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 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: |Sin(x)|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) = = a               (1) 2ia                             (2) Multiply (2) Let ,                                       (3) then We consider a is real and a >1. It means that and iz = Ln z = = = = = Ln(i)= i( 1+2k), k = 0, ±1, ±2, ... z = So, when a = 2 z = = 1.7320; Ln(3.7320) = 1.3169; Ln(0.2680) = -1.3169 Thus the equation Sin(z) = 2 has infinitely many solutions z = ± 1.3169i, where k = 0, ±1, ±2, ... None of these solutions is a real number. by Tetyana Butler