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Mathematics, math modeling and research

Complex analysis is studying the most unexpected, surprising, even paradoxical ideas in mathematics. The familiar rules of algebra and trigonometry of real numbers may break down when applied to complex numbers. There are many cases with complex functions which appear to be paradoxes. We may not, for example, use the rule square (a)  square (b) = square (ab), when a and b are negative ... Details
Complex sine (cosine) functions could be equal to 2 or 5 ... Details
A lot of mathematical paradoxes are connected with Complex Functions. Look at the right column of this page ...
We offer you Complex Analysis Free Tutorial Free Lessons

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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 Moivre's theorem; Powers of complex numbers; n-th root of complex numbers.

Lesson 7 Complex Exponential Function and Complex Logarithm Function
In this lesson: 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: 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: 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

Mathematics. It seems we are on the eve of the great discovery...

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Mathematical paradoxes

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

Infinity paradoxes
Set theory paradoxes
Zeno paradoxes
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