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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.

Properties of the modulus of the complex numbers

Complex Analysis. FreeTutorial

5.3. Properties of the modulus
      5.3.1 Proof of the properties of the modulus


5.3. Properties of the modulus

Triangle Inequality: Proof
1. |z1 + z2|<=|z1| + |z2| Proof
2. |z1 + z2|>=|z1| - |z2| Proof
3. |z1 - z2|>=|z1| - |z2| Proof
4. |z1 + z2 + z3|<=|z1| + |z2| + |z3|
5. |z1z2| = |z1||z2| Proof

Top

5.3.1 Proof of the properties of the modulus

Proof of the Triangle Inequality #1:

1. |z1 + z2|<=|z1| + |z2|

|z1 + z2|= square 1.
|z1| + |z2|= square 2.

We have to prove that
square 1

square 2 is true.
Square both sides.
equation
square 3.

2x1x2 +2y1y2 <= square 4

Square both sides again.
2
x1x2
y1y2 <= x12y22 + y12x22 and we get
0<=(y1x2 - x1y2)2.
It is true because x1, x2, y1, y2 are all real.

Back to Properties

Proof of the Triangle Inequality #2:

2. |z1 + z2|>=|z1| - |z2|

We have to prove that square 1

square 5 is true.
Square both sides.
>=
- square 4.

2x1x2 +2y1y2 >=-square 4.

Multiply both sides by (-1/2).

-(x1x2 +y1y2) <=square 1.

Square both sides.
- x12x22 - 2x1x2y1y2 - y12y22<= x12x22 + x12y22 + y12x22+ y12y22 and we get
0<= (y1x2 + x1y2)2 + 2x12x22 + 2y12y22.
It is true because x1, x2, y1, y2 are all real, and squares of real numbers are >=0.

Proof of the Triangle Inequality #3:

3. |z1 - z2|>=|z1| - |z2|

We have to prove that square 1>=

square 5 is true.
Square both sides.
>=
- square 4.

-2x1x2 -2y1y2 >=-square 4.

Multiply both sides by (-1/2).

(x1x2 +y1y2) <=square 1.

Square both sides again.
2
x1x2
y1y2 <= x12y22 + y12x22 and we get
0<=(y1x2 - x1y2)2.
It is true because x1, x2, y1, y2 are all real.

4. |z1 + z2 + z3|<=|z1| + |z2| + |z3|

Proof: By the triangle inequality,
|z1 + (z2+z3)|<=|z1| + |z2+z3|<=|z1| + |z2| + |z3|

5. |z1z2| = |z1||z2|

Proof: |z1z2| = |(x1+y1i)(x2+y2i)| =
square 1 = square (x**2 + y**2)square (x**2 + y**2) = |z1||z2|.

by Tetyana Butler

Mathematical paradoxes
Possibly the greatest paradox is that mathematics has paradoxes...
Complex functions paradoxes
Infinity paradoxes
Set theory paradoxes
We will add more
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