1. Introduction –Function of A Complex Variable
2. Analytic Functions(C-R Equations)
3. Harmonic and Orthogonal Properties Of Analytic Functions
4. Construction of Analytic Functions
5. Conformal Mapping
6. Bilinear Transformation

**ANALYTIC FUNCTIONS**

1.
Introduction –Function of A Complex Variable

2.
Analytic Functions(C-R Equations)

3.
Harmonic and Orthogonal Properties Of Analytic Functions

4.
Construction of Analytic Functions

5.
Conformal Mapping

6.
Bilinear Transformation

**ANALYTIC
FUNCTIONS**

**1** **Introduction:
Analytic Functions**

**1.1 Function of Complex Variable**

Many
complicated integrals of real functions are solved with the help of complex
variable. They are very useful in solving large number of engineering and
science problems

**1.2
Complex Variable:**

**1.3
Function of Complex Variable:**

z=x+
i y and w=u+ iv are two complex variable. If for each value of z in a given
region R of the complex plane there corresponds one or more values of w, then w
is called a function of z and it is denoted by w=f(z)=u(x, y)+iv(x, y)where u(x, y) ,v(x ,y) are real functions of the
real variable x and y.

**1.4
Single Valued Function**

If for each value of z in R, there is
correspondingly only one value of w, the w is called a single valued function
of z.

**1.5 Multiple Valued Function**

If for each value of z in R, there is
correspondingly more than one value of w, the w is called a multiple valued
function of z.

**1.6 Neighbourhood of a Point :**

Neighbourhood of a point is a small
circular region excluding he points on the boundary with centre at

**2 Analytic function**

**2.1 Limit
of The Function**:

**2.2
Continuity:**

**2.3 **notes

**2.4
Differentiabil ty at the Point**

**2.5 Note:**

**2.6 Analytic (Or) Holomorphic (Or) Regular
Function**

A
function is said to be analytic at a point if its derivative exists not only at
the point but also in some neighbourhood of that point.

**2.7 Entire Function**:

A
function which is analytic everywhere in the finite plane is called an entire
function.

**2.8 The
Necessary Condition For f(z) To Be Analytic:(Cauchy-Riemann Equations)**

i. Cartesian
form: The necessary condition for a complex function f(z)=u(x,y)+iv(x,y)

**2.9
Problems Based on Analytic Function-Necessary Conditions (C-R Equations)**

1.Show
that the function f(z)=xy+iy is continuous everywhere but not differentiable
anywhere.

Hence f(z) is not differentiable anywhere though it is
continuous everywhere

2. Show that the function f(z)=e^{z} is differentiable everywhere in the complex
plane.

**2.10 Tutorial problems**

**3 Harmonic and Orthogonal
Properties Of Analytic Functions**

**3.1 Laplace Equation:**

**3.2
Properties Of Analytic Functions And Harmonic Conjugate**

**3.3 Problems Based On Harmonic
Conjugate**

**3.3 Problems Based On Harmonic Conjugate**

**4 Constructions
of Analytic Functions (Milne- Thomson Method)**

**4.5 Tutorial problems**

**5 Conformal Mapping**

**5.1 Definition:**

The
transformation w=f(z) is called as **conformal mapping** if it preserves
angle between every pair of curves through a point, both in magnitude and sense

The
transformation w=f(z) is called as **Isogonal** mapping if it preserves
angle between every pair of curves through a point in magnitude but altered in
sense

**5.2
Standard Transformations**

**1. Translation:**

The
transformation w=C+z ,where C is a complex constant ,represents a
translation

**2. Magnification:**

The
transformation w=Cz ,where C is a real constant ,represents magnification

**3. Magnification And Rotation:**

The
transformation w=Cz,where C is a complex constant ,represents magnification and
Rotation

**4. Magnification , Rotation And
Translation:**

The
transformation w = Cz + D ,where C,D are complex constant ,represents
Magnification, Rotation and Translation

**5. Inversion And Reflection:**

The
transformation w=1/z represents inversion w.r.to the unit circle |z| = 1,
followed by reflection in the real axis

**5.3 Problems Based on
Transformation**

1.
Find the image of the circle |z| = 1 by the transformation w=z+2+4i

5.4
Tutorial problems

6
**Bilinear Transformation**

**6.1
Definition:**

**6.2 Note:**

x. If one
of the point is the point at infinity the quotient of those difference which
involve this point is replaced by 1

**6.3 Problems
based on Bilinear Transformation**

**6.4 Tutorial problems**

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