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Methodical directions for solving problems and tasks
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Problem 1. Show this number 
on the complex plane, and write it in algebraic, trigonometric and exponential forms.
Solution. Definition. The complex number z i s an ordered pair of real numbers (x, y) and write z=(x, y). The first number x is called real part of complex number z and denoted by x=Re z.
Second number y is called imaginary part of complex number z and denoted by y=Im z.
Each complex number corresponds to a point of the coordinate plane Oxy.
In the considered problem 
.
Definition. Two complex numbers that differ only in the sign of the imaginary part are called conjugate.
As is well known, except for a Cartesian coordinate system, the position of a point on the plane can be specified by polar coordinates 
.
Let 
are the polar coordinates of the point M, corresponding to the complex number z=(x, y).
Definition. Polar radius of point M, that is the number 
called the module of a complex number z=(x,y) and denoted by |z|:
Definition. The polar angle of the point M, that is the angle 
is called the argument of a complex number z=(x, y) and denoted by Arg z,
As is well known, the polar angle is not defined uniquely, and up to a term of the form 
.
The value of , satisfying the following condition 
is called the principal value of the argument and denoted by 
.
If satisfies the following condition 
, then
In the problem 
and
A complex number (0, 1) is called the imaginary unit and denoted by the symbol i.
Thus, 
.
Notation of a complex number in the form z=x+iy is called the algebraic form of complex number.
The algebraic form of the given complex number is the following form:
Notation of a complex number in the form 
is called the trigonometric form of a complex number.
Notation of a complex number in the form 
is called the exponential form of a complex number.
The trigonometric and exponential form of a complex number are as follows:
.
Problem 2. Find the Re z and Im z, if 
.
Solution. Addition and multiplication of complex numbers in algebraic form can also be performed, as in the case of ordinary polynomials and successive substitution 
In practical calculations the quotient is found by multiplying numerator and denominator by the conjugate denominator (so get rid of the imaginary in the denominator):
.
Thus, 
.
Date: 2016-02-19; view: 397; Нарушение авторских прав