Complex Analysis Notes

Polar form of a complex number solutions

  Polar form of a complex number. If z= x + iy is a complex number. Then z = r ( cosθ + sinθ ) is called polar form or trigonometric form of a complex number.   By comparing real and imaginary parts of a complex number,  we get x = r cosθ           (1) y = r sinθ           (2)   By squaring above equations and adding we get       EXAMPLE       Express in polar form.     Solution   With and y = −1 we obtain   Now   and    which is an angle whose terminal side is in the first quadrant.But   since the point lies in the third quadrant, we take the solution So, Therefore, polar form of the number is   Solved Questions of polar form Polar of   click here Polar for of       click here Polar for of   click here Polar for of        click here Polar for of    click here Polar for of         click here CHALLENGE   write the given complex number in polar form first using an argument θ = Arg(z) 1.       2 2.    −10 3.     −3i 4.       6i 5.       1 + i 6.     …

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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 }                               …

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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  

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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.

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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.      …

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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.                             

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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          

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