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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.

Trigonometry of the complex numbers

Sin(z) = 2 Is it a paradox?

Complex Analysis. FreeTutorial

The real sine and cosine functions are bounded:
1, |Cos(x)|1

The complex sine and cosine functions are not bounded if they are defined over the set of all complex numbers. Complex sine (cosine) functions could be equal to 2 or 5.

Let us find such complex z that Sin(z) = a where a is real and a >1.
For example
Sin(z) = 2 or Sin(z) = 5.

Sin(z) = complex exponentials= a               (1)

complex exponential difference2ia                             (2)

Multiply (2) complex exp

complex exponentials equation

complex equation

Let t = complex exp,                                       (3)

then equation

solving complex equation solving equation

We consider a is real and a >1. It means that solving root and

solving equation continue

solving problem

solving exp

iz = Lnsolving complex root

z = solving complex logarithm =

= solving complex logarithm continue =

= solving complex logarithm for sine

Ln(i)= i( 1+2k), k = 0, ±1, ±2, ...

z = solution for complex sine

So, when a = 2

z = sine = 2

= 1.7320;
Ln(3.7320) = 1.3169;
Ln(0.2680) = -1.3169

Thus the equation Sin(z) = 2 has infinitely many solutions z = sine = 2± 1.3169i, where k = 0, ±1, ±2, ... None of these solutions is a real number.

by Tetyana Butler

Paradoxes of Complex functions
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