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. Roots of complex numbers Complex Analysis. FreeTutorial

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14.1 n-th root of complex numbers

To find the n-th root of a complex number w 0 we have to solve the equation
zn = w.              (1.26)

Let w = r(cos +i sin ),
z = (cos +i sin ).

Then (1.26) takes the form n(cos n +i sin n ) = r(cos +i sin ). (1.27)

The equation (1.27) is fulfilled if n = r and n = and we obtain the root
z = , (1.28)
where is the positive n-th root of the positive number r.

(1.28) is not the only solution of (1.27). It is also fullfield for all angles, satisfied , where k is any integer.
However, only the values k = 0, 1, ༦ont size="+1">n - 1 give different values of z.
The complete solution of the (1.27) is given by
z = ,
k = 0, 1, ༦ont size="+1">n - 1.           (1.29)

Every non zero complex number has exactly n distinct n th roots.
It means that every number has two square roots, three cube roots, four fourth roots, ninety ninetieth roots, and so on.
They have the same modulus and their arguments differ by , k = 0, 1, ༦ont size="+1">n - 1.

Geometrically, the n th roots are the vertices of a regular polygon with n sides.
To find n th roots it is a four-step process.
1. Plot your number r(cos +i sin ), that you want to take the root of.
2. Plot a circle of radius . It is the length of all roots.
3. The first root has direction /n.
4. The other n roots are equally spaced on the circle of radius .

14.1.1 Roots of unity

The roots of equation
zn = 1             (1.30)
are called n-th roots of unity.
The solution of the (1.30) is given by
z = cos( ) + i sin( ),
k
= 0, 1, ༦ont size="+1">n - 1.           (1.31)

We remember that every non zero complex number has exactly n distinct n th roots. It means that unity has two square roots, three cube roots, four fourth roots, and so on.
They have the same modulus = 1 and their arguments differ by , k = 0, 1, ༦ont size="+1">n - 1.

The n-th roots of unity are located on the unit circle of the complex plane. They form the vertices of a n-sided regular polygon with one vertex on 1.

Figure 1.29 shows all distinct roots of unity zn = 1 for n = 1, 2, ..., 14 Figure 1.29. Roots of unity

There is one first root of unity, equal to 1.
The square roots of unity (n = 2) are 1 and -1.
The third (cubic) roots of unity are 1, .
The fourth roots of unity are 1, i, -1, -i.

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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 Hilbertਯtel Ross-Littlewood paradox Paradox of wizard and mermaid Paradox of enchantress and witch Paradox of Tristram Shandy Barber paradox Achilles and tortoise Contact us