goldch8 - Contents VIII Complex Numbers 1 VIII.A Algebraic...

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Unformatted text preview: Contents VIII Complex Numbers 1 VIII.A Algebraic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 VIII.B Advantages and Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 VIII.C Complex Conjugate and Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 VIII.D Exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 VIII.E Polar Coordinates and the Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 VIII.F Complex Valued Functions of a Real Variable . . . . . . . . . . . . . . . . . . . . . . . . . . 5 VIII.G Product, Power and Quotient Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 VIII Complex Numbers VIII.1 VIII Complex Numbers Definition: A complex number is an expression of the form x + iy where x and y are real numbers. If z = x + iy is a complex number, then we call x and y the real and imaginary parts of z and write x = Re z and y = Im Z. Thus both Re z and Im z are real numbers. In the special case that y = Im z is zero we write simply z = x + i 0 = x. ( z is called real) If instead x = Re z is zero we write z = 0 + iy = iy. ( z is called purely imaginary) In particular we write 0 = 0 + i i = 0 + i 1 . The set of all complex numbers is denoted by C . It contains the set R of all real numbers: R = { z C | Im z = 0 } . VIII.A Algebraic Operations We add (or subtract ) complex numbers by adding (or subtracting) real and imaginary parts: ( x + iy ) ( x 1 + iy 1 ) = ( x x 1 ) + i ( y y 1 ) . We multiply complex numbers by the rule: ( x + iy )( x 1 + iy 1 ) = ( xx 1- yy 1 ) + i ( xy 1 + x 1 y ) . (1) This may seem a little bizarre at first, and quite hard to remember. However it is easy to remember if you follow two simple rules. Firstly a special case of (1): i 2 = (0 + i 1)(0 + i 1) = (0- 1) + i (0 + 0) =- 1; i.e.i....
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This note was uploaded on 10/04/2011 for the course MATH 16121 taught by Professor Rachelbelinsky during the Spring '11 term at Georgia State University, Atlanta.

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goldch8 - Contents VIII Complex Numbers 1 VIII.A Algebraic...

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