Lecture 3 By: Dr Tarek Abdolkader

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ةكلمملا ةيدوعسلا ةيبرعلا

يلاعلا ميلعتلا ةرازو -

ىرقلا مأ ةعماج

ةيملاسلإا ةرامعلا و ةسدنهلا ةيلك ةيئابرهكلا ةسدنهلا مسق

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ADVANCED ENGINEERING MATHEMATICS

KINGDOMOF SAUDI ARABIA Ministry of Higher Education

Umm Al-Qura University

College of Engineering and Islamic Architecture Electrical Engineering Department

Lecture 3

By: Dr Tarek Abdolkader

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OUTLINE

3.1 Differentiability and Analyticity

• Differentiability of a complex function

• Differentiation rules

• Analyticity of a complex function

3.2 Cauchy-Riemann Equations

• necessary conditions for analyticity

• sufficient conditions for analyticity

• Analyticity for functions in polar form

3.3 Harmonic Functions:

• Laplace’s equation and harmonic functions

• Harmonic conjugates 3.4 Applications:

Lecture 3 Analytic Functions

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1. What is the definition of a derivative of a complex function f ( z )?

2. What are the rules of differentiation for complex functions?

3. How can you prove that a complex function is not differentiable?

4. What is the definition of analyticity of a complex function at a point?

5. What is the difference between saying that a complex function f ( z ) is differentiable or analytic at a point z 0 ?

6. What is Entire function?

7. Is a polynomial complex function an entire function?

8. Is a rational complex function an entire function?

9. What is singular point?

10. If a complex function is differentiable, can you guarantee that it is continuous?

11. If a complex function is continuous, can you guarantee that it is differentiable?

12. What is L’Hopital rule and when can you use it?

13. Regarding differentiability and analyticity, what are the differences between complex functions and real functions?

At the end of this section you should be able to answer the following questions:

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The Derivative:

There are two definitions of a derivative of a complex function f ( z ) at a point z 0 :

where Δ z = z - z 0 .

• The function is said to be differentiable at z if the limit in

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Show that the function f ( z ) = x + 4 iy is not differentiable at any point z.

See Example 3 page 144 in the textbook

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• Analyticity at a point is not the same as differentiability at a point.

Analyticity at a point is a neighborhood property ; in other words, analyticity is a property that is defined over an open set

• For example the function f ( z ) = | z | 2 is differentiable at z = 0, but however, it is not analytic at z = 0.

• Any analytic function is differentiable, but not every differentiable function is analytic.

• If the functions f and g are analytic in a domain D , then, the sum f ( z )

+ g ( z ), difference f ( z ) - g ( z ), and product f ( z ) g ( z ) are analytic. The

quotient f ( z )/ g ( z ) is analytic provided g ( z ) ≠ 0 in D.

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Entire Function:

A function that is analytic at every point z in the complex plane is said to be an entire function.

Singular point:

A point z at which a complex function w = f ( z ) fails to be analytic is

called a singular point of f .

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Continuity, Differentiability, and Analyticity:

Continuity

Analyticity

Differentiability

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1. What is the necessary condition for Analyticity of f ( z ) = u ( x , y ) + i v ( x , y )?

2. What is Cauchy-Rieman equations?

3. What is the sufficient condition for Analyticity of f ( z ) = u ( x , y ) + i v ( x , y )?

4. How can you use Cauchy-Rieman equations in polar coordinates?

5. What is the value of the derivative of f ( z ) = u ( r , θ ) + i v ( r , θ )?

At the end of this section you should be able to answer the following questions:

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u v u v

   

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Show that the complex function f ( z ) = 2 x

²

+ y + i ( y

²

− x ) is not

analytic at any point.

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Using Cauchy-Riemann conditions, we have either of four cases:

Conditions are Decision

not satisfied at any point Neither differentiable nor analytic at any point

not satisfied except at certain points The function is differentiable at these points but not analytic at any point

satisfied except at certain points (singular points)

The function is differentiable and analytic at all points except the singular points

Conditions are satisfied at all points

The function is differentiable

and analytic at all points (The

function is entire function)

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Show that the complex function is analytic at any domain not containing z =0:

Note: Since analyticity implies differentiability, theorem 3.5 is also applied for differentiability

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Show that the complex function f ( z ) = 2 x

²

+ y + i ( y

²

− x ) is

differentiable at the line y = 2x.

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1

u v

r r 

 

  

Cauchy- Riemann conditions for polar for of u and v are:

( ) i u v 1 i v u

f z e i e i

r r r

 

         

                 1

v u

r r 

 

   

Cauchy- Riemann conditions for polar form:

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1. What is harmonic function?

2. What is the relation between harmonic function and Laplace’s equation?

3. What is the relation between real harmonic functions and complex analytic function?

4. What are harmonic conjugate functions?

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2 2

2 2 0

x y

 

   

 

Laplace’s Equation:

A real-valued function ɸ is said to be satisfies Laplace’s equation if,

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Show that the real and imaginary parts of the complex function f ( z )

= z

²

are harmonic.

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If f(z) = u + i v is an analytic function, then u and v are called harmonic conjugate of each other, i.e., u is a harmonic conjugate of v and v is a harmonic conjugate of u.

Harmonic Conjugates:

(a) Verify that the function u ( x, y ) = x

³

− 3 xy

²

− 5 y is harmonic in the entire complex plane.

(b) Find the harmonic conjugate function of u .

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1. What is relation between the level curves of the functions u and v of an analytic function f ( z ) = u + i v ?

2. What is the complex potential?

3. How do you express electrostatic field in terms of complex potential?

4. How can you solve simple Dirichlet problems including Laplace’s equation?

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Orthogonal Families:

If f(z) = u + i v is an analytic complex

function, then the family of curves

u(x, y) = c 1 and v(x, y) = c 2

are orthogonal.

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For the function f(z) = z 2 = u + i v, show that the family of

curves u(x, y) = c1 and v(x, y) = c2 are orthogonal and draw

them.

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Complex Potential:

• if a potential function φ ( x , y ) satisfies Laplace’s equation in some domain D , it is harmonic.

• There exists a harmonic conjugate function ψ ( x, y ) defined in D so that the complex function:

Ω(z) = φ ( x , y ) + i ψ ( x, y ) is an analytic in D.

• The complex function Ω(z) is called the complex potential.

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Dirichlet Poroblem:

• The problem of finding the potential function φ ( x , y ) which

satisfies Laplace’s equation in some domain D , such that it

equals a function g on the boundary C of D is called a

Dirichlet problem

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Solve the Dirichlet problem illustrated in Figure beside.

The domain D is a vertical infinite strip defined

by − 1 < x < 1, −∞ < y < ∞ ; the boundaries of D

are the vertical lines x = − 1 and x = 1.

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