ECE 316 ENGINEERING MATHEMATICS
PART I: LECTURE (1)
Ayman A. Arafa
ayman.arafa@ejust.edu.eg
Office: B8-F2-04
1
COURSE INFORMATION
Credit Hours: 3 Cr. Hrs.
Contact Hours: 2 Hrs. lecture and 2 Hrs. tutorial per week
Prerequisite: MTH 121 Mathematics (2) (Calculus + Linear Algebra)
Co-requisite: None
Attendance Policy: A student should attend more than 75% of the course.
(Max 3 absences)
GRADING POLICY
Assessment
Assignments
Quizzes in Tutorial
Two Quizzes in
Lecture
Mid-term Exam
Final Exam
Percentage
10%
10%
10%
Marks
30
30
30
30%
40%
90
120
ECE 316 ENG. MATHEMATICS
Part I: Special Functions
 Gamma function, Beta function, Series solution of differential equations with variable
coefficients, Bessel’s differential equation, series solution of Bessel’s differential equation,
Bessel function of order n of the second kind, recurrence formulas, Bessel functions of order
n of the third kind, or Hankel functions of order n, modified Bessel functions, Legendre’s
differential equation, Legendre coefficients
Part II: Laplace transform and its applications.
 Definition of Laplace transform, Laplace transform of basic functions, properties of Laplace
transform, inverse Laplace transform, applications of Laplace transform.
Part III: Vector Calculus
 Vector calculus: vector differential calculus, vector integral calculus
TEXT AND REFERENCE BOOKS
Text Book:
-1- “Advanced Engineering Mathematics,” Dennis G. Zill & Warren
S. Wright; 5th edition, Jones & Bartlett Learning, 2012
-2- “Differential Equations for Engineers,” Xie, W. C.; Cambridge
University Press, 2010.
Reference Book:
- “Advanced Engineering Mathematics with Matlab,” Dean G.
Duffy, CRC Press, Taylor and Francis Group, 3rd Ed., 2011.
OUTLINE
➢ Improper Integral
➢ Special Functions Defined by Integrals
➢
Gamma Function
➢
Beta Function
6
IMPROPER INTEGRAL
• Improper integrals are definite integrals that cover an unbounded area. For
example,
∞
1
න 2 𝑑𝑥
𝑥
1
• An improper integral is a definite integral that has either one or both limits infinite
or an integrand that approaches infinity at one or more points in the range of
integration.
• Hence improper integrals are of two types.
IMPROPER INTEGRAL TYPE-I:
(INFINITE LIMITS OF INTEGRATION)
• In this kind of integrals one or both limits of integration are infinity. For example,
∞
1
න 2 𝑑𝑥
𝑥
1
is an improper integral. It can be viewed as the limit
𝑏
1
lim න 2 𝑑𝑥
𝑏→∞
𝑥
1
IMPROPER INTEGRAL TYPE-II:
(DISCONTINUOUS INTEGRAND) OR
(INTEGRANDS WITH VERTICAL ASYMPTOTES)
• In this type of improper integrals the endpoints are finite, but the integrand function
is unbounded at one (or two) of the endpoints. For example,
1
1
න
𝑑𝑥
𝑥
0
is an improper integral. It can be viewed as the limit
1
1
lim+ න
𝑑𝑥
𝑎→0
𝑥
𝑎
REMARKS
• Not all improper integrals have a finite value, but some of them
definitely do.
• When the limit exists we say the integral is convergent, and when
it does not we say it is divergent.
GAMMA FUNCTION
The Gamma function, denoted by Γ 𝑛 , is a defined by the improper integral
∞
Γ 𝑛 = න 𝑒 −𝑥 𝑥 𝑛−1 𝑑𝑥
0
The integral is convergent for 𝑛 > 0.
Example: Γ 2 =
∞
‫׬‬0 𝑒 −𝑥 𝑥 𝑑𝑥,
Γ
1
2
=
∞ −𝑥 −1
‫׬‬0 𝑒 𝑥 2 𝑑𝑥
The Gamma function has numerous properties that will be covered in the next slides
PROPERTIES OF GAMMA FUNCTION
Property 1:
Γ 1 =1
Property 2:
Γ 𝑛 + 1 = 𝑛Γ 𝑛 ,
𝑛>0
Property 3:
Γ 𝑛 + 1 = 𝑛!
Property 4:
If 𝑛 is Positive Integer
∞
Γ 𝑛 = 2න
0
2 2𝑛−1
−𝑡
𝑒 𝑡
𝑑𝑡
PROPERTIES OF GAMMA FUNCTION
Property 5:
1
Γ
= 𝜋
2
Property 6:
Property 7:
𝜋
Γ 𝑛 Γ 1−𝑛 =
sin 𝜋𝑛
1
1
Γ 𝑛 = න ln
𝑦
0
𝑛−1
𝑑𝑦
PROPERTIES OF GAMMA FUNCTION
Property 8:
∞
Γ 𝑛 = 𝛼 𝑛 න 𝑒 −𝛼𝑥 𝑥 𝑛−1 𝑑𝑥
0
Property 9:
∞
Γ 𝑛+1 =න𝑒
1
−𝑦 𝑛
𝑑𝑦
0
Property 10:
Γ 𝑛 = ∞, if n is Negative Integer
PROPERTIES OF GAMMA FUNCTION
Property 11: Legendre Duplication Formula
22𝑛−1
1
Γ 2𝑛 =
Γ 𝑛 Γ 𝑛+
2
𝜋
If 𝑛 is a positive integer, then Γ 𝑛 +
1
2
=
2𝑛 !
22𝑛 𝑛!
𝜋
THE GRAPH OF GAMMA FUNCTION
EXAMPLES

e
(1) Show that
−x4
dx =
0
1 1
 
4  4
1 − 34
Let x = y  4 x dx = dy  dx = y dy
4
4
3

  e −x
0

4
1 − y − 34
dx =  e y dy
4 0
=
1 1
 , using the definition.
4 4
EXAMPLES
1
(2)
Find
 (x ln
x )5 dx
0
using ln x = -y, we get x = e - y , dx = -e - y dy
and y ranges from  to 0 when x = 0 to 1


0
0
  (x ln x )5 dx = −  y 5e − 6 y dy,
interchanging the lower and the upper limits

5
t
dt
=  e− t
,
6
6
0
=−
1
5!
(
)

6
=
−
66
66
choosing 6y = t
PROBLEMS
Evaluate the following in the simplified
∞ exact−form.
𝑥
6.
𝑥
𝑒
𝑑𝑥
‫׬‬
3Γ(6)
5
0
1.
×Γ
∞ ln 𝑥 3
Γ 4
2
7. ‫׬‬1
𝑑𝑥
∞ 7 −𝑥
2
𝑥
2.‫׬‬0 𝑥 𝑒 𝑑𝑥
1
1 𝛼−1
∞ 4 1−𝑥
8. ‫׬‬0 ln
𝑑𝑥
3. 2 ‫𝑥𝑑 𝑒 𝑥 ׬‬
4.
5.
0
∞ 3 −4𝑥
𝑑𝑥
‫׬‬0 𝑥 𝑒
𝑥2
∞ 3 −
‫׬‬0 𝑥 𝑒 2 𝑑𝑥
𝑥
−5
9. Γ
2
101
10. Γ
2
PROBLEMS
By using techniques involving the Gamma function, show that
1
න ln 𝑥 𝑛 𝑑𝑥 = −1 𝑛 𝑛!,
𝑛∈ℕ
0
By using techniques involving the Gamma function, show that
∞ −𝑘
𝑒 𝑥2
න
0
𝑥6
𝑑𝑥 =
3 𝜋
5
8𝑘 2
,
𝑘≠0
BETA FUNCTION
The Beta function, denoted by 𝛽 𝑝, 𝑞 , can be expressed as a definite integral with 0 and 1 as
limits, and is given by
1
𝛽 𝑝, 𝑞 = න 𝑥 𝑝−1 1 − 𝑥
0
The integral is convergent for 𝑝, 𝑞 > 0
𝑞−1 d𝑥
PROPERTIES OF BETA FUNCTION
Property 1: Symmetry
𝛽 𝑝, 𝑞 = 𝛽 𝑞, 𝑝
Property 2:
𝜋
2
𝛽 𝑝, 𝑞 = 2 න sin2𝑝−1 𝜃 cos 2𝑞−1 𝜃 d𝜃
0
Property 3: Relation with Gamma Function
Γ 𝑝 Γ 𝑞
𝛽 𝑝, 𝑞 =
Γ 𝑝+𝑞
PROPERTIES OF BETA FUNCTION
Property 4:
∞
𝑦 𝑞−1
𝛽 𝑝, 𝑞 = න
d𝑦
𝑝+𝑞
1+𝑦
0
Property 5:
𝛽 𝑝 + 1, 𝑞
𝛽 𝑝, 𝑞 + 1
𝛽 𝑝, 𝑞
=
=
𝑝
𝑞
𝑝+𝑞
EXAMPLES
(1)
1 5𝑥
𝑑𝑥
‫׬‬0
5
1−𝑥
2 1
,
5 2
=𝛽
1 −45
Letting x = t, we get x = t , dx = t dt
5
and t = 0 to 1 when x = 0 to 1
1
5
1
∴න
0
1
5𝑥
1−
5
𝑥5
𝑑𝑥 =
1ൗ
න 5𝑡 5
1−𝑡
0
1
= t
0
−3
5
(1 − t )
−1
2
1 −4ൗ
−1ൗ2
𝑡 5 𝑑𝑡
5
2 1
dt = 𝛽 ,
5 2
by definition.
EXAMPLES
(2) Evaluate
πൗ
2
න
0
πൗ
2
න
tanθdθ
0
πൗ
2
tanθdθ = න
1ൗ
−1ൗ2
2
sin θcos
θdθ
0
3
1
Γ
Γ
3 1
𝜋
4
4
𝛽 ,
=
=
= 2𝜋
𝜋
4 4
Γ 1
sin
4
1 3 1
= β , ,
2 4 4
by definition
PROBLEMS
Evaluate the following in the simplified exact form.
1.
1
‫׬‬0 7 𝑥 5
𝜋
2
2. ‫׬‬0
3.
1−𝑥
4
d𝑥
sin5 𝜃 cos4 𝜃 d𝜃
1 4
‫׬‬0 𝑥
1 − 𝑥 2 d𝑥
𝜋
2
4. ‫׬‬0
5.
6.
sin 𝑥 d𝑥
4 𝑥3
‫׬‬0 4−𝑥 d𝑥
𝜋
𝜃
𝜃
5
2
‫׬‬0 sin 2 cos 2
d𝜃
PROBLEMS
By using techniques involving the Beta function, show that
1
න 𝑥 𝑞−1
1−
𝑥 𝑛 𝑝−1 𝑑𝑥
1 𝑞
= 𝛽 ,𝑝 ,
𝑛 𝑛
0
By using techniques involving the Beta function, show that
1
2
න 1 − 𝑥 𝑑𝑥 =
(𝑛 + 2)(𝑛 + 1)
𝑛
0
𝑛≠0