc(n) cos nωt
Fourier integral
12
(
limt0
f(t) + limt0
f(t))
=12π
∫ ∞
−∞eiωt
∫ ∞
−∞f(u)e−iωu du dω
Fourier integral transform
f(ω) = F f(t) =1√2π
∫ ∞
−∞e−iωuf(u) du
f(t) = F−1
f(ω)
=1√2π
∫ ∞
−∞eiωtf(ω) dω
18
NUMERICAL FORMULAE
Iteration
Newton Raphson method for refining an approximate root x0 of f(x) = 0