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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 Rational Functions

Complex Analysis. FreeTutorial

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1) Definition of the rational function   

2) Möbius (or homographic or fractional linear) transformations

3) Fractional-linear function

4) Zhukovskii function

Definition of the rational function

In complex analysis, a rational function is the ratio of two polynomials and with complex coefficients:

rational function formula,          (1)

where 0, and and have no common factor. The coefficients of and polynomials are called the coefficients of the rational function. The function is called irreducible when and have no common zeros. Every rational function can be written as an irreducible fraction (1).

If has degree m and has degree n, then the degree of is the number N = max{m, n}. Some literature offers the degree of is the pair (m, n).

A rational function of degree (m, n) with n = 0 is a polynomial. It is also called an entire rational function. Otherwise it is called a fractional-rational function.

When m <n, the fraction is called proper. It is called improper when m >n. An improper fraction can be uniquely written as

improper fraction rational function formula,          (2)

where is a polynomial, called the integral part of the fraction , and is a proper fraction.

Möbius (or homographic or fractional linear) transformations

Rational functions with degree 1 are called Möbius (or homographic or fractional linear) transformations.

Möbius, homographic, fractional linear transformation formula,          (3)

satisfying ad - bc 0.


Fractional-linear function

Fractional-linear function is a function of the type

Fractional-linear function formula,

where are complex or real variables,
are complex or real coefficients, and

If ,

the fractional-linear function is an integral-linear function.

L(z) is a constant, if the rank of the matrix

matrix is equal to one.

A proper fractional-linear function (3) is obtained if the rank of A is equal to two, and if



Zhukovskii function

Zhukovskii function is the rational function

Zhukovskii function formula, where z is complex.

It was discovered by Russian scientist N. E. Zhukovskii (often spelled "Joukowski"). Zhukovskii function is important for applications in fluid mechanics, particularly in constructing and studying the Kutta ꨵkovskii airfoils (aerofoils).

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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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