Advanced Math

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ADVANCED ENGINEERING MATHEMATICS

Complex Numbers :

z = x + jy = r(cos θ + j sin θ) = r cjs θ = r∠θ = rejθ

where: j2 = −1 or j = √−1

Power of Complex Numbers :

z n = (x + jy)n = r n (cos nθ + j sin nθ) = r n cjs nθ = r n ∠nθ = r n ejnθ

Roots of Complex Numbers :

since for any whole number k,

sin (θ + 360°k) = sin θ
cos (θ + 360°k) = cos θ

therefore, in general form

1 1 θ + 360°k 1 θ + 360°k 1 j(θ+2πk)
n n
√z = z n = r n cjs ( ) = rn ∠ = rn e
n n
where: k = 0, when taking the first root
k = 1, when taking the second root
k = 2, when taking the third root
k = n – 1, when taking the nth root

Trigonometric and Hyperbolic Functions of Complex Numbers :

1. cosh (jθ) = cos θ 5. sin (x ± jy) = sin x cos jy ± cos x sin jy

2. sinh (jθ) = j sin θ = sin x cosh y ± j cos x sinh y
3. cos ju = cosh u 6. cos (x ± jy) = cos x cos jy ∓ sin x sin jy
4. sin ju = j sinh u = cos x cosh y ∓ j sin x sinh y

Laplace Transform:

The Laplace Transform of a function f(t), denoted by ℒ[f(t)] is defined as a function of a

variable s by the the integral :
∞
F(s) = ℒ[f(t)] = ∫ f(t)e−st dt
0
where t > 0 and s is any number, real or complex.

Formulas:

a k s
1. ℒ[a] = 4. ℒ[sin kt] = 7. ℒ [cosh kt] =
s s2
+ k2 s2 − k2
n! s Γ(p + 1 ) where p > −1
2. ℒ[t n ] = , n = 1, 2, … 5. ℒ[cos kt] = 2
s + k2
8. ℒ [t p ] = p+1
,
and non˗integrals
s n+1 s
1 k
3. ℒ[e± at ] = 6. ℒ[sinh kt] =
s∓a s2 − k2

Important Theorems on Laplace Transforms :

Theorem 1. (Linearity Theorem)

ℒ[a f(t) + b g(t)] = a ℒ[f(t)] + b ℒ[g(t)]

Theorem 2. (First Shifting Theorem) Theorem 3. (Second Shifting Theorem)

ℒ[eat f(t)] = ℒ[f(t)]s→s−a ℒ[f(t − a)u(t − a)] = e−as ℒ[f(t)]

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Theorem 4. (Transforms of Derivative)
ℒ[f ′ (t)] = sℒ[f(t)] − f(0)
ℒ[f ′′ (t)] = s 2 ℒ[f(t)] − [s f(0) + f ′ (0)]
ℒ[f ′′′ (t)] = s 3 ℒ[f(t)] − [s 2 f(0) + s f ′ (0) + f ′′ (0)]
ℒ[f (n)(t)] = s n ℒ[f(t)] − [s n−1 f(0) + s n−2 f ′ (0) + ⋯ + sf (n−2) (0) + f (n−1) (0)]
Theorem 5. (Transform of Integral)

If f(t) is of exponential order and at least piecewise continuous, then

t
1 1 0
ℒ [∫ f(t)dt] = ℒ[f(t)] + ∫ f(t)dt
c s s c

Theorem 6. Theorem 7.
If ℒ[f(t)] = ϕ(s), then lim f(t)
If t→0+ t exists, and if ℒ[f(t)] = ϕ(s) ,
ℒ[t f(t)] = −ϕ′ (s) then
∞
f(t)
ℒ [ ] = ∫ ϕ(s)ds
t s

Inverse Laplace Transform :

If ℒ[f(t)] = F(s), then f(t) = ℒ −1 [F(s)]

Provided that ℒ[f(t)] exists.

Gamma Function :
∞
The gamma function denoted by Γ(n) is defined by ∫0 t n−1 e−t dt which is convergent for n > 0.
A recursion or recurrence formula for the gamma function is Γ(n + 1) = nΓ(n) = n!

Note:

1. If n < 0 but n ≠ −1, −2, −3, … 3. For n = 0, −1, −2, −3, …

Γ(n + 1) Γ(n) = ∞
Use Γ(n) =
n
1
2. Γ ( ) = √π
2

Sequence and Series :

Sequence of Numbers – defined as a succession Series – defined as the indicated sum of a

of numbers formed sequence of numbers.
according to some fixed rule

For example: nth term For example, for the sequence a1, a2, a3, …, an
the corresponding series is a1 + a2 + a3 + … + an
1. 2, 4, 6, 8, 10, 12, …, 2n, …
2. 1, 4, 9, 16, 25, …, n2, …

Type of Series :

1. Finite Series - the number of terms are 2. Infinite Series – the number of terms are
limited unlimited

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Power Series :

A power series in x – a has the form

+∞

∑ cn (x − a)n = c0 + c1 (x − a) + c2 (x − a)2 + ⋯ + cn (x − a)n + ⋯

n=0

When a = 0, and the series become a power in x, which is

+∞

∑ Cn x n = C0 + C1 x + C2 x 2 + ⋯ + Cn x n + ⋯
n=0

For example:

1
= 1 + x + x2 + x3 + ⋯
1−x
General Method: for expanding a function in a power series in x and in (x – a) is given below. Note the
requirement that the function and its derivative of all orders must exist at x = 0 or at
1
x = a. Thus, , ln x, and cot x cannot be expanded in power of x
x

Maclaurin’s Series :

- power series expansion of f(x) about x = 0.

- power series in x.

f ′′ (0) 2 f ′′′ (0) 3 f (n) (0) n

f(x) = f(0) + f ′ (0)x + x + x + ⋯+ x +⋯
2! 3! n!
or
+∞ (n)
f ( 0)
f(x) = ∑ xn
n!
n=0

Taylor’s Series :

- power series expansion of f(x) about x = a. This series, which includes Maclaurin’s series as
a special case (a = 0).
- power series in x – a

f ′′ (a) f ′′′ (a) f (n) (a)

f(x) = f(a) + f ′ (a)(x − a) + (x − a)2 + (x − a)3 + ⋯ + (x − a)n + ⋯
2! 2! n!

Fourier Series :

- is a series used to represent a periodic wave in either exponential or trigonometric form. The
trigonometric form is in terms of sine and cosine functions. The series has the form,

A0
F ( t) = + A1 cos t + A2 cos 2t + A3 cos 3t + ⋯ + An cos nt + B1 sin t + B2 sin 2t + B3 sin 3t + ⋯ + Bn sin nt
2
For a particular periodic wave, the coefficient of the series are determined by means of the
following equations:
1 2π 1 2π 1 2π
A0 = ∫ F(t)dt An = ∫ F(t) cos nt dt Bn = ∫ F(t) sin nt dt
π 0 π 0 π 0

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REE – Apr. 2017 / Aug. 2019
1. Simplify i^(39)
A. −i B. 1 C. −1 D. i

REE – Sep. 2011 / Sept. 2015 / Sept. 2018

2. If z1 = 1 – i, z2 = −2 + 4i, z3 = sq. rt. of 3 – 2i, evaluate Re { 2z13 + 3z22 – 5z32 }.
A. 35 B. 35i C. −35 D. −35i

REE – Sept. 2017

3. Give one indicated root of (2 square root of 3 – 2i)^1/2.
A. 2 cis 330° B. 2 cis 67.5° C. 2 cis 165° D. 2 cis 167.5°

REE – Apr. 2007

4. Solve for one value of x in x cubed – 8 = 0.
A. 1 + i sq. rt. of 3 B. −3 C. −2 D. −1 – i sq. rt. of 3

REE – Apr. 2002

5. Evaluate cosh (i pi/4).
A. 1.414214∠270 degrees C. 1.414214∠180 degrees
B. 0.707107∠0 degree D. 0.707107∠90 degrees

REE – Apr. 2001

6. Evaluate sinh (5 + j5).
A. 23.15 – j78.28 B. 21.05 – j71.16 C. 25.47 – j64.69 D. 19.14 – j86.11

REE – Aug. 2019

7. Obtain L{t^n}.
A. n! / s2(n – 1) B. n! / s^n C. (n +1)! / s^(n + 1) D. n! / s^(n + 1)

REE – Sept. 2012

8. Evaluate L { sint cost }.
A. ½ (s^2 + 4) B. 1 / (s^2 + 4) C. 1 / (s^2 + 1) D. ½ (s^2 + 1)

REE – Apr. 1997

2 1 −1 2
9. Matrix | | + 2 Matrix | | equals
−1 3 1 1
−2 4 −1 2 2 1 0 5
A. Matrix | | B. Matrix | | C. Matrix | | D. Matrix | |
2 2 1 1 −1 3 1 5
REE – Apr. 1997
3 1 2
10. Transpose the Matrix |−2 −1 0 |.
0 2 −1
−1 2 0 3 −2 0 3 1 2 1 3 2
A. | 0 −1 −2 | B. | 1 −1 2 | C. | 0 2 −1 | D. |−1 −2 0 |
2 1 3 2 0 −1 −2 −1 0 2 2 −1

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