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

1. Submitted To: Sadia Salma
Lecturer,Dept. of Civil engineering
Topics: Complex Analysis and Its Applications.
Submitted By: Mijanur Rahman

2. ❖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. ▪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
i. all n-degree polynomials with real coefficients then haven roots, making algebra "complete";
ii. 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 𝑖2
= −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 orcomplex conjugate isdefined as 𝒛=𝒙−𝒊𝒚.

5. ✓Cartesian or Rectangular Form :-𝑧=𝑥+𝑖𝑦
✓Polar Form :-𝑧=𝑟(cos𝜃+𝑖sin𝜃) 𝑜𝑟 𝑧=𝑟∠𝜃
✓Exponential Form :-𝑧 = 𝑟𝑒𝑖𝜃
✓Modulus of complex number (|z|) OR mod(z) OR 𝑟 = 𝑋2 + 𝑌2
✓Argument OR Amplitude of complex number (𝜃) OR arg (𝑧) OR amp(z)=tan−1 𝑥
𝑦

6. ❖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.
This Photo by Unknown Author is licensed under CC BY-SA

7. 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. 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 with
real parts and imaginary parts with imaginary parts.

9. Let a+ib and c+id be two numbers, then
(a+ib)*(c+id)=ac-bd+i(ad+bc)
Proof: (a+i)*(c+id)=𝑎𝑐 + 𝑖𝑎𝑑 + 𝑖𝑏𝑐 + 𝑖2
𝑏𝑑
=ac+i(ad+bc)+(-1)bd
=(ac-bd)+i(ad+bc)

10. •Two complex numbers which differ only in the sign of
imaginary are called conjugate of each other.
•A pair of complex number a+ib and a-ib are said to be
conjugate of each other.

11. Let a+ib and c+id be two numbers, then
𝑎+𝑖𝑏
𝑐+𝑖𝑑
=
𝑎𝑐+𝑏𝑑
𝑐2+𝑑2 +
𝑏𝑐−𝑎𝑑
𝑐2+𝑑2 𝑖

12. Let x+iy be a complex number.
Putting x= 𝑟 cos𝜃 and y= 𝑟 sin𝜃, so r= 𝑥2 + 𝑦2
cos𝜃 =
𝑥
𝑥2+𝑦2
and sin𝜃 =
𝑦
𝑥2+𝑦2

14. Complex analysis, traditionally known as the theory of functions of a
complex variable, is the branch of mathematical analysis that
investigates functions of complex numbers.
•A complex variable can be assume any complex value.
•We use z to represent a complex variable z=x+iy

15. •F(z) is a function of complex variable z and is denoted by w
W=f(z)
W=u+iv
When u and v are the real and imaginary parts of f(z)

16. CONTINUITYCONTINUITY IN TERMS OF REAL & IMAGINARY NUMBER

20. The necessary condition for f(z) to be analytic

22. 1. Solutions to 2-D Laplace equation by means of conformal mapping.
2. Quantum mechanics.
3. Series expansions with analytic continuation.
4. Transformation between special functions,
5. Control Theory
6. Signal analysis
7. Fluid Dynamics

23. 8. Electromagnetism:
9. Contour integrals :
i. Evaluate definite integrals & series.
ii. Invert power series.
iii. Form infinite products.
iv. Asymptotic solutions.
v. Stability of oscillations.
vi. Invert integral transforms.
10. 2-D designing of buildings and cars

24. ☺ ☺