Find Laurent series for the function(a) f(2) = in each of the following domains(i) 0< |z| < 1;(ii) 1< |z|;1(iii) 0 < |z + 1| < 1;(iv) 1< |z+ 1|.(a) g(2) = Te+DE-2 on annulus domain(i) 1< |z| < 2;(ii) 0 < |z + 1| < 3;(iii) 0 < |z – 2| < 3;(iv) 2 < |z+ 1|.(b) g(z) = -21E- on annulus domain(i) 0 < |z – 2| < 1;(ii) 0 < |z – 1| < 1;item g(z) =22-22+2on annulus domainz-2(i) 1< |z – 1|;(ii) 0< |z – 2|;(c) g(2) = 2² cos() in |z| > 0.(d) g(z) = 울그 for |z| > 1.Apply online for aCredit Card andlive the best you.APPLY NOWIT CAN BE.*Ts&Cs apply. Auth FSPNCRCP15.

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Using basic real analysis, solve Find the limit superior and limit inferior of each of the following sequences:
(a) Let a, = 1 and a = =√1+a, for all ne N. (i) Show that {a} is increasing and bounded above by 2. (ii) Show that {a} converges to 1+√5 State clearly any result you assume.
Lemma 4. Let {yn} be a positive solution of equation (1.1) with the corresponding sequence {z„} e So for n > N. Then: (i) (1– p)zn N; (ii) {zn¢n} is increasing for all n > N. Proof. Assume that {yn} is a positive solution of equation (1.1) with the corres- ponding sequence {zn} E So. Then z, is positive, Zn 2 yn, and Yn = Zn – Pnyo(n) 2 (1– p)zn, n>N> no, so (i) is proved. It easy to see that z, E So implies lim b,(Az,)“ = 0; n00 otherwise we would eventually have Az, > 0 contradicting z, E So. Similarly, lim a„A(b,(Azn)“) = 0. A summation of equation (1.1) then yields 9szs+1 Zn+1 LIs. s=n s=n s=n Summing once more, we obtain as t=s s=n or Azn 2-Zn+1Qn. Hence, A(zn&n) = PnAzn + Zn+1Ao, > Zn+1(A0n – OnQn) =0 since {0,} is a solution of the difference equation (Ao, – Qnºn) = 0. Therefore, {z,9n} is increasing and this completes the proof of the lemma.
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3. Prove the root test: Let {an} be a sequence of real numbers, and define p(n) = |a,|'/n. Prove the following: (i) If there exists c 1 so that p(n) N, then Ean converges absolutely. (ii) If for all N there exists n > N so that p(n) > 1, then an diverges.
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Find Laurent series for the function (a) f(2) = ; in each of the following domains (i) 0< |z| < 1; (ii) 1< |z]; (iii) 0 < |z + 1| < 1; (iv) 1< |z+ 1|.