complex function
TRANSCRIPT
Complex Function
N. B. Vyas
Department of Mathematics,Atmiya Institute of Tech. and Science,
Rajkot (Guj.)
N.B.V yas − Department of Mathematics, AITS − Rajkot (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|
Circles
A circle with centre z0 = (x0, y0)εC and radius pεR+ isrepresented by |z − z0| = p
Interior and exterior part of the circle |z − z0| = p
The set {zεC, pεR+/|z − z0| < p} indicates the interior part ofthe circle |z − z0| = p whereas {zεC, pεR+/|z − z0| > p} indicatesexterior part of it.
N.B.V yas − Department of Mathematics, AITS − Rajkot (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)εC and radius pεR+ isrepresented by |z − z0| = p
Interior and exterior part of the circle |z − z0| = p
The set {zεC, pεR+/|z − z0| < p} indicates the interior part ofthe circle |z − z0| = p whereas {zεC, pεR+/|z − z0| > p} indicatesexterior part of it.
N.B.V yas − Department of Mathematics, AITS − Rajkot (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)εC and radius pεR+ isrepresented by |z − z0| = p
Interior and exterior part of the circle |z − z0| = p
The set {zεC, pεR+/|z − z0| < p} indicates the interior part ofthe circle |z − z0| = p
whereas {zεC, pεR+/|z − z0| > p} indicatesexterior part of it.
N.B.V yas − Department of Mathematics, AITS − Rajkot (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)εC and radius pεR+ isrepresented by |z − z0| = p
Interior and exterior part of the circle |z − z0| = p
The set {zεC, pεR+/|z − z0| < p} indicates the interior part ofthe circle |z − z0| = p whereas {zεC, pεR+/|z − z0| > p} indicatesexterior part of it.
N.B.V yas − Department of Mathematics, AITS − Rajkot (4)
Curves and Regions in Complex Plane
Circular Disk
The open circular disk with centre z0 and radius p is given byzεC, pεR+/|z − z0| < p.
The close circular disk is given by{zεC, pεR+/|z − z0| ≤ p}
Neighbourhood
An open neighbourhood of a point z0 is a subset of Ccontaining an open circular disk centered at z0 . MathematicallyNp(z0) = {zεC, pεR+/|z − z0| < p}A punctured or deleted neighbourhood of a point z0 containall the points of a neighbourhood of z0, excepted z0 itself.Mathematically {zεC, pεR+/0 < |z − z0| < p}
N.B.V yas − Department of Mathematics, AITS − Rajkot (5)
Curves and Regions in Complex Plane
Circular Disk
The open circular disk with centre z0 and radius p is given byzεC, pεR+/|z − z0| < p. The close circular disk is given by{zεC, pεR+/|z − z0| ≤ p}
Neighbourhood
An open neighbourhood of a point z0 is a subset of Ccontaining an open circular disk centered at z0 . MathematicallyNp(z0) = {zεC, pεR+/|z − z0| < p}A punctured or deleted neighbourhood of a point z0 containall the points of a neighbourhood of z0, excepted z0 itself.Mathematically {zεC, pεR+/0 < |z − z0| < p}
N.B.V yas − Department of Mathematics, AITS − Rajkot (5)
Curves and Regions in Complex Plane
Circular Disk
The open circular disk with centre z0 and radius p is given byzεC, pεR+/|z − z0| < p. The close circular disk is given by{zεC, pεR+/|z − z0| ≤ p}
Neighbourhood
An open neighbourhood of a point z0 is a subset of Ccontaining an open circular disk centered at z0 .
MathematicallyNp(z0) = {zεC, pεR+/|z − z0| < p}A punctured or deleted neighbourhood of a point z0 containall the points of a neighbourhood of z0, excepted z0 itself.Mathematically {zεC, pεR+/0 < |z − z0| < p}
N.B.V yas − Department of Mathematics, AITS − Rajkot (5)
Curves and Regions in Complex Plane
Circular Disk
The open circular disk with centre z0 and radius p is given byzεC, pεR+/|z − z0| < p. The close circular disk is given by{zεC, pεR+/|z − z0| ≤ p}
Neighbourhood
An open neighbourhood of a point z0 is a subset of Ccontaining an open circular disk centered at z0 . MathematicallyNp(z0) = {zεC, pεR+/|z − z0| < p}
A punctured or deleted neighbourhood of a point z0 containall the points of a neighbourhood of z0, excepted z0 itself.Mathematically {zεC, pεR+/0 < |z − z0| < p}
N.B.V yas − Department of Mathematics, AITS − Rajkot (5)
Curves and Regions in Complex Plane
Circular Disk
The open circular disk with centre z0 and radius p is given byzεC, pεR+/|z − z0| < p. The close circular disk is given by{zεC, pεR+/|z − z0| ≤ p}
Neighbourhood
An open neighbourhood of a point z0 is a subset of Ccontaining an open circular disk centered at z0 . MathematicallyNp(z0) = {zεC, pεR+/|z − z0| < p}A punctured or deleted neighbourhood of a point z0 containall the points of a neighbourhood of z0, excepted z0 itself.Mathematically {zεC, pεR+/0 < |z − z0| < p}
N.B.V yas − Department of Mathematics, AITS − Rajkot (6)
Curves and Regions in Complex Plane
Annulus
The region between two concentric circles with centre z0 of radiip1 and p2(> p1) can be represented by p1 < |z − z0| < p2.
Such aregion is called open circular ring or open annulus.
N.B.V yas − Department of Mathematics, AITS − Rajkot (7)
Curves and Regions in Complex Plane
Annulus
The region between two concentric circles with centre z0 of radiip1 and p2(> p1) can be represented by p1 < |z − z0| < p2. Such aregion is called open circular ring or open annulus.
N.B.V yas − Department of Mathematics, AITS − Rajkot (7)
Curves and Regions in Complex Plane
Annulus
The region between two concentric circles with centre z0 of radiip1 and p2(> p1) can be represented by p1 < |z − z0| < p2. Such aregion is called open circular ring or open annulus.
N.B.V yas − Department of Mathematics, AITS − Rajkot (8)
Curves and Regions in Complex Plane
Open Set
Let S be a subset of C. It is called an open set if for eachpoints z0εS, there exists an open circular disk centered at z0which 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 bejoined by finitely many line segments such that each point on theline segment is a point of A
N.B.V yas − Department of Mathematics, AITS − Rajkot (9)
Curves and Regions in Complex Plane
Open Set
Let S be a subset of C. It is called an open set if for eachpoints z0εS, there exists an open circular disk centered at z0which 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 bejoined by finitely many line segments such that each point on theline segment is a point of A
N.B.V yas − Department of Mathematics, AITS − Rajkot (9)
Curves and Regions in Complex Plane
Open Set
Let S be a subset of C. It is called an open set if for eachpoints z0εS, there exists an open circular disk centered at z0which 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 bejoined by finitely many line segments such that each point on theline segment is a point of A
N.B.V yas − Department of Mathematics, AITS − Rajkot (10)
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 boundarypoints included).
Bounded region
A region is said to be bounded if it can be enclosed in a circleof finite radius.
N.B.V yas − Department of Mathematics, AITS − Rajkot (11)
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 boundarypoints included).
Bounded region
A region is said to be bounded if it can be enclosed in a circleof finite radius.
N.B.V yas − Department of Mathematics, AITS − Rajkot (11)
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 boundarypoints included).
Bounded region
A region is said to be bounded if it can be enclosed in a circleof finite radius.
N.B.V yas − Department of Mathematics, AITS − Rajkot (11)
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 boundarypoints included).
Bounded region
A region is said to be bounded if it can be enclosed in a circleof finite radius.
N.B.V yas − Department of Mathematics, AITS − Rajkot (12)
Function of a Complex Variable
If z = x+ iy and w = u+ iw are two complex variables andif to each point z of region R there corresponds at least onpoint w of a region R we say that w is a function of z andwe write w = f(z)
If for each value of z in a region R of the z-plane therecorresponds a unique value for w then w is called singlevalued function.
E.g.: w = z2 is a single valued function of z.
If for each value of z if more than one value of w exists thenw 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) areknown as real and imaginary parts of the function w.
E.g.: f(z) = z2 = (x+ iy)2 = (x2 − y2) + i(2xy)
∴ u(x, y) = x2 − y2 and v(x, y) = 2xy
N.B.V yas − Department of Mathematics, AITS − Rajkot (13)
Function of a Complex Variable
If z = x+ iy and w = u+ iw are two complex variables andif to each point z of region R there corresponds at least onpoint w of a region R we say that w is a function of z andwe write w = f(z)
If for each value of z in a region R of the z-plane therecorresponds a unique value for w then w is called singlevalued function.
E.g.: w = z2 is a single valued function of z.
If for each value of z if more than one value of w exists thenw 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) areknown as real and imaginary parts of the function w.
E.g.: f(z) = z2 = (x+ iy)2 = (x2 − y2) + i(2xy)
∴ u(x, y) = x2 − y2 and v(x, y) = 2xy
N.B.V yas − Department of Mathematics, AITS − Rajkot (13)
Function of a Complex Variable
If z = x+ iy and w = u+ iw are two complex variables andif to each point z of region R there corresponds at least onpoint w of a region R we say that w is a function of z andwe write w = f(z)
If for each value of z in a region R of the z-plane therecorresponds a unique value for w then w is called singlevalued function.
E.g.: w = z2 is a single valued function of z.
If for each value of z if more than one value of w exists thenw 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) areknown as real and imaginary parts of the function w.
E.g.: f(z) = z2 = (x+ iy)2 = (x2 − y2) + i(2xy)
∴ u(x, y) = x2 − y2 and v(x, y) = 2xy
N.B.V yas − Department of Mathematics, AITS − Rajkot (13)
Function of a Complex Variable
If z = x+ iy and w = u+ iw are two complex variables andif to each point z of region R there corresponds at least onpoint w of a region R we say that w is a function of z andwe write w = f(z)
If for each value of z in a region R of the z-plane therecorresponds a unique value for w then w is called singlevalued function.
E.g.: w = z2 is a single valued function of z.
If for each value of z if more than one value of w exists thenw 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) areknown as real and imaginary parts of the function w.
E.g.: f(z) = z2 = (x+ iy)2 = (x2 − y2) + i(2xy)
∴ u(x, y) = x2 − y2 and v(x, y) = 2xy
N.B.V yas − Department of Mathematics, AITS − Rajkot (13)
Function of a Complex Variable
If z = x+ iy and w = u+ iw are two complex variables andif to each point z of region R there corresponds at least onpoint w of a region R we say that w is a function of z andwe write w = f(z)
If for each value of z in a region R of the z-plane therecorresponds a unique value for w then w is called singlevalued function.
E.g.: w = z2 is a single valued function of z.
If for each value of z if more than one value of w exists thenw 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) areknown as real and imaginary parts of the function w.
E.g.: f(z) = z2 = (x+ iy)2 = (x2 − y2) + i(2xy)
∴ u(x, y) = x2 − y2 and v(x, y) = 2xy
N.B.V yas − Department of Mathematics, AITS − Rajkot (13)
Function of a Complex Variable
If z = x+ iy and w = u+ iw are two complex variables andif to each point z of region R there corresponds at least onpoint w of a region R we say that w is a function of z andwe write w = f(z)
If for each value of z in a region R of the z-plane therecorresponds a unique value for w then w is called singlevalued function.
E.g.: w = z2 is a single valued function of z.
If for each value of z if more than one value of w exists thenw 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) areknown as real and imaginary parts of the function w.
E.g.: f(z) = z2 = (x+ iy)2 = (x2 − y2) + i(2xy)
∴ u(x, y) = x2 − y2 and v(x, y) = 2xy
N.B.V yas − Department of Mathematics, AITS − Rajkot (13)
Function of a Complex Variable
If z = x+ iy and w = u+ iw are two complex variables andif to each point z of region R there corresponds at least onpoint w of a region R we say that w is a function of z andwe write w = f(z)
If for each value of z in a region R of the z-plane therecorresponds a unique value for w then w is called singlevalued function.
E.g.: w = z2 is a single valued function of z.
If for each value of z if more than one value of w exists thenw 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) areknown as real and imaginary parts of the function w.
E.g.: f(z) = z2 = (x+ iy)2 = (x2 − y2) + i(2xy)
∴ u(x, y) = x2 − y2 and v(x, y) = 2xy
N.B.V yas − Department of Mathematics, AITS − Rajkot (13)
Function of a Complex Variable
If z = x+ iy and w = u+ iw are two complex variables andif to each point z of region R there corresponds at least onpoint w of a region R we say that w is a function of z andwe write w = f(z)
If for each value of z in a region R of the z-plane therecorresponds a unique value for w then w is called singlevalued function.
E.g.: w = z2 is a single valued function of z.
If for each value of z if more than one value of w exists thenw 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) areknown as real and imaginary parts of the function w.
E.g.: f(z) = z2 = (x+ iy)2 = (x2 − y2) + i(2xy)
∴ u(x, y) = x2 − y2 and v(x, y) = 2xy
N.B.V yas − Department of Mathematics, AITS − Rajkot (14)
Limit and Continuity of f(z)
A function w = f(z) is said to have the limit l as zapproaches a point z0 if for given small positive number ε wecan find positive number δ such that for all z 6= z0 in a disk|z − z0| < δ we have |f(z)− l| < ε
Symbolically, we write limz→z0
f(z) = l
A function w = f(z) = u(x, y) + iv(x, y) is said to becontinuous at z = z0 if f(z0) is defined andlimz→z0
f(z) = f(z0)
In other words if w = f(z) = u(x, y) + iv(x, y) is continuousat 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 Mathematics, AITS − Rajkot (15)
Limit and Continuity of f(z)
A function w = f(z) is said to have the limit l as zapproaches a point z0 if for given small positive number ε wecan find positive number δ such that for all z 6= z0 in a disk|z − z0| < δ we have |f(z)− l| < εSymbolically, we write lim
z→z0f(z) = l
A function w = f(z) = u(x, y) + iv(x, y) is said to becontinuous at z = z0 if f(z0) is defined andlimz→z0
f(z) = f(z0)
In other words if w = f(z) = u(x, y) + iv(x, y) is continuousat 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 Mathematics, AITS − Rajkot (15)
Limit and Continuity of f(z)
A function w = f(z) is said to have the limit l as zapproaches a point z0 if for given small positive number ε wecan find positive number δ such that for all z 6= z0 in a disk|z − z0| < δ we have |f(z)− l| < εSymbolically, we write lim
z→z0f(z) = l
A function w = f(z) = u(x, y) + iv(x, y) is said to becontinuous at z = z0 if f(z0) is defined andlimz→z0
f(z) = f(z0)
In other words if w = f(z) = u(x, y) + iv(x, y) is continuousat 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 Mathematics, AITS − Rajkot (15)
Limit and Continuity of f(z)
A function w = f(z) is said to have the limit l as zapproaches a point z0 if for given small positive number ε wecan find positive number δ such that for all z 6= z0 in a disk|z − z0| < δ we have |f(z)− l| < εSymbolically, we write lim
z→z0f(z) = l
A function w = f(z) = u(x, y) + iv(x, y) is said to becontinuous at z = z0 if f(z0) is defined andlimz→z0
f(z) = f(z0)
In other words if w = f(z) = u(x, y) + iv(x, y) is continuousat 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 Mathematics, AITS − Rajkot (15)
Limit and Continuity of f(z)
A function w = f(z) is said to have the limit l as zapproaches a point z0 if for given small positive number ε wecan find positive number δ such that for all z 6= z0 in a disk|z − z0| < δ we have |f(z)− l| < εSymbolically, we write lim
z→z0f(z) = l
A function w = f(z) = u(x, y) + iv(x, y) is said to becontinuous at z = z0 if f(z0) is defined andlimz→z0
f(z) = f(z0)
In other words if w = f(z) = u(x, y) + iv(x, y) is continuousat 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 Mathematics, AITS − Rajkot (16)
Differentiation of f(z)
The derivative of a complex function w = f(z) a point z0 iswritten as f ′(z0) and is defined bydw
dz= f ′(z0) = lim
δz→0
f(z0 + δz)− f(z0)
δzprovided limit exists.
Then f is said to be differentiable at z0 if we write thechange δz = z − z0 since z = z0 + δz
∴ f ′(z0) = limz→z0
f(z)− f(z0)
z − z0
N.B.V yas − Department of Mathematics, AITS − Rajkot (17)
Differentiation of f(z)
The derivative of a complex function w = f(z) a point z0 iswritten as f ′(z0) and is defined bydw
dz= f ′(z0) = lim
δz→0
f(z0 + δz)− f(z0)
δzprovided limit exists.
Then f is said to be differentiable at z0 if we write thechange δz = z − z0 since z = z0 + δz
∴ f ′(z0) = limz→z0
f(z)− f(z0)
z − z0
N.B.V yas − Department of Mathematics, AITS − Rajkot (18)
Analytic Functions
A single - valued complex function f(z) is said to beanalytic at a point z0 in the domain D of the z−plane, iff(z) is differentiable at z0 and at every point in someneighbourhood of z0.
Point where function is not analytic (i.e. it is not singlevalued 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 Mathematics, AITS − Rajkot (19)
Cauchy-Riemann Equation
f is analytic in domain D if and only if the first partialderivative of u and v satisfy the two equations
∂u
∂x=∂v
∂y,∂u
∂y= −∂v
∂x−−−−− (1)
The equation (1) are called C-R equations.
N.B.V yas − Department of Mathematics, AITS − Rajkot (20)
Example
Ex. Find domain of the following functions:
11
z2 + 1
Sol. Here f(z) =1
z2 + 1f(z) is undefined if z = i and z = −i∴ Domain is a complex plane except z = ±i
2 arg
(1
z
)Sol. Here f(z) = arg
(1
z
)1
z=
1
x+ iy=
1
x+ iyxx− iyx− iy
=x− iyx2 + y2
∴1
zis undefined for z = 0
Domain of arg
(1
z
)is a complex plane except z = 0.
N.B.V yas − Department of Mathematics, AITS − Rajkot (21)
Example
3z
z + z̄
Sol. Here f(z) =z
z + z̄f(z) is undefined if z + z̄ = 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 Mathematics, AITS − Rajkot (22)