Complex Exponential Function || Lec 17 || Professor Maqsood Ali Abbas

#Exponentail_Function #Complex_Exponential_function #Coplex_function #maqsood #urdu #hindi #bs #ms #complexanalysis

[NisarAhmad-vjsw]

What is Complex exponential function?
The function e^z defined by,
e^z=e^xcosxy+ie^xsiny,
Or..e^z=e^x(cosy+isiny)
What is modulus?
|e^z|=e^x is a module of exponential function
What is argument??
Arg(e^z)=y + 2nπ

Is a argument of exponential function
What is conguate
Arg e^z-(e^z-bar)=e^z-(z -bar)

[NisarAhmad-vjsw]

Nisar Ahmad Roll no 2214 eve
What is entire function?
The function that is analytic every point in z plane is called entire function

[cr-]

Masah Akram 1041
Qno1:Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that.

[mariaabdulghafoor]

Q complex algorithm:
The multipled value function in z
defined by: ln z= loge^|z| is called complex algorithm.

[knowledgeofmathematicsonli]

Muhamad Arslan 2246
1) Complex Exponential function
The function e^z defined by
e^z=e^xcosy+ie^xsiny
is called the Complex Exponential function.
2) Entire function
The exponential function e^z is entire and it's derivative is given by
d/dz.e^z=e^z
3) Modulus of exponential function
|e^z|=e^x
4) Argument of exponential function
arg(e^z)=y+2nπ,
5) Algebraic Properties of e^z
e^0=1
e^z1+e^z2=e^z1+z2
e^z1/e^z2=e^z1-z2

[tabeerchand]

Tabeer chand 2237 evening
Complex algorithm:
The multipled value function in z definite by: lnz = loge^|z| is called complex algorithm.

[tabeerchand]

Tabeer chand 2237 evening
Complex exponential function
The complex exponential function e^z is periodic with a pure imaginary period 2pie i

[abdulrehmanjb]

Ayesha Riaz 1042
Q.1. Entire function
In complex analysis an entire function is a complex valued function that is holomorphic at all finite points over the whole complex plane.
Q.2. Complex Exponential Function.
The function e^z is defined by
e^x(cos y +i sin y ) is called complex Exponential function.
Q.3. Modulus of complex Exponential function.
The polar form of a complex no. is w=e^z=r (cos theta +i sine theta)
r is the modulus of complex function |e^z|=e^x
Q.4. The argument of complex function is
arg (e^z)=y+2npi where

[haseebagill]

Haseeba Asif 2230
Q(1. Entire function:
The function f(z) which is analytic at every point 'z' is called entire function.
Q(2): The function e^z defined by
e^z=e^xcosy+ie^xsiny
is called complex exponential function
Q(3):
i). |e^z| = e^x, is modulus of complex exponential function.
ii). arg(e^z) = y+2nπ,
iii). conjugate of complex exponential function is equal to e^z— ( z-bar ).

Q(4). Properties of e^z
i) e⁰ = 1
ii) e^z1e^z2=e^z1+z2
iii) e^z1/e^z2=e^z1-z2
iv) (e^z1)^n= e^nz1
Q(5): A periodic function is a function that repeats its values at regular intervals, for example, the trigonometric functions, which repeat at intervals of 2π radians.
Q(6): The period of complex exponential function e^z is a pure imaginary period, i,e 2πi.

Q(7) The multiple-valued function ln z defined by
ln z = loge |z| + iarg(z) .

[nasurallahkhan]

Q.1: Complex exponential function
The function e^x defined by
e^x = e^x cosy + i e^x siny
is called complex exponential function
Q.2: properties :-
if z1 and z2 are complex number (i) e^0 =1
(ii) e^z1.e^z2= e^z1+z2
(iii) e^z1/e^z2= e^z1+Sr
(iv) (e^z1)^n = e^nz1 n=
Q.3 The complex exponential function e^x is periodic with a pure imaginary period 2i pi
e^z+2i pi= e^a is called periodicity

Q.4 : complex logarithmic
The multiple valued lnz defined by lnz = log€|z|+i arg|z| is called complex logarithmic

[tabeerchand]

Entire function:
A function that is analaytic at any point z in complex plane is called entire function.

[nasurallahkhan]

saba noor 1038
Q.5 Entire function :-
A function that is analytic at every point z in complex plane is known as entire function
Q.6 Modulus of complex exponential function
|e^z| = e^x
Q.7 Argument of complex exponential function
e^zbar = e^x cosy - ie^x siny =e^x cos(-y)+ i e^x sin(-y)= e^x-iy = e^z-

[muqadasamin]

Muqaddas amin 2251 evening

Q(1):
Entire function:
The function f(z) which is analytic at every point 'z' is called entire function.

Q(2):
The function e^z defined by
e^z=e^xcosy+ie^xsiny
is called complex exponential function.

Q(3):
i). |e^z| = e^x, is modulus of complex exponential function.
ii). arg(e^z) = y+2nπ,
iii). conjugate of complex exponential function is equal to e^z— ( z-bar ).

Q(4):
Properties of e^z
i) e⁰ = 1
ii) e^z1e^z2=e^z1+z2
iii) e^z1/e^z2=e^z1-z2
iv) (e^z1)^n= e^nz1

Q(5):
A periodic function is a function that repeats its values at regular intervals, for example, the trigonometric functions, which repeat at intervals of 2π radians.

Q(6):
The period of complex exponential function e^z is a pure imaginary period, i,e 2πi.

Q(7):
The multiple-valued function ln z defined by
ln z = loge |z| + iarg(z) .

[muhammadtayyab]

M Tayyab Roll no 1034 Morning
Q.1.
Entire Function?
A function thatis analytic at every point z in the complex plane is said to be an entire function.
Q.2.
Complex Exponential Function?
The function ex defined by
e^z=e^xcos v+ie^xsin v is called complex exponential function.
Q.3.
The mode of Complex Exponential Function is
|e^x|=e^x.
Q.4.
The argument of complex exponential function is
arg (e^x)=y + 2nπ, n=0, +-1, +-2.
Q.5.
The conjugate of complex exponential function is e^zbar=e^xcos y -iex Siny = e^x+it=ez-.
Q.6.
The complex exponential function e^z is period with a pure imaginary period 2πi.

[jawadrana]

Muhammad Jawad 1019
Q.1
Entire function:
A function that is analytic at every point z in a complex plane is known as entire function.
Polynomial function
P(z)=a^n
Where n is non negative integer is an entire function.
Q.2
Complex exponential function
The function
ez=excosy+iexsiny is called complex exponential function
Q.3
Modulus, argument and conjugate
The modulus, argument and conjugate of a complex function are easily determined
ez=excosy+iexsiny and if we Express complex number in polar form
W=excosy +iexsiny=r(cos theta +sin thetha).

[AliKhan-lyob]

Ans03:
The modulus argument and conjugate of complex function are easily determined by e^z=e^xcosy+I e^sin
And if we express complex function in polar form w=e^xcosy+I e^using =r (cos theta + I sin thita)

[zaidking]

Muqaddas Saeed (1021) Morning
Q#1:⭐A function that is analytic at every point z in the complex plane is said to be an entire function.
Q#2:Complex exponential function:
⭐The function ez defined by e^z=e^xcos y+ie^xsiny is called the complex exponential function.
Q#3:⭐The mod of complex exponential function is |e^z|=e^x
Q#4:⭐The argument of complex exponential function is arg(e^z)=y+2nπ, n=0±1±2
Q#5:⭐The conjugate of complex exponential function is e^zbar=e^x cos y- ie^x sin y=e^x cos (-y)+ie^x sin(-y)=e^x-iy=ez-
Q#6:⭐ Algebraic properties of ez if z1 and z2 are complex numbers then (i) e^0=1(ii) e^z1e^z2=e^z1+z^2
Q#7:⭐The complex exponential function e^z is periodic with a pure imaginary period 2πi.
Q#8: Complex Logarithm?
⭐The multiple valued function in z defined by:lnz =loge^|z|+i arg (z) is called complex logarithm.

[shaffiq]

Muhammad Shaffiq Ellahi(1020)
Q1 :
A function that is analytic at every point z in the
complex plane is said to be an entire function.
Q2:
Complex Exponential Function
The function ez defined by
e^z = e^x cos y + ie^x sin y

is called the complex exponential function
Q3:
The mod of complex Exponential function is
|e^z|= e^x
Q4:
The argument of complex Exponential function is
arg(e^z)= y + 2nπ, n = 0, ±1, ±2,
Q5:
The conjugate of complex Exponential function is
e^zbar = e^x cos y − ie^x sin y = e^x cos(−y)+ ie^x sin(−y) = e^x−iy = ez¯
Q6:
Algebraic Properties of ez
If z1 and z2 are complex numbers, then
(i) e^0 = 1
(ii) e^z1 e^z2 = e^z1+z^2
(iii) e^z1/e^z2 = e^z1−z^2
(iv) (e^z1 )
n = enz1, n = 0, ±1, ±2, ...
Q7:
The complex exponential function e^z is periodic with a pure imaginary
period 2πi.
Q8:
Complex Logarithm
The multiple-valued function ln z defined by:
lnz = loge^ |z| + i arg(z)
is called the complex logarithm.

[faizaanwar]

♦1:Entire function:
A function that is analytic at every point z in complex plane is said to be an entire function.
♦2:Complex Exponential function:
The function e^z defined by e^z=e^xCosy+ie^xSiny
is called the Complex Exponential Function.
♦3:Modulus:
If we express the complex number w=e^z in polar form
w=e^xCosy+ie^xSiny
=r(Cosθ+iSinθ)
then r=e^x
|e^z|=e^x
♦4:Argument:
arg(e^z)=y+2nπ
where
♦5:Conjugate
(e^z)bar=e^(z bar)
♦6:Properties:
If z1 and z2 are complex numbers, then
🌟e^0=1
🌟e^z1.e^z2=e^(z1+z2)
🌟e^z1/e^z2=e^(z1-z2)
🌟(e^z1)^n=e^(nz1),
♦7:Periodicity:
The real complex function is not periodic, but the complex exponential function is periodic because it is defined using the real cosine and sine functions, which are periodic.
e^(z+2πi)=e^z
♦8:Period:
The complex exponential function e^z is periodic with a pure imaginary period 2πi.
♦9:Complex Logarithm function:
The multiple-valued function ln z defined by
ln z=log e|z|+iarg(z)
is called complex logarithm.

[Introverta]

Ghulam Fareed (1013) morning
Q1:
A function that is analytic at every point z in the
complex plane is said to be an entire function.
Q2:
Complex Exponential Function
The function ez defined by
e^z = e^x cos y + ie^x sin y

is called the complex exponential function
Q3:
The mod of complex Exponential function is
|e^z|= e^x
Q4:
The argument of complex Exponential function is
arg(e^z)= y + 2nπ, n = 0, ±1, ±2,
Q5:
The conjugate of complex Exponential function is
e^zbar = e^x cos y − ie^x sin y = e^x cos(−y)+ ie^x sin(−y) = e^x−iy = ez¯
Q6:
Algebraic Properties of ez
If z1 and z2 are complex numbers, then
(i) e^0 = 1
(ii) e^z1 e^z2 = e^z1+z^2
(iii) e^z1/e^z2 = e^z1−z^2
(iv) (e^z1 )
n = enz1, n = 0, ±1, ±2, ...
Q7:
The complex exponential function e^z is periodic with a pure imaginary
period 2πi.
Q8:
Complex Logarithm
The multiple-valued function ln z defined by:
lnz = loge^ |z| + i arg(z)
is called the complex logarithm