advmath_lecturenotes

7/30/2019 AdvMath_LectureNotes

1/9

ADVANCED MATHEMATICSLECTURE NOTES

ADVANCED ENGINEERING MATHEMATICS

COMPLEX NUMBERS

Complex Number in Rectangular Form

biaz +=

where:z : complex variablea : real partb : imaginary part

i : imaginary unit, 1i =

Notes:

1. If 0b = , then bia+ is equivalent to the real number a.2. If 0a = and 0b , then bia+ is equivalent to the pure imaginary number bi.

Powers of Imaginary Unit

1i = 1i2 = ii3 = 1i4 =

Complex Conjugate

biaz =

where:

z : complex conjugate

Operations with Complex Numbers

OperationsAddition i)db()ca()dic()bia( +++=+++

Subtraction db()ca()dic()bia( +=++

Multiplication ad()bdac()dic)(bia( ++=++

Division

22dc

i)adbc()bdac(

dic

bia

+

++=

+

+

Operations on Complex Conjugate NumbersLet biaz += and biaz =OperationsAddition a2)bia()bia(zz =++=+

Subtraction bi2)bia()bia(zz =+=

Multiplication 22 ba)bia()bia(zz +=+=

jkl, ece


2/9


Division

22

22

ba

babi2a

bia

bia

z

z

+

+=

+=

Equality of Two Complex Numbers

dicbia +=+ iff ca = and db =

Argand Diagram

22 baz +=a

btan 1=

where:z : modulus or absolute value

: argument or amplitude

Forms of Complex Numbers1. Trigonometric Form

=+= rcis)sini(cosrz

where:

22 bar +=r

acos =

r

bsin =

2. Polar Form

= rz

where:22 bar +=

: argument or amplitude (degrees)

a

btan 1=

3. Exponential Form= irez

where:22

bar+=

: argument or amplitude (radians)

4. Logarithmic Form

+= irlnzln

Properties of Complex NumbersLet z and w represent two complex numbers.

jkl, ece


3/9


1. wzzw = 2.w

z

w

z= , 0x 3. wzwz ++

4. wzwz + 5. wzwz +=+ 6. wzzw =

7.w

z

w

z =

Operations on Complex Numbers in Polar Form

Multiplication of Complex Numbers Division of Complex Numbers

)(rr)r)(r( 21212211 += )(

r

r

r

r21

2

1

22

11 =

Powers of Complex Number Roots of Complex Number

)n(r)r(nn

=n

)360(krr nn

+=

where: where:n : exponent n : nth root

1nk = , (0, 1, 2, )

Exponential, Trigonometric and Hyperbolic Functions of a Complex Number

+= sinicosei

=

sinicosei

2

eecos

ii +

=i2

eesin

ii =

2

eecosh

+

=2

eesinh

=

Inverse Trigonometric and Hyperbolic Functions of a Complex Number

= 21 z1izlnizsin

= 1zzlnizcos 21

+

=iz1

iz1ln

2

iztan 1

+= 1zzlnzsinh 21

= 1zzlnzcosh 21

+

=z1

z1ln

2

1ztanh

1

MATRICESMatrixis a rectangular array of elements arranged in m rows and n columns.

jkl, ece


4/9


mn4m3m2m1m

n224232221

n14131211

a...aaaa

.....

.....

a...aaaa

a...aaaa

A =

where:aij : elements of the matrixm : number of rowsn : number of columns

A Submatrix of A is the matrix with certain rows and columns removed.

Notes:1. The size of a matrix is determined by the number of rows and columns.2. The expression m x n is the dimension or order of the matrix.

Properties of Matrices

Let A, B, and C are m x n matrices and c and d are scalarsMatrix Property

Commutative Property of Addition ABBA +=+Associative Property of Addition C)BA()CB(A ++=++Associative Property of Multiplication C)AB()BC(A =

Associative Property of Scalar Multiplication )dA(cA)cd( =

kB(A)AB(kB)kA( ==

Scalar Identity AA1 =Distributive Property

dAcAA)dc(

cBcA)BA(c

+=++=+

CBCA)BA(C

BCACC)BA(

+=+

+=+

Transpose of a MatrixTranspose of a Matrix(AT) is the matrix resulting from an interchange of rows and columns of a

given matrix A.

Properties of Transpose of a Matrix Operation:

jkl, ece


5/9


Let A and B are m x n matrices and k is scalar.

1. A)A( TT = 2. TT )A(k)kA( = 3. IIT = 4.TTT BA)BA( +=+

5. AAT =

Minor of a Matrix

The ijM of the element ija of a matrix A in the ith row and the jth column is the determinant of

the matrix obtained by deleting the ith row and jth column of A.

Cofactor of a Matrix

The cofactor ijC of the entry ija is the signed minor by ijji

ij M)1(C+

= .

Inverse of a Matrix

If the determinant of a matrix A is not equal to zero, then the inverse of matrix A is defined by

A

adjAA 1 = .

Property of Inverse Matrix:

IAAAA 11 ==

where:I : identity matrix

Note:A matrix has an inverse that is; it is nonsingular, if its determinant is nonzero.

Types of Matrices:1. Square Matrix

A Square Matrixis a matrix with the same number of rows and columns )nm( = .

2. Identity or Unit MatrixIdentity or Unit Matrixis a square matrix in which all the elements in the leading diagonal are 1

and the remaining all elements are zero.

3. Diagonal MatrixDiagonal Matrixis a square matrix in which all the elements except those in the main diagonal

are zero.

4. Scalar MatrixScalar Matrixis a diagonal matrix in which all the elements in the main diagonal are equal to

some scalar except 0 or 1.

5. Symmetric MatrixSymmetric Matrixis a square matrix whose transpose is equal to itself.

6. Skew Symmetric Matrix

jkl, ece


6/9


Skew Symmetric Matrixis a square matrix whose transpose is equal to the negative of itself.

7. Row MatrixRow Matrixis a matrix having a single row.

8. Column MatrixColumn Matrixis a matrix having a single column.

9. Null MatrixNull Matrixis a matrix whose all elements are zero.

10. Singular MatrixSingular Matrixis a matrix whose determinant is zero.

11. Triangular MatrixTriangular Matrixis a matrix with zeros in all entries above or below the diagonal.

a. Lower Triangular Matrix

Lower Triangular Matrixis a matrix with all the elements above the diagonal are zeros.b. Upper Triangular MatrixUpper Triangular Matrixis a matrix with all the elements below the diagonal are zeros.

Operations with Matrices:1. Matrix Addition

The sum of two matrices is obtained by adding their corresponding entries or elements. Thematrices to be added must be of the same order.

2. Matrix SubtractionMatrix subtraction is analogously defined as matrix addition. The difference of two matrices is

obtained by subtracting their corresponding entries or elements. The matrices to be subtracted mustbe of the same order.

3. Matrix MultiplicationThe process of matrix multiplication is, conveniently referred to as row-by-column multiplication.

This definition requires that the number of columns of A be the same as the number of rows of B. Inthat case, the matrices A and B are said to be conformable. Otherwise the product is undefined.

nm

A

pn

B

pm

AB

=

equalorder of AB

4. Matrix DivisionDivision of matrices is done by multiplying the numerator by the inverse matrix of thedenominator.

DETERMINANTThe Determinantof a matrix is a number assigned to every square matrix.

Properties:

jkl, ece


7/9


1. If the rows of one matrix are the same as the columns of another, and in the same order, the twodeterminants are equal.

2. If two columns (or rows) of a matrix are interchanged, the value of the resulting determinant isequal to the negative of the value of the given determinant.

3. If two columns (or rows) of a matrix are identical or proportional, the value of the determinant iszero.

4. If the elements of a column (or row) of a matrix are multiplied by k, the value of the determinant ismultiplied by k.

5. If the matrix has a row or column whose entries are all zeros, the determinant is zero.6. If the corresponding rows or columns of a matrix are interchanged, its determinant is unchanged.7. The determinant of a product of two matrices is the product of their determinants.8. The determinant of a triangular matrix is the product of the elements on the main diagonal.9. The value of the determinant is not changed if a column of a matrix is replaced by the column

plus a multiple of another column. Similarly for rows.

LAPLACE TRANSFORM

==0

stdte)t(f)]t(f[L)s(F

where:s : any number (real or complex)

0t >

Laplace Transform of Some Elementary Functions

f(t) F(s) f(t) F(s) f(t) F(s)

1

s

1 ktsine at

22 k)as(

k

+

atcosh

22as

s

t

2s

1 ktcose at

22 k)as(

as

+

atsint222 )as(

as2

+

tn

1ns

!n+

atsin22

as

a

+

atcost

222

22

)as(

as

+

ate

as

1

atcos

22as

s

+

)tsin( +

22 ks

cosksins

+

+

atnet

1n)as(

!n+

atsinh 22 as

a

)tcos( +

22 ks

cosksins

+

ECE BOARD PROBLEMS01. ECE Board November 1992

The following Fourier series equation represents a periodic __________ wave.

...x3sinix2sinixsini...x3cosix2cosixcosii)x(i 3232 ++++++++=A. cosine B. cotangent C. sine D. tangent

jkl, ece


8/9


Answer:

02. ECE Board April 1993

Find the product MN of the following matrices:

fed

cbaM = ,

lk

ji

hg

N =

A.

)flejdh()fkeidg()clbjsh()ckbiag(

++++

++++C.

)fkeidg()ckbiag(

++

++

B.

)fl()ej()dh(

)ck()bi()ag(D.

)flejdh(

)clbjah(

++

++

Solution:

)flejdh()fkeidg(

)clbjah()ckbiag(MN

++++

++++

=

Answer:

03. ECE Board April 1994The scalar product of A and B is equal to the product of the magnitudes of A and B and the __________ of theangle between them.

A. cosine B. sine C. tangent D. value in radians

jkl, ece


9/9


Answer:

04. ECE Board November 1995 / ECE Board November 1997In Complex Algebra, we use a diagram to represent a complex plane commonly called __________.

A. Argand diagram B. De Moivres diagram C. Funicular diagram D. Venn diagram

Answer:

05. ECE Board March 1996When the corresponding elements of two rows of a determinant are proportional, then the value of thedeterminant is __________.

A. multiplied by the ratio B. one C. unknown D. zero

Answer:

06. ECE Board November 1996In any square matrix, when the elements of any two rows are exactly the same, the determinant is

__________.A. negative integer B. positive integer C. unity D. zero

Answer:

07. ECE Board April 2000

Solve for x in i814)i42)(yix( +=++ .A. 3 B. 4 C. 8 D. 14

Solution:

i814i)y2x4()y4x2(

i814y)1(4i)y2x4(x2

i814yi4iy2ix4x2

i814)i42)(yix(

2

+=++

+=+++

+=+++

+=++

By Inspection:

14y4x2 = or 7y2x = eq1

i8i)y2x4( =+ or 4yx2 =+ eq2

eq1 and eq2

3x

15x58y2x4)(

7y2x

24yx2

7y2x

=

=

=++

=

=+

=

Answer:

jkl, ece

advmath_lecturenotes

Documents