Upper bound Lower Bound Problems Complex Analysis Lec 5 by Maqsood Ali Abbas

for BS & MS classes
#complex_analysis #lower_bound #BS #MS

[UET_ZAHIDJAMIL]

Assalam o ALIKUIM Sir your teaching method is your good and i aknowledge the method of your teaching Iam the student of UET Taxila .Thank you Very much sir. 😇😇

[hoorainfatima]

Tanseer Fatima (1002) morning

Bounded function in complex number
Ans1: A function f defined on some set X with complex values is called bounded if the set of its values is bounded.
Equation of line in complex analysis:
Ans2:In complex analysis, we know that the equation for a line is
R e [ ( ( m + i ) z + b ) ] = 0Equation of line in geometry
Ans3:The equation of a line is typically written as
y=mx+b
where m is the slope and b is the y-intercept.
Hyperbola:
And 4:The set of all points such that the difference between the distances to two focal points is constant.
*All hyperbolas have an eccentricity value greater than 1.
Standard equation of hyperbola:
The standard equation of a hyperbola is given as:
[(x2 / a2) – (y2 / b2)] = 1

[natashashafaqat]

Natasha shafaqat 1026
1)in geometry:
On the coordinate plane a line can be describe by equation
Y=mx+b
Where m is slope of line and b is intercept.
In complex analysis:
The general equation of straight line is in the form of
a-z-az+b=0 where
a€complex number and b€real number.
2) hyperbola is one of three kinds of conic sections, formed by the intersection of a plane and a double cone.
3)x^2/a^2-y^2/b^2=1 is the equation of hyperbola.
4) a function f defined on some set X with real or complex value is called bounded if the set of its value is bounded.in other words, there exist a real number M such that
|If (x)|<=M
For all x in X.
A function which is not bounded is called unbounded.

[jawadrana]

M.jawad 1019
Q.1
A function f defined on some set x with complex values is called bounded if the set of values is bounded.
Q.2
There are many different forms of equation of line. The simplest forms are
Z=x+i(mx+b) where x€R and z€C.
Q.3
A hyperbole is a type of smooth curve lying in a plane defined by its geometric properties for which it is the solution.
Q.4
Equation of hyperbola is
X^2/a^2- y^2/b^2=1

[qandyqueen]

QANDEEL TARIQ(1004)Morning
Q1:Bounded function in complex analysis:A function f is defined on some set x with or complex values is called bounded if the set of its value is bounded.In other words there exist a real number M such that |f(x)|≤M for all x in X.
NOTE:Boundedness depends on domain of the function
Q2:Equation of line in complex analysis:
The general equation of a straight line is of the form
Z=x+i(mx+b)
where a€complex number and b€ real number
(2): Equation of line in geometry:
A straight line on the co-ordinate plane can be described by the equation
Y=mx +b. Where m is the slope of line and b is the intercept.
Q3:Hyperbola:
If cone is cut by a plane and cutting plane is parallel to the axis of cone and intersect both its nappes, then curve of intersection is hyperbola.
EQUATION:
(1):Standard form of hyperbola at origin (0, 0) and transverse Axis on x-axis is
x²/a²–y²/b²=1
This Equation uses when vertices are located at (±a, 0) and foci at
(±c, 0).
(2) standard form of hyperbola at centre(0, 0) and transverse axis on y axis is
Y²/a²–x²/b²=1
this equation uses when vertices are located at( 0, ±a) and foci at (0, ±c).

Abdul Hanan (1016)
👉Q#1
A function f defined on some set X with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number M such that
|f(x)|<=M
for all x in X. A function that is not bounded is said to be unbounded.

👉Q#2

🔶 Equation of line in Geometry
A straight line on the coordinate plane can be described by the equation
y = mx+b
where m is the slope of the line and b is the intercept.

🔶Equation of line in complex analysis:
The general equation of a straight line is of the form
a–z-az–+b=0
where a€complex number and b€ real number

👉Q#3
A plane curve generated by a point so moving that the difference of the distances from two fixed points is a constant.
The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone.

👉Q#4
Equation of Hyperbole is
x^2/ a^2 - y^2/b^2 = 1

[mfahid]

A bounded function defined on some set x with real or complex values is bounded.

Equation of line is described as
|z-p|=|z-q| where p q are two complex number.

The hyperbola in complex plane has a simple expression x^2-y^2=1

Hyperbola is defined as a curve in a plane . It is defined by equation kf solution or by geometric sequences.

Equation of hyperbola is written as
x^2/a^2-y^2/b^2=1

[saim]

Saima parveen (1008)
Q:1Bounded function :
Let a function be defined as
f(x):A from A and we can define two real numbers m and M such that m<=f(x)<=M for all x€A then f(x) is called upper and lower bound of f(x)
Q:2 Equation of line in geometry:
Y=mx+c where m is slope and c is y-intercept
The equation of line in complex analysis is
z=x+i(mx+b) where x€R and z€c

[shaffiq]

Muhammad Shaffiq Ellahi(1020)
Q#1
A function f defined on some set X with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number M such that
|f(x)|<=M
for all x in X. A function that is not bounded is said to be unbounded.

Q#2


A plane curve generated by a point so moving that the difference of the distances from two fixed points is a constant.
The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone.

Q#3
Equation of line in Geometry
A straight line on the coordinate plane can be described by the equation
y = mx+b
where m is the slope of the line and b is the intercept.
Equation of line in complex analysis:
The general equation of a straight line is of the form
a–z-az–+b=0
where a€complex number and b€ real number
Q#4
Equation of Hyperbole is
x^2/ a^2 - y^2/b^2 = 1

[masoomshah]

Q1)
A function f defined on some set X with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number M such that. for all x in X. 
Q2)
a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches
Q3)
he equation for the hyperbola will have the form x 2 /a 2 − y 2 /b 2 = 1
Q4)
The equation of line in complex analysis is ˉaz+aˉz+b=0
The equation of line in Real analysis is y=mx+c

[muhammadanas]

Muhammad Anas (1043) Morning
Q.1: Bounded function:-
A function "f" defined on some set X with real and complex value is called bounded function. There exists a real number M such that for all x in X.
Q.2: Eq. of line in complex analysis:-
The equation of line in complex analysis is: a(bar)z+az(bar)+b=0, where "a" is complex number and "b" is real number.
Eq of line in geometry:-
The equation of line in geometry is: y=mx+c, where "m" is the slope of line and "b" is the intercept.
Q.3: Define Hyperbola:-
A hyperbola is a conic section defined as the locus of all points in plane the difference of whose distances and from two fixed points sepaarated by a distance is a given positive constant.
Eq of Hyperbola: (x^2/a^2)- (y^2/b^2)=1

[rimshazahid]

Rimsha Zahid 2204
Q1.
A function f defined on some set
X with real or complex values is
called bounded if the set of its
values is bounded. In other words,
there exists a real number number
M such that, for all x in X.
Q2.
The general equation of straight
line is the form of ⁻az+a⁻z+b=0,
where a is a complex number b is
a real number.
Q3.
The equation of straight line is
y=mx+p.
Q4.
A hyperbola is a type of smooth
curve living in plane, defined by its
geometric properties or by
equations for which it is the solution set. A " hyperbola" is formed by the intersection of a plane perpendicular to the base of double cone.
Q5.
Equation of hyperbola is :
" x∧2/a∧2-y∧2/b∧2=1 "

[fiazahmad]

Fiaz Ahmad 2218
Equation of lines?
Equation of straight line which is vertical or parallel to Y- axis is x=a, where a is x- coordinate of the points of the line. For example the eqution of line which is parallel to x- axis and contains the points (2, 3) is y=3.

[haseebagill]

Haseeba Asif 2230
Q1.A bounded function in complex analysis is a function that its range can be included in a closed interval, i.e for some real numbers 'a' and 'b' you get
a<_f(x)<_b for all 'x' in domain of 'f'
For example f(x)=sinx is a bounded function for all values of 'x', -1<_sinx<_1
Q2.Equation of line in geometry is y=mx+c where 'm' is slope and 'c' is y_intercept and
Equation of line in complex plane in parameteric form is z=x+i(mx+c) where x€R, z€c
Q3.Hyperbola is the set of all points the difference of whose distance from two fixed points (foci) is a contant.
Equation of hyperbola is x^2/a^2 -y^2 /b^2=1.

[cr-]

Masah Akram 1041 morning
Q1:Bounded function in complex number?
A function f defined on some set x with complex values is called Bounded if the set of values is Bounded.
Q2:Equation of line in geometry?
In complex analysis we known that the equation for a line is
Re[(m+I) z+b] =0 Eqution of a line in geometry.
Q3:
The Eqution of a typically written as
Y=mx+b
Where m is a slope and b is the y - intercept.
Q4:Hyperbola?
The set of all points such that the difference between the distance to two focal point is constant.
Equation of hyperbola:
X2/a2-y2/b2=1

[learningwaytofuture]

Muhammad Kamran 1033 morning
Q#1(Ans):Bounded function:
Bounded function is a function that its range can be included in a closed interval. That is for some real numbers a and b you get a≤f(x)≤b for all x in the domain of f. For example: f(x)=sinx is bounded because for all values of x, −1≤sinx≤1.
Q#2(Ans): Equation of line in complex analysis:
The general equation of a straight line is of the form:
z=x+i(mx+b).
Equation of line in Geometry:
A straight line on the co-ordinate plane can be describe by the equation: y=mx+c
Q#(Ans):Hyperbola:
Hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone. The plane may or may not be parallel to the axis of the cone.
Q#4(Ans):Standard Equation of Hyperbola:
x2/a2 - y2/b2=1.

[amnanaz]

Amna Naz (1027)
Ans (1) A function f defined on some set x with complex values is called bounded if the set of its values is bounded.
Ans (2) Equation of line in complex analysis Re((m+i)z+b)=0
Ans (3) The set of all points such that the difference between the distance of two focal points is called hyperbola.
Equation: (×^2/a^2)_(y^2/b^2)=1
Ans (4) Equation of line y=m×+b
Where m is slope and b is y intercept.

[alikanggaming]

Ali Raza 2220 Eve:
Equation of lines?
Equation of straight line which is vertical or parallel to Y- axis is x=a, where a is x- coordinate of the points of the line. For example the eqution of line which is parallel to x- axis and contains the points (2, 3) is y=3.

[abdulrehmanjb]

Ayesha Riaz 1042 (Morning)
Q.1. Bounded equation
A function f defined on some set X with real or complex values is called bounded if the set of its values is bounded. In other words there exist a real number M such that
|f(x)|<=M
For all x in X
Q.2. Equation of line in real analysis.
The general equation of a straight line is y= mx+c where m is the gradient and y=c is the value where the line cuts the y-axis.
Q.3. Equation line in complex analysis.
Equation of line in complex is
Z=x+i (mx+b)
Where x €R and Z €C.
Q.4. Definition and equation of hyperbola.
A hyperbola ia a conic section formed by intersecting a right circular cone with a plane at an angle such that both both halves of the cone is intersected.
The equation of hyperbola is.
y^2/a^2-x^2/b^2=1.

[arooshamubeen]

Aroosha Mubeen (2225) eve

1_ A function f defined on some set X with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number M such that. for all x in X.

2_ Equation of line in complex analysis:
"The general equation of a straight line is of the form ˉaz+aˉz+b=0, where a is complex number and b is real number".
Equation of line in geometry: y= mx+b

3_ A "hyperbola" is formed by the intersection of a plane perpendicular to the bases of a double cone.

Equation of Hyperbola:
x^2/a^2 - y^2/b^2 = 1