Limit & Continuity Complex Analysis Lec 10 by Maqsood Ali Abbas

#complex_Analysis #Limit #Continuity #math #maqsood #BS #bs #MS #ms #physics

[nomanawais]

Numan Awais 22 09. Countinuty of polynomial function: Every polynomial function is continuos everywhere on from negative infinity to positive infinity.

[aliabbas]

Bounded function and property :if a complex function is countinous on closed region R, the f is bounded on R. There is real constant M>0 such that
|f(z)| <M for all in R

[ranasohail]

Amir Sohail (2208)
Limit of a Function
For f(x) a , the  of f as x approaches infinity is L, denoted. means that for all, there exists c such that whenever x > c. Or, symbolically: . Similarly, the  of f as x approaches negative infinity is L, denoted. means that for all there exists c such that whenever x < c.
Rational Function;
In , a  is the ratio of two polynomials and with  coefficients:, (1) where 0, and and have no common factor. The coefficients of and polynomials are called the coefficients of the . The  is called irreducible when and have no common zeros.
Complex Function:
Let the  variable z be  by z = x + iy where x and y are real  and i is, as. usual, given by i2 = −1. Now let a second  variable w be  by w = u + iv where u. and v are real .
Polynomial Function:
A  is a  such as a quadratic, a cubic, a quartic, and so on, involving only non-negative integer powers of x. We can give a general defintion of a , and  its degree.

[natashashafaqat]

Natasha shafaqat 1026
1)a rational function R(z) is the ratio of two polynomial P(z) and Q(z) with complex coefficient
R(z)=P(z)/Q(z)
Where p(z) and Q(z) have no common factor and Q(z) is non zero, the coefficient of P(z) and Q(z) are called the coefficient of rational function.
2)a continuous function is a function that has not have any discontinuities.
3) polynomial function is a function that involves only non negative integer power of a variable in a equation.eg quadratic and cubic equation.
4)if a complex function f is a continuous on a closed and bounded region R, then f is bounded on R that is there real constant M>0 such that
F(z)<M for all z in R
5) rational function are continuous on their domain.
6) supposed that f(z)=u(x, y)+iv(x, y), z°=x°+iy° and L°=u°+iv° the limit z to z° f(z)=L, if and only if
Limit (x, y) to (x°, y°) v(x, y)=v°

[aliabbas]

in complex analysis:The limit of a function

The limit of w = f(z) as z → z0 is a number l such that |f(z) − l| can be made as small as we wish by making |z − z0| sufficiently small. In some cases the limit is simply f(z0), as is the case for w = z2 − z. For example, the limit of this function as z → i is f(i) = i2 − i = −1 − i.

[aliabbas]

continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, sufficiently small changes in the input of a continuous function result in arbitrarily small changes in its output. If not continuous, a function is said to be discontinuous.

[nasurallahkhan]

Saba Noor 1038
Q.1 : The limit of function at a point a in its domain (if it exist ) is the value that the function approaches as its argument approaches . informally, A function is said to have a limt L at a if is possible to make the function arbitary close to L choosing values closer and closer to a
Q.2 A rational function is defined as the quotient of polynomials in which the denominator has a degree of at least 1 . another words there must be varible in the denominator
Also rational function is defined as p(x)/q(x) where p(x)and q(x) are polynomials and q(x)not equll to zero
Q.3 A polynomial function is quadratic, a cubic, quartic and so on involving only non -negative integer power of x
Also Algebraic expression consisting of one or more term in each of which the exponent of the varible is zero or postive integers
Q.4 A function f is defined on some set X with real or complex value called bounded function if the set of its value is bounded .
In other words there exists a real no M such that for all x in X

[gadgetshubhere]

Muqaddas Mustafa 1022
Q:Bounded function
A function f defined on some set X with real or complex values is called bounded if the set of its values is bounded.
Q:Continuous function
Complex function defined we can now also define the concept of continuity if a function at a point .If it is continuous for every point then is said to be continuous on.If is not continuous at then is said to be Discontinuous at.

[tabeerchand]

Tabeer chand 2237 evening
Limitof complex function:
The limit of w= f(z) as z→z0 is a number 1 such that|f(z) - 1| can be made as small as we wish by making |z-z0| sufficiently small. In some cases the limit is simply f(z0) as in the case for w=z2-z
For example:
The limit of this function as z→i isf(i)=i2 -i=-1-i.

[amanali]

Aman Ali 2210
Limit of a function in real analysis
Limit ki definition in calculas
Complex function ki definition
Non existent criterion
Properties of complex function
Real and imaginary part's of limit
Criterion
Continuous function
Polynomial funcation
Bounded property

[aliabbas]

A polynomial function is a function such as a quadratic, a cubic, a quartic, and so on, involving only non-negative integer powers of x. We can give a general defintion of a polynomial, and define its degree.

[anayasubhan]

Insha Akhtar 1009
A function f defined on some set x with real or complex values is called bounded if the set of its values is bounded

[muhammadanas]

Q.1: Limit of a complex function:-
The limit of f as Z tends to z° exist and is equal to L, written as lim z to z° f(z) = L,
if for every e>0 there exist a delta>0 such that |f(z)-L| <e whenever 0<|z-z°|<delta .

Q.2: criterion for non-existence of limit:-
If f approaches two complex numbers L 1≠L 2 for two different curves or paths through z°, then limit z to z° f(z) does not exist.
Q.3: Real and imaginary parts of limit:-
Suppose that f(z)= u(x, y) + iv(x, y), z°= x°+iy° and L=u°+v°i . Then limit z toz° f(z) = L if and only if
Limit (x, y) to (x°, y°) u(x, y) = u° and limit (x, y) to (x°, y°) v(x, y) = v°
Q.4: Rational function:-
A rational function is any function which can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and denominator are polynomials.
Q.5: Polynomial function:-
A polynomial function is a function that involves only nonnegative integer powers or only positive integer exponents of a variable in an equation.

[ifrariaz]

Ifra Riaz 1023 Morning
Q#1: The limit of f as Z tends to z° exist and is equal to L, written as lim z to z° f(z) = L, if for every e>0 there exist a delta>0 such that |f(z)-L| <e whenever 0<|z-z°|<delta .

Q#2:
If f approaches two complex numbers L1≠L2 for two different curves or paths through z°, then limit z to z° f(z) does not exist.

Q#3:
Suppose that f(z)= u(x, y) + iv(x, y), z°= x°+iy° and L=u°+v°i . Then limit z toz° f(z) = L if and only if

Limit (x, y) to (x°, y°) u(x, y) = u° and limit (x, y) to (x°, y°) v(x, y) = v°
Q#4:A rational function is any function which can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and denominator are polynomials.

Q#5:
A polynomial function is a function that involves only nonnegative integer powers or only positive integer exponents of a variable in an equation.

[maliusman]

1023 ( morning )
What is polynomial function?
A polynomial function is a function that involves only non-negative integer powers or only positive integer exponents of a variable in an equation like the quadratic equation, cubic equation, etc. For example, 2x+5 is a polynomial that has exponent equal to 1.

[cr-]

Masah Akram 1041 morning
Q1:A polynomial function is a function that involves only non-negative integer power or only positive integers exponent of a variable in an equation like the quadratic equation.
Example 2x+5 is a polynomial that has exponent equal to 1.

[umergee]

Ayesha Siddiqa (1025) Morning
Properties of complex numbers:
Suppose that f and g are complex functions.If lim f(z)=L and lim g(z)=M, then
1.lim cf(z)=cL, 'c' a complex constant
2.lim (f(z)+g(z))=L+M
3.limf(z).g(z)=L.M
4.lim f(z)\g(z)=L/M, provided M not equal to 0

[tabeerchand]

Tabeer chand 2237
Countinuty of rational function:
Rational function aur countinuos for all real.numbers except all those where the dinominator is zero. If the denominator of the rational functionf(x) is zero at x=a then it contains some number of factors of(x-a).

[preparationofmathsubjectof]

Sumia 1028 Morning
Q#1 Limit of a function in real analysis :
Lim (x approache x') f(x) = L
If for every€>0 there exist a ∆>0 such that if f(x) - L factorial<€ where 0<factorial x-x' factorial<®∆
2 Criterion for non existence of limit:
If f approaches two complex Numbers L1 doesn't equal to L2 for two different curves or paths though Z°, then limit z to Z° f(z) does not exist.
Q#3
Real and imaginary parts of limit:
Suppose that f(z) =u ( xy) +IV(x, y) z° = x° + iy and L= u°+v°i. Then limit z to z°
F(z)=L if and only if
Limit (x, y) to ( x°, y° ) u ( x, y) =u° and the Limit (x, y) to ( x°, y°) v( x, y)=v°
4.Rational function :
A rational function is any function which can be defined by a rational fraction, Which is an algebraic fraction such that both the numerator and denominator are polynomials.5. Polynomial function:
A polynomial function is a function that involves only nonnegative integer Powers or only positive integer exponents of a variable in an equation.

[aliabbas]

Rational Function:A rational function is simply a complex function that's a ratio of two polynomials. So, out of the examples that we have this one which is the generating function for the number of strings having no occurrences of peak consecutive zeros is rational.