Complex Analysis Notes

Locus of a complex number

Locus of complex number play a very important role in the complex analysis. Today we learn how to represent and find the locus of a complex number. We can also prepare Multiple choice questions about this topic and complex analysis. Let P(Z) be the property that satisfied by a complex number z = x + iy. Let there is a complex number that may satisfy the condition |z| =2. Then the set A = { z : |z| = 2 } ⇒ { ( x, y} : | z | = } ⇒ { ( x, y} : | z | = } is called the locus of the complex number z satisfying | z | =2. From above the locus of complex number z represents a circle with the center at ( 0, 0 ) and radius r = 2. EXAMPLE: Find the locus of a complex number z = x + iy, such that for a fixed point , |z-z1 |= aSolution: As A = { z ; |z-z1 | =a } ⇒ { ( x, y} : | | = a } …

Modulus and Argument of a complex number

Modulus: It is defined as Argument: Arg(Z): It is defined as it is also known as amplitude of Z symbolically amp(Z) Note: If |Z|=0 then x=0 and y=0 The Argument θ of a complex number is not unique. If θ be the value of argument so also is 2nπ + θ where n=0,± 1, ±2,... Principal Value Argument.: The value of argument which satisfy the inequality complex analysis mcqs

Polar form of a complex number

The polar coordinates of a P are (r,θ) where r is known as modulus and θ is the argument of a complex number Z. From above figure a = r cosθ b = r sinθ or The plane is known as Argand plane or Argand Diagram or complex plane or Gaussian plane.

Operations on Complex Numbers

Let Z1=( x1, y1 ) = Addition: The sum of two complex numbers is Subtraction: The difference of two complex numbers is Multiplication: The product of two complex numbers is = Division: The Quotient of two complex Numbers is = Remark: A set of complex numbers form a field. …

Need Of Complex Numbers

Need Of Complex Numbers Some quadratic equations which have no solutions in real numbers. For Example 1. 2. 3. In order to find the solutions of above given or similar quadratic equations, the symbol " i " was used. Euler was the first person to introduce the symbol " i ". where Gauss ( 1777-1855) was a German Mathematician and was a first person to prove in satisfactory manner that there are some algebraic equations which have real coefficients have complex roots in the form of . Note: complex and irrational roots always occur in pairs.

properties of complex numbers w.r.t distributive laws

Distributive laws Left Distributive Law Right Distributive Law

Properties of complex numbers w.r.t multiplication

Properties of complex Numbers If be the three complex numbers then Associative law of multiplication Multiplicative identity.. Commutative law of multiplication For each non-zero

Definition ( Complex number)

Definition( Complex Number) A complex number is an element (x, y) of the set = { } obeying the following rules of addition and multiplication If then 1. 2. as Z= 3+4i

Properties of complex Numbers with respect to addition

There are following properties of complex numbers. Associative, additive identity, additive inverse and commutative properties. If be the three complex numbers then . Associative property of addition 2. Z + 0 = 0 + Z = Z Additive identity 3. Z + ( -Z ) = ( 0,0) = 0 Additive inverse property 4. . . . . . . . . . . Commutative property of addition READ Related topics Properties of complex Numbers with respect to multiplication. Definitions of complex numbers