Basic concept of Complex Function

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 |
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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.
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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.
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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}
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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 .
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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}
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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}
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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 .
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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.
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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.
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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.
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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.
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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
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17. Curves and Regions in Complex Plane
Domain
An open connected set is called a domain.
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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.
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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).
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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.
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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
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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
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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
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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
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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 )
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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 .
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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
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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
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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.
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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.
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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
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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
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