let fcz) = C2+2) e, and C:l2l=3 be a circle inPositive Sense. Then.i- write the lauren series fer f in o2 -112i1 - Find the redidu residue of fii- Use the resi due to Ealuate fcz) dz

Given- Let fz = z2 + z e1z-1, and C: z = 3 be a circle in positive sense. Then To find- Write the…

Answered: 7-1 let f(z) = (z²+z) e²-1, and C:…

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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2 COS examine the co vergence of series and the character of its multiplication.
Part (B) - What is the type of the following series? Prove it. cos(nn) √n n=2
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.
Investigate Taylor series expansion. Explain how the Taylor series is used in your calculator to compute values of functions such as sin(x) , cos(x) , and ex , where x represent any value. Write a brief report explaining your findings, and give examples.
For each series, determine whether it converges absolutely. converges conditionally, or diverges. (a) (-1)" sin(n) n3" (-2)"-3" (b) iMiM
For the series 2-1 (k+5)? calculate lim, An+1 -0 00 (Use symbolic notation and fractions where needed.) an+1 lim %3D an
Complex analysis Find the first three terms of Laurent series for z/Logz in |z −1| < 1, where Log(z) is the principal branch.
20x3 Find the power series for f(x) in the form An. (1 – x4)* n=1 5 Hint: First, find the power series for g(x) Then differentiate. x4 (Express numbers in exact form. Use symbolic notation and fractions where needed.) An =
cosh(2r) – 1 6. (a) Show that sinh?(z) = 2 (b) Find the nth term of the following sequence {1,8, 10, 26, 46, · - - } for all n > 1.
Using basic real analysis 1, solve the question
Steps: 1.What is the f" (x) of the function? 2. After getting the f"(x), what is the f"(a) of the function as a=1? + co ?(a) 3. After getting the f"(a), input it into the (numerator) of Taylor series expansion formula: E (x – a)", for n! - (x - a)", a=1. 4. After getting the Taylorseries expansion, find the intervalof convergence (using Ratio Test). f(x)= log,(2x – 1) 2 f'(x)= In (6) (2x-1) 4 P"(x)= - In (6) (2x-1)2 16 p"(x}= In (6) (2x-1)3 96 p''(x)= - In (6) (2x-1)4 f"(x)=???? To get the f"(x), find the trend among the derivatives.

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7-1 let f(z) = (z²+z) e²-1, and C: 121-3 be a circle in Positive Sense. Then. i-write the laurent series for f in 02-1/²00 - Find the redu residue of f ii- Use the residue to Evaluate (fcz) dz