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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.

De Moivreযrmula

Complex Analysis. FreeTutorial

12.1 De Moivre's theorem
      12.1.1 De Moivre's theorem for fractional powers

12.1 De Moivre's theorem

It was showen (1.19) and (1.20) that the product of two complex numbers:
z1 = r1(cos+i sin),
z2 = r2(cos+i sin)
can be written in the form
z1z2 =r1r2[(coscos - sinsin) +
+ i(sincos + cossin)].

If z1 = z2 = z = r(cos+i sin), the product
z1z2 = z2 = r2(cos(2)+i sin(2)). (1.23)

The n th power of z, written zn, is equal to
zn = rn(cos(n)+i sin(n)),       (1.24)
where n is a positive or negative integer or zero.

For r = 1 we obtain De Moivreযrmula:
(cos+i sin)n = cos(n)+i sin(n). (1.25)
The formula connects complex numbers and trigonometry.
It is easy to use this formula to find explicit expressions for the n-th roots of unity.


12.1.1 De Moivre's theorem for fractional powers

De Moivre's theorem is not only true for the integers. It can be extended to fractions.
Let n = p/q in (1.24). Then we have

zp/q = rp/q{cos((p/q))+i sin((p/q))}. (1.26)

For r = 1 we obtain De Moivreযrmula for fractional powers:
(cos+i sin)p/q = cos((p/q))+i sin((p/q)). (1.27)

Example 1
Calculate (cos(/4)+i sin(/4))1/3.

By De Moivre's theorem for fractional powers
(cos(/4)+i sin(/4))1/3 = cos(/12)+i sin(/12).
It is the first cube root of cos(/4)+i sin(/4).

By the Fundamental theorem of algebra, the equation of degree n has n roots. A complex number has 3 cube roots.
cos(/12+2/3)+i sin(/12+2/3) is the second cube root of cos(/4)+i sin(/4).

cos(/12+4/3)+i sin(/12+4/3) is the third cube root of cos(/4)+i sin(/4).

by Tetyana Butler

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