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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 Analysis
Free Tutorial

Content

Chapter 1. Complex numbers
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

4. Trigonometric and algebraic form conversion
            4.1 Conversion from trigonometric to algebraic form
            4.2 Conversion from algebraic to trigonometric form
            4.3 Examples of the conversion from algebraic to trigonometric form

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5. The algebra of complex numbers
      5.1 Arithmetic operations with complex numbers
            5.1.1 Addition
            5.1.2 Subtraction
            5.1.3 Multiplication
            5.1.4 Division
      5.2 Properties of the complex numbers
            5.2.1 Proof of the properties of the complex numbers
      5.3 Properties of the modulus of the complex numbers
            5.3.1 Proof of the properties of the modulus of the complex numbers

6. Geometric interpretation of addition & subtraction

      6.1 Geometric addition
      6.2 Geometric subtraction

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7. Powers of imaginary unit i
     7.1 Positive integer powers of i
     7.2 Negative integer powers of i
     7.3 Formulas for any integer powers of i

8. Geometric interpretation of multiplication
     8.1 The modulus and argument of the product
     8.2 Multiplication of complex numbers as stretching (squeezing) and rotation
            8.2.1 Multiplying a complex number by a real number
            8.2.2 Multiplying a complex number by imaginary unit i
            8.2.3 Multiplying a complex number by -i
            8.2.4 Multiplying a complex number by positive integer powers of i
            8.2.5 Multiplying a complex number by negative integer powers of i

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

10. Definition and notation of Reciprocals
      10.1 Modulus and argument of Reciprocals
      10.2 Geometric interpretation of Reciprocals

11. Division as multiplication and reciprocation
      11.1 The modulus and argument of the quotient
      11.2 Example of division

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12. De Moivre's theorem
      12.1 De Moivre's theorem for fractional powers
13. Integer powers of complex numbers
      13.1 Modulus |z|<1
      13.2 Modulus |z|>1
      13.3 Modulus |z|=1

14. n-th root of complex numbers
      14.1 Roots of unity

Chapter 2. Complex elementary functions

Complex Exponential Function

Complex Logarithm Function

Complex Trigonometric Functions
1) Definitions
     1.1 Right triangle and unit-circle definitions
     1.2 Definitions via series
     1.3 Definitions via complex exponentials
     1.4 Definitions via differential equations

2) A difference between the real and complex trigonometric functions
     2.1 Relationship to exponential function
     2.2 The complex sine and cosine functions are not bounded

3) Identities
     3.1 Periodic identities
     3.2 Even and odd identities
     3.3 Pythagorean identity
     3.4 The sum and difference formulas
     3.5 The double-angle formulas
     3.6 More identities
4) Calculus


Complex Inverse Trigonometric Functions
1) The notations

2) Range of usual principal value

3) Definitions as infinite series

4) Logarithmic forms
     4.1 Natural logarithm's expressions
     4.2 Logarithmic formulas and connections

     4.3 Logarithmic formulas. Proofs

5) Derivatives of inverse trigonometric functions

6)Indefinite integrals of inverse trigonometric functions

Complex Hyperbolic Functions

Complex Inverse Hyperbolic Functions
1) The notations

2) Definitions    

3) Derivatives of inverse hyperbolic functions

4) Indefinite integrals

Complex Power Function
1) Computation of complex power function
     1.1 The complex power function is a multi-valued function
     1.2 Computation of the roots of the complex value

     1.3 Computation of the complex z and real a
     1.4 The power of a real number to a non-integer power    

     1.5 Raising a complex number to a complex power

2) The rules for combining quantities containing powers

3) Derivatives of complex power function

4) Indefinite integral of complex power function

Complex Rational Functions
1) Definition of the rational function   

2) Möbius (or homographic or fractional linear) transformations

3) Fractional-linear function

4) Zhukovskii function

Chapter 3. Complex functions paradoxes

An improper use of SQRT(-1)
Sin(z) = 2 Is it a paradox?
Bernoulli's Sophism
Paradox of Bernoulli and Leibniz

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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