A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1.[1] In this expression, a is the real part and b is the imaginary part of the complex number. If {\displaystyle z=a+bi} {\displaystyle z=a+bi}, then {\displaystyle \Re z=a,\quad \Im z=b.} {\displaystyle \Re z=a,\quad \Im z=b.}
Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers are a field extension of the ordinary real numbers, in order to solve problems that cannot be solved with real numbers alone.
In this lecture, we will discuss:
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1.[1] In this expression, a is the real part and b is the imaginary part of the complex number. If {\displaystyle z=a+bi} {\displaystyle z=a+bi}, then {\displaystyle \Re z=a,\quad \Im z=b.} {\displaystyle \Re z=a,\quad \Im z=b.}
Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers are a field extension of the ordinary real numbers, in order to solve problems that cannot be solved with real numbers alone.
In this lecture, we will discuss:
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Complex analysis and its application
2.Contents,Complex number
Different forms of complex number
Types of complex number
Argand Diagram
Addition, subtraction, Multiplication & Division
Conjugate of Complex number
Complex variable
Function of complex variable
Continuity
Differentiability
Analytic Function
Harmonic Function
Application of complex Function
3.Complex Number,For most human tasks, real numbers (or even rational numbers) offer an adequate description of data. Fractions such as 2/3 and 1/8 are meaningless to a person counting stones, but essential to a person comparing the sizes of different collections of stones. Negative numbers such as -3 and-5 are meaningless when measuring the mass of an object, but essential when keeping track of monetary debits and credits.
All numbers are imaginary (even "zero“ was contentious once). Introducing the square root(s) of minus one is convenient because
all n-degree polynomials with real coefficients then haven roots, making algebra "complete";
it saves using matrix representations for objects that square to-1 (such objects representing an important part of the structure of linear equations which appear in quantum mechanics ,heat,diffusion,optics,etc) .The hottest contenders for numbers without purpose are probably the p-adic numbers (an extension of the rationales),and perhaps the expiry dates on army ration packs.
4.Complex Number is defined as an ordered pair of real number X & Y and is denoted by (X,Y)
It is also written as 𝒛=𝒙,𝒚=𝒙+𝒊𝒚,where 𝑖^2=−1
𝑥 is called Real Part of z and written as Re(z)
Y is called imaginary part of z and written as Im(z).
-If R(z) = 0 then 𝑧=𝑖𝑦, is called Purely Imaginary Number.
-If I(z) = 0 then 𝑧=𝑥, is called Purely Real Number.
-Here 𝑖can be written as (0, 1) = 0 ±1𝑖
Note:-−𝒂= 𝑎−1=𝑖𝑎
-If 𝑧=𝑥+𝑖𝑦is complex number then its conjugate or complex conjugate is defined as 𝒛=𝒙−𝒊𝒚.
5.DIFFERENT FORMS OF COMPLEX NUMBER
Cartesian or Rectangular Form :-𝑧=𝑥+𝑖𝑦
Polar Form :-𝑧=𝑟(cos𝜃+𝑖sin𝜃) 𝑜𝑟 𝑧=𝑟∠𝜃
Exponential Form :-𝑧=𝑟𝑒^𝑖𝜃
MODULUS & ARGUMENT OF COMPLEX NUMBER
Modulus of complex number (|z|) OR mod(z) OR 𝑟=√(𝑋^2+𝑌^2 )
Argument OR Amplitude of complex number (𝜃) OR arg (𝑧) OR amp(z)=tan^(−1)(𝑥/𝑦)
6.Argand Diagram
Mathematician Argand represent a complex number in a diagram known as Argand diagram. A complex number x+iy can be represented by a point P whose co–ordinate are (x,y).The axis of x is called the real axis and the axis of y the imaginary axis. The distance OP is the modulus and the angle, OP makes with the x-axis, is the argument of x+iy.
7.Addition of Complex Numbers
Let a+ib and c+id be two numbers, then
(a+ib)+(c+id)=(a+c)+i(b+d)
Procedure: In addition of complex numbers we add real parts with real parts and imaginary parts with imaginary parts.
8.Subtraction of Complex Numbers
Let a+ib and c+id be two numbers, then
(a+ib)-(c+id)=(a-c)+i(b-d)
Procedure: In subtraction of complex numbers we subtract real parts w
Complex analysis and its application
2.Contents,Complex number
Different forms of complex number
Types of complex number
Argand Diagram
Addition, subtraction, Multiplication & Division
Conjugate of Complex number
Complex variable
Function of complex variable
Continuity
Differentiability
Analytic Function
Harmonic Function
Application of complex Function
3.Complex Number,For most human tasks, real numbers (or even rational numbers) offer an adequate description of data. Fractions such as 2/3 and 1/8 are meaningless to a person counting stones, but essential to a person comparing the sizes of different collections of stones. Negative numbers such as -3 and-5 are meaningless when measuring the mass of an object, but essential when keeping track of monetary debits and credits.
All numbers are imaginary (even "zero“ was contentious once). Introducing the square root(s) of minus one is convenient because
all n-degree polynomials with real coefficients then haven roots, making algebra "complete";
it saves using matrix representations for objects that square to-1 (such objects representing an important part of the structure of linear equations which appear in quantum mechanics ,heat,diffusion,optics,etc) .The hottest contenders for numbers without purpose are probably the p-adic numbers (an extension of the rationales),and perhaps the expiry dates on army ration packs.
4.Complex Number is defined as an ordered pair of real number X & Y and is denoted by (X,Y)
It is also written as 𝒛=𝒙,𝒚=𝒙+𝒊𝒚,where 𝑖^2=−1
𝑥 is called Real Part of z and written as Re(z)
Y is called imaginary part of z and written as Im(z).
-If R(z) = 0 then 𝑧=𝑖𝑦, is called Purely Imaginary Number.
-If I(z) = 0 then 𝑧=𝑥, is called Purely Real Number.
-Here 𝑖can be written as (0, 1) = 0 ±1𝑖
Note:-−𝒂= 𝑎−1=𝑖𝑎
-If 𝑧=𝑥+𝑖𝑦is complex number then its conjugate or complex conjugate is defined as 𝒛=𝒙−𝒊𝒚.
5.DIFFERENT FORMS OF COMPLEX NUMBER
Cartesian or Rectangular Form :-𝑧=𝑥+𝑖𝑦
Polar Form :-𝑧=𝑟(cos𝜃+𝑖sin𝜃) 𝑜𝑟 𝑧=𝑟∠𝜃
Exponential Form :-𝑧=𝑟𝑒^𝑖𝜃
MODULUS & ARGUMENT OF COMPLEX NUMBER
Modulus of complex number (|z|) OR mod(z) OR 𝑟=√(𝑋^2+𝑌^2 )
Argument OR Amplitude of complex number (𝜃) OR arg (𝑧) OR amp(z)=tan^(−1)(𝑥/𝑦)
6.Argand Diagram
Mathematician Argand represent a complex number in a diagram known as Argand diagram. A complex number x+iy can be represented by a point P whose co–ordinate are (x,y).The axis of x is called the real axis and the axis of y the imaginary axis. The distance OP is the modulus and the angle, OP makes with the x-axis, is the argument of x+iy.
7.Addition of Complex Numbers
Let a+ib and c+id be two numbers, then
(a+ib)+(c+id)=(a+c)+i(b+d)
Procedure: In addition of complex numbers we add real parts with real parts and imaginary parts with imaginary parts.
8.Subtraction of Complex Numbers
Let a+ib and c+id be two numbers, then
(a+ib)-(c+id)=(a-c)+i(b-d)
Procedure: In subtraction of complex numbers we subtract real parts w
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2. DEFINITION
A complex number z = x+iy is defined as an
order pair (x,y), where x & y are real number
and i =
Y = imaginary part of z
X = real part of z
If x=0 & y≠0 then complex number is called
purely imaginary number.
If x≠0 & y=0 then complex number is called
purely real number.
3. ALGEBRA OF COMPLEX NUMBER
z1 = x1+iy1 z2 = x2+iy2
(1) Equality Of Complex Number:-
z1 ↔ z2 when x1 = x2 & y1 = y2
(2) Addition of complex number:-
z1 + z2 = (x1+iy1) + (x2+iy2)
z1 + z2 = (x1+x2) + i (y1+y2)
6. ARGAND DIAGRAM / COMPLEX PLANE
A complex number (z) can also be
represented as an order pair (x,y) and thus
as a point in a plane, known as complex
plane or argand diagram, where x-axis is
called real axis and y-axis is called
imaginary axis.