Mathematics Database Programming Web Design Price List     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 Top Top Top 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 4) Logarithmic forms      4.1 Natural logarithm's expressions      4.2 Logarithmic formulas and connections      4.3 Logarithmic formulas. Proofs 2) Definitions     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 Chapter 3. Complex functions paradoxes by Tetyana Butler Top  Possibly the greatest paradox is that mathematics has paradoxes... Bernoulli's sophism Paradox of Bernoulli and Leibniz Paradox of even (odd) and natural numbers Paradox of Hilbert’s hotel Ross-Littlewood paradox Paradox of wizard and mermaid Paradox of enchantress and witch Paradox of Tristram Shandy Barber paradox Achilles and tortoise Contact us