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Intro to Analysis in-class_Part_44

# Intro to Analysis in-class_Part_44 - 0 implies...

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Chapter 7 Sequences and Series of Functions 7.1 Complex Numbers 1 7.1.1 Basic properties of C Consider the plane R 2 := { ( x, y ) . . . x, y R } . This space already has a vector space structure: ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( x 1 + x 2 , y 1 + y 2 ) a ( x 1 , y 1 ) = ( ax 1 , ay 1 ) . We can also endow it with a multiplicative structure by defining ( a, b ) · ( c, d ) := ( ac - bd, ad + bc ) . This makes R 2 into a field called C . To check that this alleged field has inverses, note that ( x, y ) · 1 x 2 + y 2 ( x, - y ) = 1 . 1 May 2, 2007

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94 Math 413 Sequences and Series of Functions Typically, one writes the basis vectors of R 2 as 1 = (1 , 0) and = (0 , 1), so that general elements are z = x (1 , 0) + y (0 , 1) = x + y . Then 2 = (0 , 1)(0 , 1) = (0 - 1 , 0 + 0) = - (1 , 0) = - 1 . Theorem 7.1.1. Let p C [ x ] be a polynomial of degree n . Then p has n roots in C . That is, if p ( x ) = a 0 + a 1 x + a 2 x 2 + · · · + a n x n , a j C , then one can write p ( x ) = c 0 ( x - c 1 ) . . . ( x - c n ) , c j C . Remarkably, every known proof relies on topology (completeness) somehow.
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Unformatted text preview: 0 implies automatically that a ∈ R , not C . Earlier: C is complete, but not ordered. Theorem 7.1.2. For any real numbers a and b , ( a, 0) + ( b, 0) = ( a + b, 0) ( a, 0) · ( b, 0) = ( ab, 0) . This allows us to identify R with the subﬁeld of C consisting of the elements ( x, 0). C has another useful operation. Deﬁnition 7.1.3. z is the conjugate of z , deﬁned by z = x + y i = x-y i . This corresponds to reﬂection in the horizontal axis R . Note: : C → C is a continuous function. Theorem 7.1.4. For z,w ∈ C , 1. z + w = z + w , 2. zw = z w , 3. if z = x + i y , then z + z = 2Re( z ) = 2 x and z-z = 2 i Im( z ) = 2 y , 4. z z = | z | 2 ≥ , with equality iﬀ z = 0 ....
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