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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.

Roots of complex numbers

Complex Analysis. FreeTutorial

14.1 n-th root of complex numbers
      14.1.1 Roots of unity


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, …, n - 1 give different values of z.
The complete solution of the (1.27) is given by
z = ,
k = 0, 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, …, 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 .

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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, …, 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, …, 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

roots of unity image

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.

 

by Tetyana Butler

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