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SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 1
SRI RAMAKRISHNA INSTITUTE OF TECHNOLOGY
(AN AUTONOMOUS INSTITUTION)
COIMBATORE- 641010
UICM002 & Engineering Mathematics – II
Analytic Functions
Course Material
Introduction:
Analytic functions originated in the 19th century, mainly due to the work of
A.L. Cauchy, B. Riemann and K. Weierstrass. The theory of analytic functions was
constructed as the theory of functions of a complex variable. It is used in general theory
of functions of a complex variable.
There are different approaches to the concept of analyticity. One definition,
which was originally proposed by Cauchy, and was considerably advanced by Riemann,
is based on a structural property of the function — the existence of a derivative with
respect to the complex variable, i.e. its complex differentiability. This approach is
closely connected with geometric ideas. Another approach, which was systematically
developed by Weierstrass, is based on the possibility of representing functions by
power series; it is thus connected with the analytic apparatus by means of which a
function can be expressed. A basic fact of the theory of analytic functions is the identity
of the corresponding classes of functions in an arbitrary domain of the complex plane.
Applications:
Electrostatics is the solution of Laplace’s equation, which in two dimensions is
also the condition for analyticity of complex-valued functions. There is a considerable
body of tricks for solving two-dimensional electrostatics problems by mapping them
into simpler problems by use of conformal maps, which map analytic functions on one
complex domain to analytic functions on another by composition with another analytic
function.
Using analytical functions for real world problems, engineer makes models of
projects and then simulates its models in real world conditions. the simulation results
are then analyzed to decide whether the project is feasible and cost effective or not.
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
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Definition: Analytic Function
A function is said to be analytic at a point if its derivative exists not only at that
point but also in some neighborhood of that point.
Limit of ( ):
If ( ) is a function of and , then we say that the limit
of ( ) as tends to is and
( ) when ( ) and
( ) as and .
Continuity of ( ):
If ( ) is a single valued function of , then ( ) is said to be
continuous at if at if
( ) ( ).
Singular point:
A point at which the function ( ) fails to be analytic is called singular point
or singularity of ( ).
Necessary conditions for ( ) to be analytic [Cauchy – Riemann Equations]
The Necessary conditions for a complex function ( ) ( ) ( ) to be
analytic in a region are
Sufficient condition for ( ) to be analytic
The function ( ) ( ) ( ) is analytic in a domain if
( ) and ( ) are differentiable in and and .
the Partial derivatives and are all continuous in .
Example:
Show that the function ( ) ̅ is nowhere differentiable.
Answer:
( ) ̅
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
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C-R equations are not satisfied.
( ) ̅ is nowhere differentiable.
Example:
Check whether the function ( ) ( ) is analytic or not.
Answer:
( ) ( )
C-R equations are not satisfied.
( ) is not analytic.
Example:
Check whether the function ( ) is analytic or not.
Answer:
( ) ( )
( )
C-R equations are satisfied.
( ) is analytic.
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 4
Example:
Check whether the function ( ) ( ) is analytic or not.
Answer:
( )
( )
( )
C-R equations are satisfied.
( ) is analytic.
Example:
Prove that the function ( ) is anlaytic. Also find its derivative.
Answer:
( ) ( )
[ ]
[ ]
C-R equations are satisfied.
( ) is analytic.
To find its derivative:
( )
( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
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Example:
( )
( ) (
)
Answer:
( )
( ) (
)
( )
(
) ( )
(
) ( )
(
)
(
)
(
)(
)
(
)
(
)
(
)(
)
C-R equations are satisfied.
( ) is analytic.
Harmonic and Orthogonal properties of analytic functions –Harmonic Conjugates
Laplace equation
Harmonic function
A real function with two variables and that satisfies Laplace equation is called
Harmonic function.
Conjugate Harmonic function
If and are harmonic functions such that is analytic, then each and
are called the conjugate harmonic function to each other.
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
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Property:
Prove that the real and imaginary parts of an analytic function are harmonic
functions.
Answer:
Given ( ) is any analytic function.
It satisfies C-R equations.
(
)
(
)
( )
(
)
(
)
( )
Adding ( ) and ( ), we get
Hence real part satisfies Laplace’s equation.
(
)
(
)
( )
(
)
(
)
( )
Adding ( ) and ( ), we get
Hence imaginary part satisfies Laplace’s equation.
This proves that the real and imaginary parts of an analytic function are
harmonic functions.
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
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Property:
If is an analytic function, then the family of curves and
cut orthogonally, where and are constants.
Answer:
Assume ( ) ( )
Differentiating both sides, we get
[ ]
( ) ( )
Also take ( ) ( )
Differentiating both sides, we get
[ ]
( ) ( )
If the two curves cuts orthogonally, then the slopes .
(
) (
)
(
) (
) [ ]
Hence the proof.
Example:
If ( ) is a regular function of in a domain then the following holds.
[| ( )| ] | ( )|
Answer:
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
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( )
| ( )| | |
| ( )|
[| ( )| ] (
) ( ) *
+
( )
( )
( )
( )
( )
( )
( )
( )
(
)
*
(
)
+ [ ( ) ]
*
(
)
+
( ) *
(
)
+
( )
( ) *
(
)
(
)
+
* (
) (
)
(
)
+
* ( ) (
)
(
)
+ [ ]
[ ( )
]
[
] [ ]
( )
( ) | ( )| ( )
( )
( ) | ( )| ( )
( ) ( ) (
) | ( )| | ( )|
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
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Construction of analytic functions (Milne Thomson Method)
Case: I
To find ( ) when is given
( ) ∫
( ) ∫
( )
Case: II
To find ( ) when is given
( ) ∫
( ) ∫
( )
Example:
Show that the function is harmonic also find its analytic function.
Answer:
To prove is harmonic:
Hence is harmonic.
To find the analytic function ( ):
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
( )
( )
( ) ∫ ∫
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 10
Example: Show that the function is harmonic also find its analytic function,
conjugate.
Answer: To prove is harmonic:
Hence is harmonic.
To find the analytic function ( ):
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
( )
( )
( ) ∫ ∫( )
(
)
To find its conjugate:
( )
[ ]
( )
( )
( ) ( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 11
Example:
Show that the function is harmonic and find the analytic function; also find its
conjugate harmonic function if
Answer:
To prove is harmonic:
Hence is harmonic.
To find the analytic function ( ):
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
( )
( )
( ) ∫ ∫( )
(
)
( )
To find its conjugate:
( ) ( )
([ ] [ ])
( )
( )
( ) ( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
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Example:
Show that the function and satisfies Laplace equation.
But is not an analytic function of z.
Answer:
To prove satisfies Laplace equation:
( )
( )
Hence satisfies Laplace equation.
To prove satisfies Laplace equation:
( )
( )
Hence also satisfies Laplace equation.
To prove satisfies analytic or not:
We know that the C-R equation
and
Here and
Hence is not an analytic function.
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
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Example:
( )
Answer:
To prove is harmonic:
( )
(
)
( )( ) ( )
( )
( )
( )
(
)
( )( ) ( )
( )
( )
( )
( )
( )
( )
Hence is harmonic.
To find the analytic function ( ):
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
( )
( )
( ) ∫
∫
( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 14
Example:
Find ( ) if Also find its analytic function.
Answer:
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
( )
( )
( ) ∫ ∫
Example:
Find ( ) if ( )
Answer:
( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
∫ ∫
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 15
∫ ∫
( )
(
)
( ) [
]
[ ]
( )
[ ]
[ ]
[ ] [ ]
[ ( ) ( )]
( ) ( )
( )
Example:
Find ( ) if
Answer:
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
( )
( )
( ) ∫ ∫
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 16
Example:
Find ( ) if
Answer:
( )
( )
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
( ) ∫ ∫
(
)
( )
Example:
Show that the function ( ) is harmonic. Also find the analytic
function ( ).
Answer:
To prove is harmonic:
[ ( ) ]
( )
( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 17
is harmonic.
To find the analytic function ( ):
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
( )
( )
( )
( ) ( )
( ) ∫ ( ) ∫
∫
[( ) ( ) ]
( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 18
Example:
Show that the function ( ) is harmonic. Also find the analytic
function ( ).
Answer:
To prove is harmonic:
( )
( )
is harmonic.
To find the analytic function ( ):
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
( )
( )
( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 19
( ) ( )
( ) ∫ ( ) ∫
∫
( )
[( ) (
) ( ) (
( ) )]
( )
( )
Example:
( )
Answer:
* (
)
+
( ) ( )
( )
( )
[ ]
( )
( )
( )
( )
( )
( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 20
( )
( )
( )
( )
( )
( ) ( )
( )
( )
( )
( ) [ ]
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
∫
( ) ∫
∫
( )
[ (
)]
∫
∫
( )
Example:
Determine the analytic function if ( ). Also
find its conjugate .
Answer:
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 21
( )
( )
( ) ( )
( ) ( ) ( )
( ) ∫ ( ) ∫
∫
*( )(
) ( )(
)+
( )
( )
( )
( )
To find the conjugate :
( ) ( ) ( )
( )
( ) ( )
[ ]
[ ] [ ]
Hence [ ]
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 22
Example:
Determine the analytic function if ( ).
Answer:
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
( ) ( )
( ) ( ) ( )
( )
( ) ∫( ) ∫( )
*
( (
) (
))+
*
(
(
))+
[
]
[
]
[ ]
( )
Example:
Find the analytic function and ( )( ). Also find the
conjugate function .
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 23
Answer:
( )( ) ( )( )
( ) ( )( ) ( )
( )( ) ( )( )
( ) ( )( ) ( )
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
∫ ∫
(
) (
)
( ) ( )
To find the conjugate function :
( ) ( ) ( )
[ ]( )
( ) ( )
( )
Example:
( )
( ) ( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 24
Answer:
( )
( ) ( )
( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( )
( )
[ ]
( )
( )
( )
( )
( )
( )
( )
( )
( )
( ) ( )
( )
( )
( )
( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 25
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
∫
( ) ∫
∫
( )
[ (
)]
∫
∫
( )
( ) ( )
( )
( )
( ) ( )
( )
( )
( ) ( )
Example:
If ( ) is an analytic function and ( ), find ( )
in terms of .
Answer:
( ) ( )
( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 26
( )
( )
( )
( ) ( )
( ) ( )
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
∫ ∫
( ) ( )
( ) ( ) ( )
( )
Example:
If ( ) is an analytic function and ( )( ), find
( ) in terms of .
Answer:
( ) ( )
( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( )( )
( )( ) ( )( )
( ) ( )( ) ( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 27
( )( ) ( )( )
( ) ( )( ) ( )
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
∫ ∫
(
) (
)
( ) ( )
( ) ( ) ( )
( )
Example:
( )
( )
Answer:
( ) ( )
( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 28
( )( ) ( )
( ) * (
)
+
( )
( )
( )
( )
( )
( )( ) ( )
( )
( )
( )
( )
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
∫ ∫
∫ [
]
(
)
(
)
( )
( ) ( )
( )
( )
( ) ( )
( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 29
( )
( )
( )
Example:
If ( ) is an analytic function and ( ), find ( )
in terms of .
Answer:
( ) ( )
( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( )
( )
( ) ( )
( )
( ) ( )
By Milne Thomson Method, we have
( ) ∫
( ) ∫
( ) [ ]
∫ ∫
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 30
[ ∫
]
( ) ( )
( ) ( ) ( )
( ) ( )
( )
( )
( ) ( )
( )
( )( )
( )
( )
( )
( )
( )
Bilinear transformation
constants is known as bilinear transformation.
The condition ensures that the transformation is conformal since
If , then every point of the plane is a critical point of the
transformation.
Properties of Bilinear transformation
1. The bilinear transformation all transforms circles with lines as limiting case.
2. It preserves cross ratio of four points.
Fixed point (or) Invariant point
A fixed point of a mapping ( ) is a point whose image is the same point.
The fixed point of the transformation ( ) are obtained by solving ( ).
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 31
Cross Ratio:
( )( )
( )( )
ratio of four and .
Result:
The bilinear transformation which maps the point ( ) into a point
( ) is given by
( )( )
( )( )
( )( )
( )( )
Example:
Answer:
The invariant points of the transformation obtained by putting , we get
( )
[ ]
[
]
Solving, we get
Example:
Answer:
The invariant points of the transformation obtained by putting , we get
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 32
( )
Solving, we get
Example:
Find the bilinear transformation which maps the point ( ) into a
point ( ) respectively.
Answer:
The bilinear transformation which maps the point ( ) into a point
( ) is given by
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )
( )( )
( )( )
( )( )
( )
( ) ( )( )
[ ] ( )
( ) ( )
( )
[ ]
Example:
Find the bilinear transformation which maps the point ( ) into a point
( ) respectively.
Answer:
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 33
The bilinear transformation which maps the point ( ) into a point
( ) is given by
( )( )
( )( )
( )( )
( )( )
( ) (
)
( ) (
)
(
) ( )
(
) ( )
( ) (
)
( ) ( )
(
) ( )
( )( )
( )
( )
( )
( )
Example:
Find the mobilus transformation which maps the point ( ) into a
point ( ) respectively.
Answer:
The bilinear transformation which maps the point ( ) into a point
( ) is given by
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( ) (
)
( ) (
)
( )( )
( )( )
( ) (
)
( ) ( )
( )( )
( )( )( )
( )
( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 34
( )( )
( )( ) ( )
( )
( )( )
( )( )
( )[ ]
( )( )
( )[ ]
( )( )
( )[ ]
( )
( ) ( )
( ) ( )
( )
( ) ( )
( )
( )
Example:
Find the bilinear transformation which maps the point ( ) into a
point ( ) respectively.
Answer:
The bilinear transformation which maps the point ( ) into a point
( ) is given by
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
( )
( )
( )( )
( )( )
( )
( )
( )
( ) ( )( )
[ ]
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 35
[ ] [ ]
[ ] [ ]
( )
Example:
Find the bilinear transformation which transforms the point
into a point and respectively.
Answer:
The bilinear transformation which maps the point ( ) into a point
( ) is given by
( )( )
( )( )
( )( )
( )( )
( ) (
)
( ) (
)
( )( )
( )( )
( ) (
)
( ) ( )
( )( )
( )( )
( )
( )
( )( )
( )( )
( )( )
( )( ) ( )
( )
( )( )
( )( )
( )[ ]
( )( ( ))
( )[ ]
( )
( )[ ]
( )
( )
( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 36
Example:
Find the bilinear transformation which transforms the point
into a point and respectively.
Answer:
The bilinear transformation which maps the point ( ) into a point
( ) is given by
( )( )
( )( )
( )( )
( )( )
( ) (
)
( ) (
)
( )( )
( )( )
( ) (
)
( ) ( )
( )( )
( )( )
( )
( )
( )
( )
( )( ) ( )
( )
( )
( )
Conformal Mapping
A transformation that preserves angles between every pair of curves through a
point, both in magnitude and sense, is said to be conformal at that point.
Isogonal
A transformation, under which angles between every pair of curves through a
point are preserved in magnitude but not in sense, is said to be isogonal at that point.
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SRIT / M & H / M. Vijaya Kumar 37
Standard transformations:
( )
( )
( )
I. Translations [ ], where is a complex constant, denotes a translation.
( ) ( )
( ) ( )
Example:
Find the image of the circle | | under the transformation .
Answer:
( ) ( )
| |
it is a circle with centre ( ) and radius .
( ) ( )
Therefore the circle in plane maps a circle ( ) ( )
with centre ( ) and radius in w plane.
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
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Example:
Determine the region of the palne into which the rectangular region in the
-plane bounded by the lines and is mapped under the
transformation .
Answer:
( ) ( )
Also given the lines
( )
𝑟
𝑥
𝑦 𝑍-Plane |𝑍|
( )
𝑟
𝑢
𝑣
𝑊-Plane
( )
𝑥
𝑥
𝑦 𝑍-Plane
𝑦
𝑦
𝑥
𝑢
𝑢
𝑣 𝑊-Plane
𝑣
𝑣 𝑢
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SRIT / M & H / M. Vijaya Kumar 39
Transformation: 2 [ ], where is a real constant, denotes a translation.
( )
Example:
Find the image of the circle | | under the transformation .
Answer:
( )
| |
It is a circle with centre ( ) and radius .
(
)
(
)
( )
Therefore the circle | | in plane maps a circle | | in w plane.
( )
𝑟 𝑎
𝑥
𝑦 𝑍-Plane |𝑍| 𝑎
𝑥 𝑦 𝑎 𝑟 𝑎
( ) 𝑢
𝑣 𝑊-Plane |𝑊| 𝑎
𝑢 𝑣 ( 𝑎)
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 40
Example:
Find the image of the circle | | under the transformation .
Answer:
This result immediate follows from previous example, in which replace .
Therefore the circle | | in plane maps a circle | | in w plane.
[
]
( ) ( )
Example:
| |
Answer:
( ) ( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 41
| |
|( ) |
| ( )|
( )
It is a circle with centre ( ) and radius .
( ) [ ( ) ]
(
)
(
)
(
)
( ) (
)
(
)
| |
Example:
| |
Answer:
( )
𝑟
𝑥
𝑦 𝑍-Plane
𝑥 (𝑦 )
𝒗 𝟏
𝟒
𝒖 𝟎
𝑢
𝑣 𝑊-Plane
𝒗 𝟎 ( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 42
( ) ( )
| |
|( ) |
|( ) |
( )
It is a circle with centre ( ) and radius .
( ) [ ( ) ]
(
)
(
)
(
)
( ) (
)
(
)
(𝑥 ) 𝑦
(𝟏 𝟎)
𝑟
𝑥 (𝟎 𝟎)
𝑦 𝑍-Plane
𝒗 𝟎
𝒖 𝟎
𝑢
𝑣
𝑊-Plane
( )
𝒖 𝟏 𝟐
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 43
| |
Example:
Answer:
( ) ( )
( )
It is a circle with centre ( ) and radius 2.
( )
It is a circle with centre ( ) and radius 1.
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 44
( ) ( )
Example:
the plane into circles or straight lines in the plane.
Answer:
( ) ( )
The general equation of circle is
(
)
(
)
(
) (
)
𝒚 𝟎
𝒙 𝟎
𝑥
𝑦 𝑍-Plane
( )
𝒚 𝟏
𝟐
𝒚 𝟏
𝟒
𝒗 𝟎
𝒖 𝟎
𝑢
𝑣 𝑊-Plane
( )
( )
( ) r=2
r=1
𝒖𝟐 (𝒗 𝟏)𝟐 𝟏
𝒖𝟐 (𝒗 𝟐)𝟐 𝟐𝟐
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 45
( )
( ) (
) (
)
( ) (
) (
)
Multiply both sides by ( ), we get
( )
Case 1: , i.e., circles not passing through the origin in plane map into
circles not passing through the origin in plane.
Case 2: , i.e., circles through the origin in plane map into straight lines
not through the origin in plane.
Case 3: , i.e., straight lines not through the origin in plane map into
circles through the origin in plane.
Case 4: , i.e., straight lines through the origin in plane map into circles
through the origin in plane.
Example:
Answer:
[
]
[ ]
[ ]
[ ]
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 46
[ ] [ ]
This is lemniscate.
III. Magnification and Rotation
The transformation represents Magnification and Rotation.
where are complex numbers.
[ ]
[ ]
[ ]
( )( )
( )
Thus the transformation = corresponds to a rotation together with magnification.
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 47
Two Marks
1. State the Cauchy-Riemann equation in Cartesian coordinates satisfied by an
analytic function.
Answer:
The Necessary conditions for a complex function ( ) ( ) ( ) to be
analytic in a region are
2. State the Cauchy-Riemann equation in polar coordinates satisfied by an
analytic function.
Answer:
The Cauchy-Riemann equation in polar coordinates is
3. Verify ( ) is analytic or not.
Answer:
( ) ( )
( ) ( )
C-R equations are satisfied.
is analytic.
4. Show that | | is not analytic at any point.
Answer:
( ) | |
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 48
C-R equations are not satisfied.
| | is not analytic.
5. Find the constant and if ( ) ( ) is analytic.
Answer:
( ) ( )
By C-R equations are, we have
6. Verify the function ( ) ̅ is analytic or not.
Answer:
( ) ̅
C-R equations are not satisfied.
Hence ( ) ̅ is not analytic function.
7. Define harmonic function.
Answer:
A real function with two variables and that satisfies Laplace equation is called
Harmonic function.
8. Show that is harmonic.
Answer:
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 49
( )
( )
( ) ( )
is harmonic.
9. Verify whether the function is harmonic.
Answer:
( )
( )
( ) ( )
is harmonic.
10. Define conformal mapping.
Answer:
A transformation that preserves angles between every pair of curves through a
point, both in magnitude and sense, is said to be conformal at that point.
11. Find the image of the circle | | under the transformation .
Answer:
( )
| |
It is a circle with centre ( ) and radius .
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 50
(
)
(
)
Therefore the circle | | in plane maps a circle | | in w plane.
12. Prove that a bilinear transformation has at most two fixed points.
Proof:
By definition of bilinear transformation.
( )
To find fixed point, put , we get
( )
( )
which is quadratic equation, it has exactly two roots. It proves that a bilinear
transformation has at most two fixed points.
Answer:
To find fixed point, put , we get
( )
𝑟
𝑥
𝑦 𝑍-Plane |𝑍|
𝑥 𝑦 𝑟
( ) 𝑢
𝑣 𝑊-Plane |𝑊|
𝑢 𝑣
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
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Solving, we get
Answer:
To find fixed point, put , we get
( )
Solving, we get
Answer:
To find fixed point, put , we get
( )
Solving, we get
( )
Answer:
To find fixed point, put , we get
( )
SRIT / UICM002 - Engineering Mathematics – II / Analytic Functions
SRIT / M & H / M. Vijaya Kumar 52
( )
( )
“Mathematics is not about numbers, equations, computations, or
algorithms: it is about understanding.”
– William Paul Thurston