The document discusses various topics related to complex functions and complex analysis. It defines concepts such as distance between complex numbers, circles, circular disks, neighborhoods, annuli, open and closed sets, connected sets, domains, regions, bounded regions, single-valued and multi-valued functions, and limits and continuity of complex functions. Specific examples are provided to illustrate definitions of circles, neighborhoods, single-valued and multi-valued functions. The limit of a complex function as z approaches a point z0 is defined using the epsilon-delta definition of a limit.

1. Complex Function
N. B. Vyas
Department of Mathematics,
Atmiya Institute of Tech. and Science,
Rajkot (Guj.)
N.B.V yas − Department of M athematics, AIT S − Rajkot (2)

2. Curves and Regions in Complex Plane
Distance between two complex numbers
The distance between two complex numbers z1 and z2 is given by
|z1 − z2 | or |z2 − z1 |
N.B.V yas − Department of M athematics, AIT S − Rajkot (3)

3. Curves and Regions in Complex Plane
Distance between two complex numbers
The distance between two complex numbers z1 and z2 is given by
|z1 − z2 | or |z2 − z1 |
Circles
A circle with centre z0 = (x0 , y0 ) C and radius p R+ is
represented by |z − z0 | = p
N.B.V yas − Department of M athematics, AIT S − Rajkot (3)

4. Curves and Regions in Complex Plane
Distance between two complex numbers
The distance between two complex numbers z1 and z2 is given by
|z1 − z2 | or |z2 − z1 |
Circles
A circle with centre z0 = (x0 , y0 ) C and radius p R+ is
represented by |z − z0 | = p
Interior and exterior part of the circle |z − z0 | = p
The set {z C, p R+ /|z − z0 | < p} indicates the interior part of
the circle |z − z0 | = p
N.B.V yas − Department of M athematics, AIT S − Rajkot (3)

5. Curves and Regions in Complex Plane
Distance between two complex numbers
The distance between two complex numbers z1 and z2 is given by
|z1 − z2 | or |z2 − z1 |
Circles
A circle with centre z0 = (x0 , y0 ) C and radius p R+ is
represented by |z − z0 | = p
Interior and exterior part of the circle |z − z0 | = p
The set {z C, p R+ /|z − z0 | < p} indicates the interior part of
the circle |z − z0 | = p whereas {z C, p R+ /|z − z0 | > p} indicates
exterior part of it.
N.B.V yas − Department of M athematics, AIT S − Rajkot (4)

6. Curves and Regions in Complex Plane
Circular Disk
The open circular disk with centre z0 and radius p is given by
z C, p R+ /|z − z0 | < p.
N.B.V yas − Department of M athematics, AIT S − Rajkot (5)

7. Curves and Regions in Complex Plane
Circular Disk
The open circular disk with centre z0 and radius p is given by
z C, p R+ /|z − z0 | < p. The close circular disk is given by
{z C, p R+ /|z − z0 | ≤ p}
N.B.V yas − Department of M athematics, AIT S − Rajkot (5)

8. Curves and Regions in Complex Plane
Circular Disk
The open circular disk with centre z0 and radius p is given by
z C, p R+ /|z − z0 | < p. The close circular disk is given by
{z C, p R+ /|z − z0 | ≤ p}
Neighbourhood
An open neighbourhood of a point z0 is a subset of C
containing an open circular disk centered at z0 .
N.B.V yas − Department of M athematics, AIT S − Rajkot (5)

9. Curves and Regions in Complex Plane
Circular Disk
The open circular disk with centre z0 and radius p is given by
z C, p R+ /|z − z0 | < p. The close circular disk is given by
{z C, p R+ /|z − z0 | ≤ p}
Neighbourhood
An open neighbourhood of a point z0 is a subset of C
containing an open circular disk centered at z0 . Mathematically
Np (z0 ) = {z C, p R+ /|z − z0 | < p}
N.B.V yas − Department of M athematics, AIT S − Rajkot (5)

10. Curves and Regions in Complex Plane
Circular Disk
The open circular disk with centre z0 and radius p is given by
z C, p R+ /|z − z0 | < p. The close circular disk is given by
{z C, p R+ /|z − z0 | ≤ p}
Neighbourhood
An open neighbourhood of a point z0 is a subset of C
containing an open circular disk centered at z0 . Mathematically
Np (z0 ) = {z C, p R+ /|z − z0 | < p}
A punctured or deleted neighbourhood of a point z0 contain
all the points of a neighbourhood of z0 , excepted z0 itself.
Mathematically {z C, p R+ /0 < |z − z0 | < p}
N.B.V yas − Department of M athematics, AIT S − Rajkot (6)

11. Curves and Regions in Complex Plane
Annulus
The region between two concentric circles with centre z0 of radii
p1 and p2 (> p1 ) can be represented by p1 < |z − z0 | < p2 .
N.B.V yas − Department of M athematics, AIT S − Rajkot (7)

12. Curves and Regions in Complex Plane
Annulus
The region between two concentric circles with centre z0 of radii
p1 and p2 (> p1 ) can be represented by p1 < |z − z0 | < p2 . Such a
region is called open circular ring or open annulus.
N.B.V yas − Department of M athematics, AIT S − Rajkot (7)

13. Curves and Regions in Complex Plane
Annulus
The region between two concentric circles with centre z0 of radii
p1 and p2 (> p1 ) can be represented by p1 < |z − z0 | < p2 . Such a
region is called open circular ring or open annulus.
N.B.V yas − Department of M athematics, AIT S − Rajkot (8)

14. Curves and Regions in Complex Plane
Open Set
Let S be a subset of C. It is called an open set if for each
points z0 S, there exists an open circular disk centered at z0
which included in S.
N.B.V yas − Department of M athematics, AIT S − Rajkot (9)

15. Curves and Regions in Complex Plane
Open Set
Let S be a subset of C. It is called an open set if for each
points z0 S, there exists an open circular disk centered at z0
which included in S.
Closed Set
A set S is called closed if its complement is open.
N.B.V yas − Department of M athematics, AIT S − Rajkot (9)

16. Curves and Regions in Complex Plane
Open Set
Let S be a subset of C. It is called an open set if for each
points z0 S, there exists an open circular disk centered at z0
which included in S.
Closed Set
A set S is called closed if its complement is open.
Connected Set
A set A is said to be connected if any two points of A can be
joined by finitely many line segments such that each point on the
line segment is a point of A
N.B.V yas − Department of M athematics, AIT S − Rajkot (10)

17. Curves and Regions in Complex Plane
Domain
An open connected set is called a domain.
N.B.V yas − Department of M athematics, AIT S − Rajkot (11)

18. Curves and Regions in Complex Plane
Domain
An open connected set is called a domain.
Region
It is a domain with some of its boundary points.
N.B.V yas − Department of M athematics, AIT S − Rajkot (11)

19. Curves and Regions in Complex Plane
Domain
An open connected set is called a domain.
Region
It is a domain with some of its boundary points.
Closed region
It is a region together with the boundary points (all boundary
points included).
N.B.V yas − Department of M athematics, AIT S − Rajkot (11)

20. Curves and Regions in Complex Plane
Domain
An open connected set is called a domain.
Region
It is a domain with some of its boundary points.
Closed region
It is a region together with the boundary points (all boundary
points included).
Bounded region
A region is said to be bounded if it can be enclosed in a circle
of finite radius.
N.B.V yas − Department of M athematics, AIT S − Rajkot (12)

21. Function of a Complex Variable
If z = x + iy and w = u + iw are two complex variables and
if to each point z of region R there corresponds at least on
point w of a region R we say that w is a function of z and
we write w = f (z)
N.B.V yas − Department of M athematics, AIT S − Rajkot (13)

22. Function of a Complex Variable
If z = x + iy and w = u + iw are two complex variables and
if to each point z of region R there corresponds at least on
point w of a region R we say that w is a function of z and
we write w = f (z)
If for each value of z in a region R of the z-plane there
corresponds a unique value for w then w is called single
valued function.
N.B.V yas − Department of M athematics, AIT S − Rajkot (13)

23. Function of a Complex Variable
If z = x + iy and w = u + iw are two complex variables and
if to each point z of region R there corresponds at least on
point w of a region R we say that w is a function of z and
we write w = f (z)
If for each value of z in a region R of the z-plane there
corresponds a unique value for w then w is called single
valued function.
E.g.: w = z 2 is a single valued function of z.
N.B.V yas − Department of M athematics, AIT S − Rajkot (13)

24. Function of a Complex Variable
If z = x + iy and w = u + iw are two complex variables and
if to each point z of region R there corresponds at least on
point w of a region R we say that w is a function of z and
we write w = f (z)
If for each value of z in a region R of the z-plane there
corresponds a unique value for w then w is called single
valued function.
E.g.: w = z 2 is a single valued function of z.
If for each value of z if more than one value of w exists then
w is called multi-valued function.
N.B.V yas − Department of M athematics, AIT S − Rajkot (13)

25. Function of a Complex Variable
If z = x + iy and w = u + iw are two complex variables and
if to each point z of region R there corresponds at least on
point w of a region R we say that w is a function of z and
we write w = f (z)
If for each value of z in a region R of the z-plane there
corresponds a unique value for w then w is called single
valued function.
E.g.: w = z 2 is a single valued function of z.
If for each value of z if more than one value of w exists then
w is called multi-valued function.
√
E.g.: w = Z
N.B.V yas − Department of M athematics, AIT S − Rajkot (13)

26. Function of a Complex Variable
If z = x + iy and w = u + iw are two complex variables and
if to each point z of region R there corresponds at least on
point w of a region R we say that w is a function of z and
we write w = f (z)
If for each value of z in a region R of the z-plane there
corresponds a unique value for w then w is called single
valued function.
E.g.: w = z 2 is a single valued function of z.
If for each value of z if more than one value of w exists then
w is called multi-valued function.
√
E.g.: w = Z
w = f (z) = u(x, y) + iv(x, y) where u(x, y) and v(x, y) are
known as real and imaginary parts of the function w.
N.B.V yas − Department of M athematics, AIT S − Rajkot (13)

27. Function of a Complex Variable
If z = x + iy and w = u + iw are two complex variables and
if to each point z of region R there corresponds at least on
point w of a region R we say that w is a function of z and
we write w = f (z)
If for each value of z in a region R of the z-plane there
corresponds a unique value for w then w is called single
valued function.
E.g.: w = z 2 is a single valued function of z.
If for each value of z if more than one value of w exists then
w is called multi-valued function.
√
E.g.: w = Z
w = f (z) = u(x, y) + iv(x, y) where u(x, y) and v(x, y) are
known as real and imaginary parts of the function w.
E.g.: f (z) = z 2 = (x + iy)2 = (x2 − y 2 ) + i(2xy)
N.B.V yas − Department of M athematics, AIT S − Rajkot (13)

28. Function of a Complex Variable
If z = x + iy and w = u + iw are two complex variables and
if to each point z of region R there corresponds at least on
point w of a region R we say that w is a function of z and
we write w = f (z)
If for each value of z in a region R of the z-plane there
corresponds a unique value for w then w is called single
valued function.
E.g.: w = z 2 is a single valued function of z.
If for each value of z if more than one value of w exists then
w is called multi-valued function.
√
E.g.: w = Z
w = f (z) = u(x, y) + iv(x, y) where u(x, y) and v(x, y) are
known as real and imaginary parts of the function w.
E.g.: f (z) = z 2 = (x + iy)2 = (x2 − y 2 ) + i(2xy)
∴ u(x, y) = x2 − y 2 and v(x, y) = 2xy
N.B.V yas − Department of M athematics, AIT S − Rajkot (14)

29. Limit and Continuity of f (z)
A function w = f (z) is said to have the limit l as z
approaches a point z0 if for given small positive number ε we
can find positive number δ such that for all z = z0 in a disk
|z − z0 | < δ we have |f (z) − l| < ε
N.B.V yas − Department of M athematics, AIT S − Rajkot (15)

30. Limit and Continuity of f (z)
A function w = f (z) is said to have the limit l as z
approaches a point z0 if for given small positive number ε we
can find positive number δ such that for all z = z0 in a disk
|z − z0 | < δ we have |f (z) − l| < ε
Symbolically, we write lim f (z) = l
z→z0
N.B.V yas − Department of M athematics, AIT S − Rajkot (15)

31. Limit and Continuity of f (z)
A function w = f (z) is said to have the limit l as z
approaches a point z0 if for given small positive number ε we
can find positive number δ such that for all z = z0 in a disk
|z − z0 | < δ we have |f (z) − l| < ε
Symbolically, we write lim f (z) = l
z→z0
A function w = f (z) = u(x, y) + iv(x, y) is said to be
continuous at z = z0 if f (z0 ) is defined and
lim f (z) = f (z0 )
z→z0
N.B.V yas − Department of M athematics, AIT S − Rajkot (15)

32. Limit and Continuity of f (z)
A function w = f (z) is said to have the limit l as z
approaches a point z0 if for given small positive number ε we
can find positive number δ such that for all z = z0 in a disk
|z − z0 | < δ we have |f (z) − l| < ε
Symbolically, we write lim f (z) = l
z→z0
A function w = f (z) = u(x, y) + iv(x, y) is said to be
continuous at z = z0 if f (z0 ) is defined and
lim f (z) = f (z0 )
z→z0
In other words if w = f (z) = u(x, y) + iv(x, y) is continuous
at z = z0 then u(x, y) and v(x, y) both are continuous at
(x0 , y0 )
N.B.V yas − Department of M athematics, AIT S − Rajkot (15)

33. Limit and Continuity of f (z)
A function w = f (z) is said to have the limit l as z
approaches a point z0 if for given small positive number ε we
can find positive number δ such that for all z = z0 in a disk
|z − z0 | < δ we have |f (z) − l| < ε
Symbolically, we write lim f (z) = l
z→z0
A function w = f (z) = u(x, y) + iv(x, y) is said to be
continuous at z = z0 if f (z0 ) is defined and
lim f (z) = f (z0 )
z→z0
In other words if w = f (z) = u(x, y) + iv(x, y) is continuous
at z = z0 then u(x, y) and v(x, y) both are continuous at
(x0 , y0 )
And conversely if u(x, y) and v(x, y) both are continuous at
(x0 , y0 ) then f (z) is continuous at z = z0 .
N.B.V yas − Department of M athematics, AIT S − Rajkot (16)

34. Differentiation of f (z)
The derivative of a complex function w = f (z) a point z0 is
written as f (z0 ) and is defined by
dw f (z0 + δz) − f (z0 )
= f (z0 ) = lim provided limit exists.
dz δz→0 δz
N.B.V yas − Department of M athematics, AIT S − Rajkot (17)

35. Differentiation of f (z)
The derivative of a complex function w = f (z) a point z0 is
written as f (z0 ) and is defined by
dw f (z0 + δz) − f (z0 )
= f (z0 ) = lim provided limit exists.
dz δz→0 δz
Then f is said to be differentiable at z0 if we write the
change δz = z − z0 since z = z0 + δz
f (z) − f (z0 )
∴ f (z0 ) = lim
z→z0 z − z0
N.B.V yas − Department of M athematics, AIT S − Rajkot (18)

36. Analytic Functions
A single - valued complex function f (z) is said to be
analytic at a point z0 in the domain D of the z−plane, if
f (z) is differentiable at z0 and at every point in some
neighbourhood of z0 .
Point where function is not analytic (i.e. it is not single
valued or not) are called singular points or singularities.
From the definition of analytic function
1 To every point z of R, corresponds a definite value of f (z).
2 f (z) is continuous function of z in the region R.
3 At every point of z in R, f (z) has a unique derivative.
N.B.V yas − Department of M athematics, AIT S − Rajkot (19)

37. Cauchy-Riemann Equation
f is analytic in domain D if and only if the first partial
derivative of u and v satisfy the two equations
∂u ∂v ∂u ∂v
= , =− − − − − − (1)
∂x ∂y ∂y ∂x
The equation (1) are called C-R equations.
N.B.V yas − Department of M athematics, AIT S − Rajkot (20)

38. Example
Ex. Find domain of the following functions:
1
1 2
z +1
1
Sol. Here f (z) = 2
z +1
f (z) is undefined if z = i and z = −i
∴ Domain is a complex plane except z = ±i
1
2 arg
z
1
Sol. Here f (z) = arg
z
1 1 1 x − iy x − iy
= = x = 2
z x + iy x + iy x − iy x + y2
1
∴ is undefined for z = 0
z
1
Domain of arg is a complex plane except z = 0.
N.B.V yas −z
Department of M athematics, AIT S − Rajkot (21)

39. Example
z
3
z+z¯
z
Sol. Here f (z) =
z+z ¯
f (z) is undefined if z + z = 0
¯
i.e. (x + iy) + (x − iy) = 0
∴ 2x = 0
∴x=0
f (z) is undefined if x = 0
Domain is complex plane except x = 0
N.B.V yas − Department of M athematics, AIT S − Rajkot (22)