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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 imaginary unit i

Complex Analysis. FreeTutorial

7.1 Positive integer powers of i

7.2 Negative integer powers of i

7.3 Formulas for any integer powers of i


7.1 Positive integer powers of i

i2 = -1. What is about i3, i4, i5, im?

i 0 = 1
i 4 = 1
i 8 = 1
i 4m = 1
i 1 = i
i 5 = i
i 9 = i
i 4m+1 = i
i 2 = -1
i 6 = -1
i 10 = -1
i 4m+2 = -1
i 3 = -i
i 7 = -i
i 11 = -i
i 4m+3 = -i

where m = 0, 1, 2, 3, …
Positive powers of i are periodic with period 4.
To evaluate i m we have to replace m with its remainder on division by 4.

Example:
Evaluate i203.
We have to replace 203 with its remainder on division by 4.
203 = 4*50 + 3;
i203 = i3 = – i.

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7.2 Negative integer powers of i

i -1 = -i
i -5 = -i
i -9 = -i
i 4m+3 = -i
i -2 = -1
i -6 = -1
i -10 = -1
i 4m+2 = -1
i -3 = i
i -7 = i
i -11 = i
i 4m+1 = i
i -4 = 1
i -8 = 1
i -12 = 1
i 4m = 1

where m = -1, -2, -3, …
Negative powers of i are periodic with period 4.

i -1 = – i. The reciprocal of i is its own negation – i.

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7.3 Formulas for any integer powers of i

Let us compare formulas for positive and negative powers of i to get formulas for any integer powers of i.

i 4m = 1
i 4m+1 = i
i 4m+2 = -1
i 4m+3 = -i

where m = 0, ±1, ±2, ±3, …
Integer powers of i are periodic with period 4.
To evaluate i m we have to replace m with its remainder on division by 4.

by Tetyana Butler

Mathematical paradoxes
Possibly the greatest paradox is that mathematics has paradoxes...
Complex functions paradoxes
Infinity paradoxes
Set theory paradoxes
We will add more
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