uicm002 & engineering mathematics ii analytic functions ......srit / uicm002 - engineering...

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.



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:

( ) ̅



C-R equations are not satisfied.

( ) ̅ is nowhere differentiable.

Example:

Check whether the function ( ) ( ) is analytic or not.

Answer:

( ) ( )


( ) is not analytic.

Example:

Check whether the function ( ) is analytic or not.

Answer:

( ) ( )

( )

C-R equations are satisfied.

( ) is analytic.



Example:

Check whether the function ( ) ( ) is analytic or not.

Answer:

( )

( )

( )


( ) is analytic.

Example:

Prove that the function ( ) is anlaytic. Also find its derivative.

Answer:

( ) ( )

[ ]

[ ]


( ) is analytic.

To find its derivative:

( )

( )



Example:

( )

( ) (

)

Answer:

( )

( ) (

)

( )

(

) ( )

(

) ( )

(

)

(

)

(

)(

)

(

)

(

)

(

)(

)


( ) 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.



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.



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:



( )

| ( )| | |

| ( )|

[| ( )| ] (

) ( ) *

+

( )

( )

( )

( )

( )

( )

( )

( )

(

)

*

(

)

+ [ ( ) ]

*

(

)

+

( ) *

(

)

+

( )

( ) *

(

)

(

)

+

* (

) (

)

(

)

+

* ( ) (

)

(

)

+ [ ]

[ ( )

]

[

] [ ]

( )

( ) | ( )| ( )

( )

( ) | ( )| ( )

( ) ( ) (

) | ( )| | ( )|



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

( ) ∫

( ) ∫

( ) [ ]

( )

( )

( ) ∫ ∫



Example: Show that the function is harmonic also find its analytic function,

conjugate.

Answer: To prove is harmonic:

Hence is harmonic.



( ) ∫

( ) ∫

( ) [ ]

( )

( )

( ) ∫ ∫( )

(

)

To find its conjugate:

( )

[ ]

( )

( )

( ) ( )



Example:

Show that the function is harmonic and find the analytic function; also find its

conjugate harmonic function if

Answer:


Hence is harmonic.



( ) ∫

( ) ∫

( ) [ ]

( )

( )

( ) ∫ ∫( )

(

)

( )

To find its conjugate:

( ) ( )

([ ] [ ])

( )

( )

( ) ( )



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.



Example:

( )

Answer:


( )

(

)

( )( ) ( )

( )

( )

( )

(

)

( )( ) ( )

( )

( )

( )

( )

( )

( )

Hence is harmonic.



( ) ∫

( ) ∫

( ) [ ]

( )

( )

( ) ∫

∫

( )



Example:

Find ( ) if Also find its analytic function.

Answer:


( ) ∫

( ) ∫

( ) [ ]

( )

( )

( ) ∫ ∫

Example:

Find ( ) if ( )

Answer:

( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )


( ) ∫

( ) ∫

( ) [ ]

∫ ∫



∫ ∫

( )

(

)

( ) [

]

[ ]

( )

[ ]

[ ]

[ ] [ ]

[ ( ) ( )]

( ) ( )

( )

Example:

Find ( ) if

Answer:


( ) ∫

( ) ∫

( ) [ ]

( )

( )

( ) ∫ ∫



Example:

Find ( ) if

Answer:

( )

( )


( ) ∫

( ) ∫

( ) [ ]

( ) ∫ ∫

(

)

( )

Example:

Show that the function ( ) is harmonic. Also find the analytic

function ( ).

Answer:


[ ( ) ]

( )

( )



is harmonic.



( ) ∫

( ) ∫

( ) [ ]

( )

( )

( )

( ) ( )

( ) ∫ ( ) ∫

∫

[( ) ( ) ]

( )



Example:

Show that the function ( ) is harmonic. Also find the analytic

function ( ).

Answer:


( )

( )

is harmonic.



( ) ∫

( ) ∫

( ) [ ]

( )

( )

( )



( ) ( )

( ) ∫ ( ) ∫

∫

( )

[( ) (

) ( ) (

( ) )]

( )

( )

Example:

( )

Answer:

* (

)

+

( ) ( )

( )

( )

[ ]

( )

( )

( )

( )

( )

( )



( )

( )

( )

( )

( )

( ) ( )

( )

( )

( )

( ) [ ]


( ) ∫

( ) ∫

( ) [ ]

∫

( ) ∫

∫

( )

[ (

)]

∫

∫

( )

Example:

Determine the analytic function if ( ). Also

find its conjugate .

Answer:


( ) ∫

( ) ∫

( ) [ ]



( )

( )

( ) ( )

( ) ( ) ( )

( ) ∫ ( ) ∫

∫

*( )(

) ( )(

)+

( )

( )

( )

( )

To find the conjugate :

( ) ( ) ( )

( )

( ) ( )

[ ]

[ ] [ ]

Hence [ ]



Example:

Determine the analytic function if ( ).

Answer:


( ) ∫

( ) ∫

( ) [ ]

( ) ( )

( ) ( ) ( )

( )

( ) ∫( ) ∫( )

*

( (

) (

))+

*

(

(

))+

[

]

[

]

[ ]

( )

Example:

Find the analytic function and ( )( ). Also find the

conjugate function .



Answer:

( )( ) ( )( )

( ) ( )( ) ( )

( )( ) ( )( )

( ) ( )( ) ( )


( ) ∫

( ) ∫

( ) [ ]

∫ ∫

(

) (

)

( ) ( )

To find the conjugate function :

( ) ( ) ( )

[ ]( )

( ) ( )

( )

Example:

( )

( ) ( )



Answer:

( )

( ) ( )

( )

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( )

( )

[ ]

( )

( )

( )

( )

( )

( )

( )

( )

( )

( ) ( )

( )

( )

( )

( )




( ) ∫

( ) ∫

( ) [ ]

∫

( ) ∫

∫

( )

[ (

)]

∫

∫

( )

( ) ( )

( )

( )

( ) ( )

( )

( )

( ) ( )

Example:

If ( ) is an analytic function and ( ), find ( )

in terms of .

Answer:

( ) ( )

( )

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )



( )

( )

( )

( ) ( )

( ) ( )


( ) ∫

( ) ∫

( ) [ ]

∫ ∫

( ) ( )

( ) ( ) ( )

( )

Example:

If ( ) is an analytic function and ( )( ), find

( ) in terms of .

Answer:

( ) ( )

( )

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )( )

( )( ) ( )( )

( ) ( )( ) ( )



( )( ) ( )( )

( ) ( )( ) ( )


( ) ∫

( ) ∫

( ) [ ]

∫ ∫

(

) (

)

( ) ( )

( ) ( ) ( )

( )

Example:

( )

( )

Answer:

( ) ( )

( )

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )



( )( ) ( )

( ) * (

)

+

( )

( )

( )

( )

( )

( )( ) ( )

( )

( )

( )

( )


( ) ∫

( ) ∫

( ) [ ]

∫ ∫

∫ [

]

(

)

(

)

( )

( ) ( )

( )

( )

( ) ( )

( )



( )

( )

( )

Example:

If ( ) is an analytic function and ( ), find ( )

in terms of .

Answer:

( ) ( )

( )

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

( )

( ) ( )

( )

( ) ( )


( ) ∫

( ) ∫

( ) [ ]

∫ ∫



[ ∫

]

( ) ( )

( ) ( ) ( )

( ) ( )

( )

( )

( ) ( )

( )

( )( )

( )

( )

( )

( )

( )

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 ( ).



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



( )

Solving, we get

Example:

Find the bilinear transformation which maps the point ( ) into a

point ( ) respectively.

Answer:


( ) is given by

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )

( )( )

( )( )

( )( )

( )

( ) ( )( )

[ ] ( )

( ) ( )

( )

[ ]

Example:

Find the bilinear transformation which maps the point ( ) into a point

( ) respectively.

Answer:




( ) is given by

( )( )

( )( )

( )( )

( )( )

( ) (

)

( ) (

)

(

) ( )

(

) ( )

( ) (

)

( ) ( )

(

) ( )

( )( )

( )

( )

( )

( )

Example:

Find the mobilus transformation which maps the point ( ) into a


Answer:


( ) is given by

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( ) (

)

( ) (

)

( )( )

( )( )

( ) (

)

( ) ( )

( )( )

( )( )( )

( )

( )



( )( )

( )( ) ( )

( )

( )( )

( )( )

( )[ ]

( )( )

( )[ ]

( )( )

( )[ ]

( )

( ) ( )

( ) ( )

( )

( ) ( )

( )

( )

Example:

Find the bilinear transformation which maps the point ( ) into a


Answer:


( ) is given by

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )

( )

( )( )

( )( )

( )

( )

( )

( ) ( )( )

[ ]



[ ] [ ]

[ ] [ ]

( )

Example:

Find the bilinear transformation which transforms the point

into a point and respectively.

Answer:


( ) is given by

( )( )

( )( )

( )( )

( )( )

( ) (

)

( ) (

)

( )( )

( )( )

( ) (

)

( ) ( )

( )( )

( )( )

( )

( )

( )( )

( )( )

( )( )

( )( ) ( )

( )

( )( )

( )( )

( )[ ]

( )( ( ))

( )[ ]

( )

( )[ ]

( )

( )

( )



Example:

Find the bilinear transformation which transforms the point

into a point and respectively.

Answer:


( ) 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.



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.



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



Example:


Answer:

This result immediate follows from previous example, in which replace .


[

]

( ) ( )

Example:

| |

Answer:

( ) ( )



| |

|( ) |

| ( )|

( )


( ) [ ( ) ]

(

)

(

)

(

)

( ) (

)

(

)

| |

Example:

| |

Answer:

( )

𝑟

𝑥

𝑦 𝑍-Plane

𝑥 (𝑦 )

𝒗 𝟏

𝟒

𝒖 𝟎

𝑢

𝑣 𝑊-Plane

𝒗 𝟎 ( )



( ) ( )

| |

|( ) |

|( ) |

( )


( ) [ ( ) ]

(

)

(

)

(

)

( ) (

)

(

)

(𝑥 ) 𝑦

(𝟏 𝟎)

𝑟

𝑥 (𝟎 𝟎)

𝑦 𝑍-Plane

𝒗 𝟎

𝒖 𝟎

𝑢

𝑣

𝑊-Plane

( )

𝒖 𝟏 𝟐



| |

Example:

Answer:

( ) ( )

( )

It is a circle with centre ( ) and radius 2.

( )

It is a circle with centre ( ) and radius 1.



( ) ( )

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



( )

( ) (

) (

)

( ) (

) (

)

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:

[

]

[ ]

[ ]

[ ]



[ ] [ ]

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.



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:

( ) ( )

( ) ( )


is analytic.

4. Show that | | is not analytic at any point.

Answer:

( ) | |




| | 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:

( ) ̅


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:



( )

( )

( ) ( )

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:

( )

| |




(

)

(

)


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:


( )

𝑟

𝑥

𝑦 𝑍-Plane |𝑍|

𝑥 𝑦 𝑟

( ) 𝑢

𝑣 𝑊-Plane |𝑊|

𝑢 𝑣



Solving, we get

Answer:


( )

Solving, we get

Answer:


( )

Solving, we get

( )

Answer:


( )



( )

( )

“Mathematics is not about numbers, equations, computations, or

algorithms: it is about understanding.”

– William Paul Thurston

[email protected]; [email protected]

mailto:[email protected]

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uicm002 & engineering mathematics ii analytic functions ......srit / uicm002 - engineering...

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