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. Powers of imaginary unit i Complex Analysis. FreeTutorial

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7.1 Positive 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.

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.

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.

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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 Hilbert’s hotel Ross-Littlewood paradox Paradox of wizard and mermaid Paradox of enchantress and witch Paradox of Tristram Shandy Barber paradox Achilles and tortoise Contact us