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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.

Powers of complex numbers

Complex Analysis. FreeTutorial

13. Integer powers of complex numbers
      13.1 Modulus |z|<1
      13.2 Modulus |z|>1
      13.3 Modulus |z|=1

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13. Integer powers of complex numbers

Integer powers of complex numbers are just special cases of products. 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.
If we know a complex number z, we can find zn. The modulus |z|n is the nth power of the modulus of z, the argument of zn is n times the argument of z.
There are three cases: |z|<1, |z|>1, |z|n=1.
Let us consifer all of them.

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13.1 Integer powers of complex numbers. |z|<1

If |z|<1, then |z|n<|z|<1 for any integer n. The bigger n the less |z|n.

Example:
|z|= 0.9; arg(z) = 30°. Find |z|2, |z|3, |z|4, |z|5, |z|6.

|z|2= 0.81; arg(z) = 60°;
|z|2
= 0.73; arg(z) = 90°;
|z|2
= 0.66; arg(z) = 120°;
|z|2
= 0.59; arg(z) = 150°;
|z|2
= 0.53; arg(z) = 180°.

Each higher power is 30° further along and closer to 0. Six powers are displayed as points on a spiral in the figure 1.25. This spiral is called an exponential sprial.

exponential sprial |z|<1 image

Figure 1.25. An exponential sprial. |z|<1.

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13.2 Integer powers of complex numbers. |z|>1

If |z|>1, then |z|n>|z|>1 for any integer n. The bigger n the bigger |z|n.

Example:
|z|= 1.2; arg(z) = 30°. Find |z|2, |z|3, |z|4, |z|5, |z|6.

|z|2= 2.4; arg(z) = 60°;
|z|2
= 2.9; arg(z) = 90°;
|z|2
= 3.5; arg(z) = 120°;
|z|2
= 4.2; arg(z) = 150°;
|z|2
= 5; arg(z) = 180°.

Six powers are displayed as points on a spiral in the figure 1.26.

exponential sprial |z|>1 image

Figure 1.26. An exponential sprial. |z| >1.

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13.3 Integer powers of complex numbers. |z|=1

If |z|=1, then |z|n=1 for any integer n.

Example:
|z|= 1; arg(z) = 30°. Find |z|2, |z|3, |z|4, |z|5, |z|6.

|z|2= 1; arg(z) = 60°;
|z|2
= 1; arg(z) = 90°;
|z|2
= 1; arg(z) = 120°;
|z|2
= 1; arg(z) = 150°;
|z|2
= 1; arg(z) = 180°.

exponential sprial |z|=1 image

Figure 1.27. |z| =1

Figure 1.28 shows all 3 cases

exponential sprial image

Figure 1.28.

by Tetyana Butler

Mathematical paradoxes
Possibly the greatest paradox is that mathematics has paradoxes...
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We will add more
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