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Complex functions Tutorial
Complex analysis studies the most unexpected, surprising, even paradoxical ideas in mathematics. The familiar rules of math of real numbers may break down when applied to complex numbers.
Free Lessons

Lesson 1 Complex numbers
In this lesson: Forms and representations of the complex numbers; Modulus and arguments; Principal value of the argument.

Lesson 2 Trigonometric and algebraic form conversion
In this lesson: Complex numbers forms conversion; Examples of the conversion.

Lesson 3 The algebra of complex numbers
In this lesson: Arithmetic operations with complex numbers; Properties of the complex numbers; Geometric interpretation of addition & subtraction.

Lesson 4 Geometric interpretation of multiplication
In this lesson: The modulus and argument of the product; Multiplication of complex numbers as stretching - squeezing and rotation; Multiplying a complex number by imaginary unit i and by powers of i.

Lesson 5 Division of the complex numbers
In this lesson: Definition and notation of conjugates and reciprocals; Division as multiplication and reciprocation.

Lesson 6 Powers and roots of complex numbers
In this lesson: De Moire's theorem; Powers of complex numbers; n-th root of complex numbers.

Lesson 7 Complex Exponential Function and Complex Logarithm Function
In this lesson: Definition and notation; Complex logarithm function is a multi-valued function; Principal branch of the logarithm.

Lesson 8 Complex Trigonometric Functions and Complex Inverse Trigonometric Functions
In this lesson: A difference between the real and complex trigonometric functions; Relationship to exponential function; Identities; Derivatives and Indefinite integrals of inverse trigonometric functions.

Lesson 9 Complex Hyperbolic Functions and Inverse Hyperbolic Functions
In this lesson: The notations; Definitions; Derivatives and Indefinite integrals of inverse hyperbolic functions.

Lesson 10 Complex Power Function
In this lesson: Raising a complex number to a complex power; Derivatives and Indefinite integral of complex power function.

Lesson 11 Complex Rational Functions
In this lesson: Definition of the rational function; Möbius transformations; Fractional-linear function; Zhukovskii function.

Complex numbers
Definition, forms & representation.

Complex Analysis. FreeTutorial

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1. Algebraic form of the complex numbers

2. Geometric representation of the complex numbers
      2.1 Cartesian representation of the complex numbers

3. Polar representation of the complex numbers
      3.1 Vector representation of the complex numbers
      3.2 Modulus and argument of the complex numbers
            3.2.1 Modulus of the complex numbers
            3.2.2 Argument of the complex numbers
            3.2.3 Trigonometric form of the complex numbers
            3.2.4 Principal value of the argument

1. Algebraic form of the complex numbers

A complex number z is a number of the form
z
= x + yi,
where x and y are real numbers, and i is the imaginary unit, with the property i2= -1.
The real number x is called the real part of the complex number. It is denoted by Re(z).
The real number y is the imaginary part. It is denoted by Im(z).
For example, 2 + 3i is a complex number, with real part 2 and imaginary part 3.
If y = 0, the number z is real. The real numbers may be regarded as subset of the set of all complex numbers by considering them as a complex
z = x = x + 0i.
If x = 0, the number is purely imaginary:
z
= y = 0 + yi.
The imaginary unit i = 0 + 1i.
Zero is the only number which is at once real and purely imaginary: 0 = 0 + 0i.
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.
x
1+ y1i = x2 + y2i if x1 = x2 and y1 = y2.

The complex numbers can be defined as ordered pairs of real numbers z(x, y) or (x, y).
The imaginary unit i = (0, 1).
Zero = (0, 0).
(x, y)(y, x). For example z(2, 3)z(3, 2).

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2. Geometric representation of the complex numbers
2.1 Cartesian representation of the complex numbers

The complex numbers can be represented by points on a two-dimensional Cartesian coordinate system called the complex plane. In this way we establish a one to one correspondence between the set of all complex numbers and the set of all points in the plane. The set of all real numbers corresponds to the real axis x and the set of all purely imaginary numbers corresponds to the imaginary axis y (see Figure 1.1).

Cartesian representation image

Figure 1.1 Cartesian representation

The Cartesian representation of the complex numbers specifies a unique point on the complex plane, and a given point has a unique Cartesian representation of the complex numbers.

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3. Polar representation of the complex numbers
3.1 Vector representation of the complex numbers

Another way of representing the complex numbers is to use the vector joining the origin (0, 0) of the complex plain to the point P = (x, y). (Figure 1.2 ).

Vector representation image

Figure 1.2 Vector representation

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3.2 Modulus and argument of the complex numbers
3.2.1 Modulus of the complex numbers

The length of the vector is called the modulus or absolute value of the complex numbers z, and is denoted by |z|. It is a nonnegative real number given by the equation
|z| = square (x**2 + y**2).                (1.1)
The only complex number with modulus zero is the number (0, 0).

Some Examples

z = -2+ 4i. Modulus |z| =square-1.
z = 1+ i. Modulus |z| =square-2.
z = -1 + 0i. Modulus |z| = 1.
z = i. Modulus |z| = 1.
z = - i. Modulus |z| = 1.

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3.2.2 Argument of the complex numbers

The angle between the positive real axis and the vector is called the argument of the complex numbers z, and is denoted by Arg(z).
More exactly Arg(z) is the angle through which the positive real axis must be rotated to cause it to have the same direction as vector . We assume that the point P is not the origin, P(0, 0). If P = (0, 0), then |z| = 0 and Arg(z) is indeterminate.
Arg(z) is considered positive if the rotation is counterclockwise and negative if the rotation is clockwise.
|z| and Arg(z) are the polar coordinates of the point (x, y).
z = |z|{cos Arg(z) +i sin Arg(z)} is a polar representation of z.            (1.2)

Special values of the complex argument

Arg(i)=
Arg(-i)=
Arg(1)= 0
Arg(-1)=
Arg(1+i)=

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3.2.3 Trigonometric form of the complex numbers

Let r = |z| and = Arg(z),
x = Re(z) = r cos, y = Im(z) = r sin.

It follows that
tantan             (1.3)

Example 1:
z = 6 + 8i.
|z| = square image.
tan = 8/6 = 4/3.

z = x + yi = r(cos+i sin).            (1.4)
The identity (1.4) is called the trigonometric form of the complex number z.
In common with the Cartesian representation, the polar representation specifies a unique point on the complex plane. But unlike the Cartesian representation, a given point does not have a unique polar label. Look at the Figure 1.3 a and b. A point P has infinitely many different labels because any angles that differ by a multiple of correspond to the same direction.

a) Polar representation image 1
b) Polar representation image 2

Figure 1.3 Polar representation

The fact about angles is very important. It means that each number z = x + yi has infinite set of representation in a polar form. Each representation differ by a multiple of .

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3.2.4 Principal value of the argument

There is one and only one value of Arg(z), which satisfies the inequality
-< Arg(z).
This is the principal value of the argument of z, written arg(z). The relation between Arg(z) and arg(z) is given by
Arg(z) = arg(z) +n ,
where n ranges over all integers 0, ±1, ±2, Ö .
tan arg(z).             (1.5)

Example 2:
Principal polar representation of z is
z
= 4(cos+ i sin)
Find other instances of the polar representation of z.

Some other instances of the polar representation of z:
z = 4(cos+ i sin);
z = 4(cos(+n) + i sin(+n)).
See Figure 1.4 for this example.

Polar representation example image
Figure 1.4 Example of polar representation

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by Tetyana Butler

Mathematical paradoxes
Possibly the greatest paradox is that mathematics has paradoxes...
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