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Intro to Analysis in-class_Part_45

# Intro to Analysis in-class_Part_45 - 7.1 Complex Numbers 95...

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7.1 Complex Numbers 95 Definition 7.1.5. We can extend | x | to | z | : | z | := ( z z ) 1 / 2 or | x + iy | := p x 2 + y 2 . Note: | · | : C R + is a continuous function. Theorem 7.1.6. 1. | z | = | z | . 2. | zw | = | z || w | . Proof. HW: show | zw | 2 = | z | 2 | w | 2 and take . 3. | Re z | ≤ | z | . Proof. a 2 a 2 + b 2 and take . 4. | z + w | ≤ | z | + | w | . Proof. | z + w | 2 = ( z + w )( z + w ) = z z + z w + zw + w w = | z | 2 + 2 Re( z w ) + | w | 2 ≤ | z | 2 + 2 | z w | + | w | 2 prev = | z | 2 + 2 | z || w | + | w | 2 = ( | z | + | w | ) 2 . Note that ( x, y ) · (0 , 1) = ( - y, x ), so multiplying by corresponds to rotation by π 2 (ccw). In general, if | α | = 1, then αz corresponds to rotating z about 0. e θ = X n =0 ( θ ) n n ! = X k =0 ( θ ) 2 k (2 k )! + X k =0 ( θ ) 2 k +1 (2 k + 1)! = X k =0 ( - 1) k θ 2 k (2 k )! + X k =0 ( - 1) k θ 2 k +1 (2 k + 1)! = (cos θ, sin θ )

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96 Math 413 Sequences and Series of Functions This shows that any complex number with unit norm can be written e θ , and that any e θ has unit norm. Definition 7.1.7. An ε -ball around a point z C is B ( z, ε ) = { z + re θ . . . 0 r < ε, θ R } = { w C . . . | z - w | < ε } . This means we can still use the same notation to discuss complex-valued functions, e.g., continuity is ε > 0 , δ > 0 , | z - w | < ε = ⇒ | f ( z ) - f ( w ) | < ε. Functions of a complex variable are discussed in a different class, but we can still discuss complex-valued functions of a real variable, i.e. f : R C . If f, g are real functions, then Z ( x ) = f ( x ) + g ( x ) is a complex function. Most (but not all) theorems will remain true for complex functions. Exceptions: without order, there
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