1. The document discusses functions of complex variables, including analytic functions, Cauchy-Riemann equations, harmonic functions, and methods for determining an analytic function when its real or imaginary part is known.
2. Some key topics covered are the definition of an analytic function, Cauchy-Riemann equations in Cartesian and polar forms, properties of analytic functions including orthogonal systems, and determining the analytic function using methods like direct, Milne-Thomson's, and exact differential equations.
3. Examples are provided to illustrate determining the analytic function given its real or imaginary part, such as finding the function when the real part is a polynomial or the imaginary part is a trigonometric function.
It includes all the basics of calculus. It also includes all the formulas of derivatives and how to carry it out. It also includes function definition and different types of function along with relation.
Jacobi Iteration Method is Used in Numerical Analysis. This slide helps you to figure out the use of the Jacobi Iteration Method to submit your presentatio9n slide for academic use.
Subject Title: Engineering Numerical Analysis
Subject Code: ID-302
Contents of this chapter:
Mathematical preliminaries,
Solution of equations in one variable,
Interpolation and polynomial Approximation,
Numerical differentiation and integration,
Initial value problems for ordinary differential equations,
Direct methods for solving linear systems,
Iterative techniques in Matrix algebra,
Solution of non-linear equations.
Approximation theory;
Eigen values and vector;
It includes all the basics of calculus. It also includes all the formulas of derivatives and how to carry it out. It also includes function definition and different types of function along with relation.
Jacobi Iteration Method is Used in Numerical Analysis. This slide helps you to figure out the use of the Jacobi Iteration Method to submit your presentatio9n slide for academic use.
Subject Title: Engineering Numerical Analysis
Subject Code: ID-302
Contents of this chapter:
Mathematical preliminaries,
Solution of equations in one variable,
Interpolation and polynomial Approximation,
Numerical differentiation and integration,
Initial value problems for ordinary differential equations,
Direct methods for solving linear systems,
Iterative techniques in Matrix algebra,
Solution of non-linear equations.
Approximation theory;
Eigen values and vector;
Spline interpolation is a problem of "Numerical Methods".
This slide covers the basics of spline interpolation mostly the linear spline and cubic spline interpolation.
What is numerical differentiation?
What is finite difference?
How to apply that to boundary value problems?
#WikiCourses #Num001
https://wikicourses.wikispaces.com/Topic+Boundary+Value+Problems+-+Finite+Difference
Newton's Backward Interpolation explained with example. History of interpolation along with it's advantages and disadvantages. Applications of interpolation in computer sciences.
How to handle Initial Value Problems using numerical techniques?
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Initial+Value+Problems
https://eau-esa.wikispaces.com/Topic+Initial+Value+Problems
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Maths is aan acknowledge for all humanity's stast of technology and many resources now in all country maths is the father of technology is a wondering..maths comes into play a vital role in all form of chemistry physics and all other subjects...in our day to day life maths playa a role everything and everywhere
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Spline interpolation is a problem of "Numerical Methods".
This slide covers the basics of spline interpolation mostly the linear spline and cubic spline interpolation.
What is numerical differentiation?
What is finite difference?
How to apply that to boundary value problems?
#WikiCourses #Num001
https://wikicourses.wikispaces.com/Topic+Boundary+Value+Problems+-+Finite+Difference
Newton's Backward Interpolation explained with example. History of interpolation along with it's advantages and disadvantages. Applications of interpolation in computer sciences.
How to handle Initial Value Problems using numerical techniques?
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Initial+Value+Problems
https://eau-esa.wikispaces.com/Topic+Initial+Value+Problems
Double integral using polar coordinatesHarishRagav10
Maths is aan acknowledge for all humanity's stast of technology and many resources now in all country maths is the father of technology is a wondering..maths comes into play a vital role in all form of chemistry physics and all other subjects...in our day to day life maths playa a role everything and everywhere
TIU CET Review Math Session 4 Coordinate Geometryyoungeinstein
College Entrance Test Review
Math Session 4 Coordinate Geometry
Formulas for the Slope of a line, Midpoint, Distance between any two points,
Equations of a Line
Analytic Function, C-R equation, Harmonic function, laplace equation, Construction of analytic function, Critical point, Invariant point , Bilinear Transformation
This powerpoint presentation gives information regarding functions. Designed or grade 11 studens studying general mathematics 11. You can use this presentation to present your lessons in grade 11 general mathematics or even use this on your lesson in grade 10 mathematics about polynomial functions
Data Approximation in Mathematical Modelling Regression Analysis and Curve Fi...Dr.Summiya Parveen
Outline of the lecture:
Introduction of Regression
Application of Regression
Regression Techniques
Types of Regression
Goodness of fit
MATLAB/MATHEMATICA implementation with some example
Regression analysis is a form of predictive modelling technique which investigates the relationship between a dependent (target) and independent variable (s) (predictor). This technique is used for forecasting, time series modelling and finding the casual effect relationship between the variables. Regression analysis is an important tool for modelling and analysing data. Here, we fit a curve / line to the data points in such a manner that the differences between the distances of data points from the curve or line is minimized.
By DR. SUMMIYA PARVEEN
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Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
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Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
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Content
Functions of a complex variable, Application of complex variable, Complex algebra, Analytic
function, C-R equations, Theorem on C-R equation(without proof) ,C-R equation in polar form,
Properties of Analytic Functions (orthogonal system), Laplace Equation, Harmonic Functions,
Determination of Analytic function Whose real or Imaginary part is known :
(1) When u is given, v can be determined
(2) When v is given, u can be determined
(3) By Milne-Thompson method and Finding Harmonic Conjugate functions, Conformal
Mapping and its applications ,Define conformal Mapping, Some standard conformal
transformations:
(1) Translation
(2) Rotation and Magnification
(3) Inversion and Reflection
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1.1 Introduction— Functions of a complex variable provide us some powerful and widely
useful tools in mathematical analysis as well as in theoretical physics.
1.2 Applications—
1. Some important physical quantities are complex variables (the wave-function , a.c.
impedance Z()).
2. Evaluating definite integrals.
3. Obtaining asymptotic solutions of differentials equations.
4. Integral transforms.
5. Many Physical quantities that were originally real become complex as simple theory is made
more general. The energy En En
0
+ i (-1
the finite life time).
1.3 Complex Algebra—A complex number z(x, y) = x + iy . Where i = √−1.
x is called the real part, labeled by Re z
y is called the imaginary part, labeled by Im z
Different forms of complex number—
1. Cartesian form— ( , ) = +
2. Polar form— z(r, ) = r(cos + i sin)
3. Euler form— z(r, ) = re
Here, = + and = tan ( )
The choice of polar representation or Cartesian representation is a matter of convenience.
Addition and subtraction of complex variables are easier in the Cartesian representation.
Multiplication, division, powers, roots are easier to handle in polar form.
1.4 Derivative of the function of complex variable—
The derivative of ( ) is defined by—
( ) =
→
= lim
→
( + ) − ( )
If = + and ( ) = + ,
let consider that—
yixz ,
viuf ,
yix
viu
z
f
Assuming the partial derivatives exist. Let us take limit by the two different approaches as in
the figure. First, with y = 0, we let x0,
x
v
i
x
u
z
f
xz
00
limlim
x
v
i
x
u
For a second approach, we set x = 0 and then let y 0. This leads to –
y
v
y
u
i
y
v
i
y
u
z
f
yz
00
limlim
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If we have the derivative, the above two results must be equal. Hence—
y
v
y
u
i
x
v
i
x
u
Hence, the necessary and sufficient conditions for the derivative of the function ( ) = +
to exit for all values of z in a region R, are—
(1) , , , are continuous functions of and in R;
(2)
y
v
x
u
and
x
v
y
u
.
1.5 Analytic function— A single valued function that possesses a unique derivative with
respect to at all points on a region R is called an Analytic function of in that region.
Necessary and sufficient condition to be a function Analytic— A function ( ) = + is
called an analytic function if—
1. ( ) = + , such that and are real single valued functions of and and
, , , are continuous functions of and in R.
2.
y
v
x
u
and
x
v
y
u
.
1.6 Cauchy-Riemann equations— If a function ( ) = + is analytic in region R, then it
must satisfies the relation
y
v
x
u
and
x
v
y
u
. These equations are called as Cauchy-
Riemann equation.
Cauchy-Riemann equation in polar form—
v
r
u
r and
r
vu
r
1
1.7 Laplace’s equation— An equation in two variables and given by + = 0 is
called as Laplace’s equation in two variables.
1.8 Harmonic functions— If ( ) = + be an analytic function in some region R, then
Cauchy-Riemann equations are satisfied. That implies—
y
v
x
u
... (1)
x
v
y
u
... (2)
Now, on differentiating (1) with respect to x and (2) with respect to y—
= … (3)
= − … (4)
Assuming that = , adding equations (1) and (2)—
+ =
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Similarly, on differentiating (1) with respect to and (2) with respect to and subtracting
them—
+ =
both and satisfy the Laplace’s equation in two variables, therefore ( ) = + is called
as Harmonic function. This theory is known as Potential theory.
If ( ) = + is an analytic function in which ( , ) is harmonic, then ( , )is called as
Harmonic conjugate of ( , ).
1.9 Orthogonal system— Consider the two families of curves ( , ) = … (1) and
( , ) = … (2).
Differentiating (1) with respect to –
= − = = [by using Cauchy s Riemann equations for analytic function]
Differentiating (2) with respect to –
= − =
Here, = −1 therefore every analytic function ( ) = ( , ) + ( , ) represents two
families of curves ( , ) = and ( , ) = form an orthogonal system.
1.10 Determination of Analytic function whose real or imaginary part is known—
If the real or the imaginary part of any analytic function is given, other part can be determined
by using the following methods—
(a) Direct Method
(b) Milne-Thomson’s Method
(c) Exact Differential equation method
(d) Shortcut Method
1.10.1 Direct Method—
Let ( ) = + is an analytic function. If is given, then we can find in following steps—
Step-I Find by using C-R equation =
Step-II Integrating with respect to to find with taking integrating constant ( )
Step-III Differentiate (from step-II) with respect to y. Evaluate containing ( )
Step-IV Find by using C-R equation = −
Step-V by comparing the result of from step-III and step-IV, evaluate ( )
Step-VI Integrate ( ) and evaluate ( )
Step-VII Substitute the value of ( ) in step-II and evaluate .
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Example— If ( ) = + represents the analytic complex potential for an electric field
and = − + , determine the function .
Solution— Given that –
= − +
Hence— = 2 +
( )( ) ( )
( )
= 2 + ( )
… (1)
Again, = −2 + ( )
… (2)
So, and must satisfy the Cauchy’s-Riemann equations:
= … (3) and = − … (4)
Now, from equations (2) and (3)—
= −2 + ( )
On integrating with respect to —
= −2 ∫ − ∫ ( )
= −2 + + ( )
Differentiating with respect to —
= −2 − ( )
+ ( ) … (5)
Again, from equations (1) and (4)—
= −2 − ( )
… (6)
On comparing (5) and (6)—
( ) = 0 , so ( ) =
Hence, = −2 + + ans
1.10.2 Milne-Thomson’s Method—
Let a complex variable function is given by ( ) = ( , ) + ( , ) … (1)
Where— = + and =
̅
and =
̅
.
Now, on writing ( ) = ( , ) + ( , ) in terms of and ̅ –
( ) =
̅
,
̅
+
̅
,
̅
Considering this as a formal identity in the two variables and , and substituting = ̅ –
∴ ( ) = ( , 0) + ( , 0) … (2)
Equation (2) is same as the equation (1), if we replace by and by 0.
Therefore, to express any function in terms of , replace by and by 0. This is an
elegant method of finding ( ) , ‘when its real part or imaginary part is given’ and this
method is known as Milne-Thomson’s Method.
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Example— If ( ) = + represents the complex potential for an electric field
and = − + , determine the function . (By using Milne-Thomson’s method)
Solution— Given that— ( ) = +
∴ = + = + [by using cauchy − Riemann equation]
Hence, ∴ = + = −2 + ( )
+ 2 + ( )
by using Milne-Thomson’s method (replacing by z and by 0)—
= (2 −
1
)
( ) = + +
Hence, = Re + + = Re − + (2 ) + − +
= −2 + + ans
Example— Find the analytic function ( ) = + , whose real part is − + −
.
Solution— Let ( ) = + is an analytic function. Hence, from the question—
= − 3 + 3 − 3
Now, = + = − [by using cauchy − Riemann equation]
Hence, ∴ = − = (3 − 3 + 6 ) + (−6 − 6 )
by using Milne-Thomson’s method (replacing by z and by 0)—
= (3 + 6 ) + (0)
( ) = + 3 +
Hence, = Im[ + 3 + ] = Im[ + 3 − 3 − + 3 − 3 + 6 + ]
∴ = 3 − +
Therefore ( ) = ( − 3 + 3 − 3 ) + (3 − + ) ans
Example— Find the analytic function = + , if − = ( − )( + + ).
Solution— Given that—
− = ( − )( + 4 + )
− = 3 + 6 − 3 … (1)
− = 3 − 6 − 3 … (2)
On applying Cauchy-Riemann’s theorem in equation (2)—
− − = 3 − 6 − 3 … (3)
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Now, from equations (1) and (3)—
= 6 and = −3 + 3
Again, = + = 6 + (−3 + 3 )
By using Milne-Thomson’s method—
= (−3 )
( ) = − +
= − ( + 3 − 3 − ) +
= (3 − + ) + (3 − ) ans
1.10.3 Exact Differential equation Method—
Let ( ) = + is an analytic function.
Case-I If ( , ) is given
We know that—
= +
= − + [By C-R equations]
Let = + , therefore = − and =
Again, = − and =
Therefore − = − + = 0 [∵ is a harmonic function]
∴ = [Satisfies the condition for
exact differential equation]
Hence,
= − + terms of not containg +
Case-II If ( , ) is given
We know that—
= +
= − [By C-R equations]
Let = + , therefore = and = −
Again, = and = −
Therefore − = + = 0 [∵ is a harmonic function]
∴ = [Satisfies the condition for exact differential equation]
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Hence,
= + terms of − not containg +
Example— If = − + + , determine for which ( ) is an analytic function.
Solution— Given that— = − 3 + 3 + 1
By using exact differential equation method, can be written as—
= ∫ − + ∫ terms of not containg +
= ∫ (6 ) + ∫(−3 + 3) +
= 3 − + 3 +
Example— If = − , determine for which ( ) = + is an analytic function.
Solution— Given that— = − 3
By using exact differential equation method, can be written as—
= ∫ − + ∫ terms of not containg +
= ∫ (−3 + 3 ) + ∫(0) +
= −3 + +
1.10.4 Shortcut Method
Case-I If ( , ) is given
If the real part of an analytic function ( ) is given then—
( ) = 2
2
,
2
− (0,0) +
Where, c is a real constant.
Case-II If ( , ) is given
If the imaginary part of an analytic function ( ) is given then—
( ) = 2
2
,
2
− (0,0) +
Where, c is a real constant.
Example— If ( ) = + is an analytic function and ( , ) = , find ( ).
Solution— Given that ( ) is an analytic function and it’s real part is ( , ) = .
Hence,
( ) = 2 , − (0,0) +
( ) = 2 +
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0
( ) = 2 ( )
+ [cosh(− ) = cosh ]
( ) = 2 ( )
+ [cosh( ) = cos ]
( ) = 2 +
( ) = tan +
Example— Determine the analytic function ( ), whose imaginary part is − .
Solution— Given that = 3 − , since the complex variable function ( ) is analytic,
hence it is given by—
( ) = 2
2
,
2
− (0,0) +
( ) = 2 3
2 2
−
2
+
( ) = 2
3
8
+
1
8
+
( ) = 2
1
2
+
( ) = +
Example— Determine the analytic function ( ), whose imaginary part is ( + ) +
− .
Solution— Given that = log( + ) + − 2
= + 1
= − 2
Let ( ) = + , ∴ = + = + [By using C − R equations]
Hence, ( ) = −2 + + 1
( ) = −2 + 2 log + +
( ) = 2 log + ( − 2) +
Example— Find the orthogonal trajectories of the family of curves given by the equation—
− = .
Solution— given curve is—
= − =
The orthogonal trajectories to the family of curves = will be = such that
( ) = + becomes analytic. Hence, orthogonal trajectory = is given by—
= − + +
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1
= (– + 3 ) + (− ) +
= −
4
+
3
2
−
4
+
Example— Find the orthogonal trajectories of the family of curves given by the equation—
− = .
Solution— given curve is—
= cos − =
The orthogonal trajectories to the family of curves = will be = such that
( ) = + becomes analytic. Hence, orthogonal trajectory = is given by—
= − + +
= ( sin − ) + (− ) +
= sin −
2
−
2
+
Example— Show that the function ( , ) = is harmonic. Determine its harmonic
conjugate ( , ) and the analytic function ( ) = + .
Solution— given that ( , ) = cos
Now, = cos , = cos
= − sin , = − cos
Here, + = 0, so ( , ) is a harmonic function.
Again, since ( ) = + is an analytic function, so ( ) is given by—
( ) = 2
2
,
2
− (0,0) +
( ) = 2 cos
2
− 1 +
( ) = 2 cos
−
2
− 1 +
( ) = 2 cos
2
− 1 +
( ) = 2
1
2
+ − 1 +
( ) = ( + 1) − 1 +
( ) = +
Now, ( ) = +
( ) = +
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2
( ) = (cos + sin ) +
( ) = + ( + )
Hence, ( , ) = sin +
1.11 Conformal Mapping—
1.11.1 Mapping—
Mapping is a mathematical technique used to convert (or map) one mathematical problem
and its solution into another. It involves the study of complex variables.
Let a complex variable function = + define in - plane have to convert (or map) in
another complex variable function ( ) = = + define in - plane. This process is
called as Mapping.
1.11.2 Conformal Mapping—
Conformal Mapping is a mathematical technique used to convert (or map) one mathematical
problem and its solution into another preserving both angles and shape of infinitesimal small
figures but not necessarily their size.
The process of Mapping in which a complex variable function = + define in - plane
mapped to another complex variable function ( ) = = + define in - plane
preserving the angles between the curves both in magnitude and sense is called as conformal
mapping.
The necessary condition for conformal mapping— if = ( ) represents a conformal
mapping of a domain D in the −plane into a domain D of the −plane then ( ) is an
analytic function in domain D.
Application of Conformal mapping— A large number of problems arise in fluid mechanics,
electrostatics, heat conduction and many other physical situations.
There are different types of conformal mapping. Some standard conformal mapping are
mentioned below—
1. Translation
2. Rotation
3. Magnification
4. Inversion
5. Reflection
Mapping
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1.11.2.1 Translation— A conformal mapping in which every point in −plane is translated in
the direction of a given vector is known as Translation.
Let a complex function = + translated in the direction of the vector = + , then
that translation is given by— = + .
Example— Determine and sketch the image of | | = under the transformation = + .
Solution— Let = +
∴ + = + + = + ( + 1)
Hence, = and = − 1
Again, from the question | | = 1
∴ + = 1
+ ( − 1) = 1, which represent a circle in ( , ) plane.
1.11.2.2 Rotation— A conformal mapping in which every point on −plane, let P( , ) such
that OP is rotated by an angle in anti-clockwise direction mapped into −plane given by
= is called as Rotation.
Example— Determine and sketch the image of rectangle mapped by the rotation of ° in
anticlockwise direction formed by = , = , = and = .
Solution— given rectangle is—
= 0, = 0, = 1 and = 2
Let the required transformation is—
=
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4
= cos + sin ( + )
=
√
+
√
( + )
=
√
+
√
+ =
√
+
√
∴ =
√
and =
√
Now, for = 0—
=
−
√2
, =
√2
⟹ + = 0
For = 0—
=
√2
, =
√2
⟹ − = 0
For = 1—
=
1 −
√2
, =
1 +
√2
⟹ + = √2
or = 2—
=
− 2
√2
, =
+ 2
√2
⟹ − = −2√2
1.11.2.3 Magnification (Scaling) — A conformal mapping in which every point on −plane
mapped into −plane given by = ( > 0 is real) is called as magnification. In this
process mapped shape is either stretched( > 1) or contracted (0 < < 1) in the direction
of Z.
Example— Determine the transformation = , where | | = .
Solution— given that | | = 2 ⟹ + = 4 , represents a circle on −plane having center
at origin and radius equal to 2-units.
Let the required transformation is—
= 2 = 2( + ) = 2 + 2
+ = 2 + 2
Hence, = and = ⟹ + = 4 ⟹ + = 16 represent a circle on
−plane having center at origin and radius equal to 4-units.
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1.11.2.4 Inversion— A conformal mapping in which every point on −plane mapped into
−plane given by = is called as inversion.
Example— Determine the transformation = , where | | = .
Solution— given that— | | = 1 ⟹ + = 1 … (1)
Again, given that
= = = ( )( )
=
+ = − = − [ + = 1]
Hence, = and = −
Now, from equation (1)—
+ = 1
1.11.2.5 Reflection — A conformal mapping in which every point on −plane mapped into
−plane given by = ̅ . This represents the reflection about real axis.
Example— Determine and sketch the transformation = , where | − ( + )| = .
Solution— given that— | − 2| = 4
|( − 2) + ( − 1)| = 4
( − 2) + ( − 1) = 16 … (1)
Again, from the question— = ̅ = −
+ = −
Hence, = and = −
Now, from equation (1)—
( − 2) + ( + 1) = 16
Sketch—
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References—
1. Internet—
1.1 mathworld.wolfram.com
1.2 www.math.com
1.3 www.grc.nasa.gov
1.4 en.wikipedia.org/wiki/Conformal_map
1.5 http://www.webmath.com/
2. Journals/Paper/notes—
2.1 Complex variables and applications-Xiaokui Yang-Department of Mathematics, Northwestern University
2.2 Conformal Mapping and its Applications-Suman Ganguli-Department of Physics, University of Tennessee, Knoxville, TN 37996- November 20, 2008
2.3 Conformal Mapping and its Applications-Ali H. M. Murid-Department of Mathematical Sciences-Faculty of Science-University Technology Malaysia-
81310-UTM Johor Bahru, Malaysia-September 29, 2012
3. Books—
3.1 Higher Engineering Mathematics-Dr. B.S. Grewal- 42
nd
edition-Khanna Publishers-page no 639-672
3.2 Higher Engineering Mathematics- B V RAMANA- Nineteenth reprint- McGraw Hill education (India) Private limited- page no 22.1-25.26
3.3 Complex analysis and numerical techniques-volume-IV-Dr. Shailesh S. Patel, Dr. Narendra B. Desai- Atul Prakashan
3.4 A Text Book of Engineering Mathematics- N.P. Bali, Dr. Manish Goyal- Laxmi Publications(P) Limited- page no 1033-1100