math185f09-hw2

math185f09-hw2 - MATH 185: COMPLEX ANALYSIS FALL 2009/10...

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MATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 2 Throughout the problem set, i = - 1; and whenever we write x + yi , it is implicit that x,y R . For z C , recall that the argument of z , denoted arg( z ), is any θ R such that z = | z | e . We write C × := C \{ 0 } . 1. Let ( z n ) n =1 be a sequence of complex numbers. (a) Show that if lim n →∞ z n = z , then lim n →∞ | z n | = | z | but that the converse is not true in general. (b) Is it true that if lim n →∞ z n = z , then lim n →∞ arg( z n ) = arg( z )? (c) Show that if lim n →∞ | z n | = r and lim n →∞ arg( z n ) = θ , then lim n →∞ z n = re . 2. Which of the following limits exists? Prove your answers. lim n →∞ ± 1 + i 1 - i ² n , X n =1 i n log ± n n + 1 ² , lim z 1 1 - z 1 - z . 3. Let Ω C be a region. Let f : Ω C and z 0 Ω. (a) Suppose lim z z 0 f ( z ) = w . Prove that lim z z 0 f ( z ) = w, lim z z 0 Re f ( z ) = Re w, lim z z 0 Im f ( z ) = Im
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math185f09-hw2 - MATH 185: COMPLEX ANALYSIS FALL 2009/10...

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