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5. (10 pts) Prove whether or not the following functions are analytie in the finite complex...
1. (10 pts) For each of the following pairs of functions, indicate whether f = 0(g), f = Ω(g), or both (in which case f-6(1). You do not need to explain your answer. f(n) (n) a) n (b) n-1n+1 (c) 1000n 0.01n2 (d) 10n2 n (lg n)2 21 е) n (f) 3" (g) 4" rl. 72 i-0 2. (12 pts) Sort the following functions by increasing order of growth. For every pair of consecutive functions f(n) and g(n) in the...
be a coordinate function for 1.) (Coordinate functions) Let f: R a (a) (Exercise 3A) Prove that -f is a coordinate function as well. (5 (b) For which real numbers a, b e R is a f+b a coordinate function? (c) Let g:E → R be a coordinate function. Prove that there exists a line ( points) Justify your answer with a proof. (10 points) real number b E R, such that g-f+b or gf+b. (10 points)
Formuals: 3. A sinusoid eơt s or can be expressed as a sum of exponentials e" and e" with complex ncies s-o +yoo and s* -ju, Locate in the complex plane the complex frequencies of (10 points) the following signals: (a) e cos2t (e) 2 ut) -2t (b) e 3 (c) cos3t (d) e Complex numbers: - R034 1 n even (reje)" rkejke Trignometric Identities sin 2x=2sinxcosx sin2 x+cos2 x = 1 in 1-cos 2 cos2x=1 + cos2x sin(x±y)-sinxcosy±cos x...
(2) (a) Prove that the set G = {+1, £i} is a finite subgroup of the multi- plicative group CX of nonzero complex numbers, and that the set H = {E1} is a finite subgroup of {+1, £i}. (b) Compute the index of H in G. (c) Compute the set of left cosets G/H.
Byty 4) (20 pts) Use the Cauchy-Riemann Equations to determine if the following functions are analytic or I a) f(x) = e* (cosy + 1) + je*siny not. +
Hi, I really need help on both parts a and b of this Complex Analysis question. Thanks! 1. Define exp(iy) := cos(y) + i sin(y). a. Prove, using trigonometry, that exp(iy+iy') = exp(iy). expliy') for y, y' ER two real numbers. b. Prove directly (using Taylor series for sin and cos) that expliy) = " where n! denotes the factorial of n. Hint: you may use the fact that an infinite sum of complex numbers an converges if and only...
please prove part (b) use complex analysis and calculus of residue -dx neif a> 0 5. (a) x2+1 (b) For any real number a > 0, cos x dx ne"/a. a Hint: This is the real part of the integral obtained by replacing cos x by e
(Complex Analysis) Prove the following maximum principle for harmonic functions: Let u be harmonic in a bounded domain E and continuous in E ∪ dE. Then max(x, y)E E U dE u= sup(x, y)E E U dE u; min(x, y)E E U dE u= inf(x, y)E E U dE u. (Not the first E after the subscript (x, y) denotes element of, and the next on is the domain and the next is the derivative of the domain.)
Part I. (30 pts) (10 pts) Let fin) and g(n) be asymptotically positive functions. Prove or disprove each of the following statements T a、 f(n) + g(n)=0(max(f(n), g(n))) 1. b. f(n) = 0(g(n)) implies g(n) = Ω(f(n)) T rc. f(n)- o F d. f(n) o(f(n)) 0(f (n)) f(n)=6((f(n))2)
3. (10 pts) For each of the following functions f(n), prove the stated claim by providing constants no C1, and c2 such that for all n2 no, cig(n) S f(n) or f(n) c2g(n), and provide a calculation that shows that this inequality does indeed hold (a) f(n) 2n2 3n3-50nlgn10 0(n3) O(g(n)) (b) f(n)-2n log n + 3n2-10n-10-Ω ( 2)-0(g(n))