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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.

Conjugate complex numbers

Complex Analysis. FreeTutorial

9.1 Definition and notation of the conjugates
      9.1.1 Geometric interpretation of the conjugates
9.2 Properies of the conjugate complex numbers
      9.2.1 Proof of the properies of the conjugates

9.1 Definition and notation of the conjugates

A complex number z is a number of the form z = x + yi.
Its conjugate is a number of the form = x - yi.
The complex number and its conjugate have the same real part.
Re(z) = Re().
The sign of the imaginary part of the conjugate complex number is reversed.
Im(z) = - Im().

The conjugate numbers have the same modulus and opposite arguments.
|z| = ||,
arg(z) = - arg().
Any complex number multiplied by its complex conjugate is a real number, equal to the square of the modulus of the complex numbers z.
z = (x + yi)(x - yi) = x2+ y2 = |z|2. (1.15)

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Example 15:
z = 4 + 3i.
Find , |z|, ||, z, |z|2, arg(z) and arg().

= 4 - 3i;
|z| = = = 5;
|| = = 5;
z
= (4 + 3i)(4 - 3i) = 42+ 32 = 25;
|z|2 = 25
arg(z) = = arctan(3/4);
arg() = arctan(-3/4).

Example 16:
If arg(z) = , then what is arg()?
arg() = -.

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9.1.1 Geometric interpretation of the conjugates

If the complex number z = x + yi has polar coordinates (r,), its conjugate = x - yi has polar coordinates (r, -). As seen in the Figure1.6, the points z and are symmetric with regard to the real axis.

conjugate image

Figure1.6

Multiplying a complex number x + yi = (r,) by its conjugate x - yi = (r, -) gives the nonnegative number r2.

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9.2 Properies of the conjugate complex numbers

1. The conjugate complex numbers have the same modulus:
|z| = ||.

2. The conjugate complex numbers have opposite arguments:
arg(z) = - arg().
arg(z) = arg() = 0 for real positive numbers.
arg(z) = arg() = for real negative numbers.


3. The conjugate of a sum of the complex numbers is equal to a sum of the conjugates:
= + .

4. The conjugate of a product of the complex numbers is equal to a product of the conjugates:
= .

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9.2.1 Proof of the properies of the conjugates

1. Show that |z| = ||.

z = x + yi; = x - yi.

|z| = ;

|| = = = |z|.

Back to Properies

2. Show that arg(z) = - arg().

z = x + yi; = x - yi.
Re(z) = Re(); Im(z) = - Im().

The formulas (1.7) for arg(z) is at Complex_Numbers.htm
(1.7)

Let us consider all 4 quadrants.

Case 1: z lies in the first quadrant, lies in the fourth quadrant. x>0.

Case 2: z lies in the fourth quadrant, lies in the first quadrant. x>0.
In both cases x>0.
arg(z) =.
arg() = = - = - arg(z ).

Case 3: z lies in the second quadrant. x<0, y>0,
arg(z) = + ;
lies in the third quadrant. x<0, y<0,
arg() = - = - (+ ) = - arg(z).

Case 4: z lies in the third quadrant. x<0, y<0,
arg(z) = - .
lies in the second quadrant. x<0, y>0,
arg() = + = - ( - ) = - arg(z).

In all cases arg(z) = - arg().

Back to Properies

3. Show that = + .

Proof:
z1 = x1+ y1i,
z
2 = x2+ y2i.

conjugate image conjugate image 1
= x1 - y1i + x2 - y2i
conjugate image 2conjugate image 3 = + .

Back to Properies

4. Show that conjugate image 5 = .

Proof:
z1 = x1+ y1i,
z
2 = x2+ y2i.

conjugate image 5 conjugate image 6

conjugate image 7

conjugate image 8

conjugate image 8= .

by Tetyana Butler

Mathematical paradoxes
Possibly the greatest paradox is that mathematics has paradoxes...
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We will add more
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