Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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5. (a) State the Residue Theorem. Your answer should include all the conditions required for the theorem to hold. (4 marks) (b) Let y be the square contour with vertices at -3, -3i, 3 and 3i, described in the anti-clockwise direction. Evaluate に dz. You must check all of the conditions of any results that you use. (5 marks) (c) Evaluate L You must check all of the conditions of any results that you use. ཙ x sin(Tx) x²+2x+5 da. (11 marks)
3. (a) Lety: [a, b] C be a contour. Let L(y) denote the length of y. Give a formula for L(y). (1 mark) (b) Let UCC be open. Let f: U→C be continuous. Let y: [a,b] → U be a contour. Suppose there exists a finite real number M such that |f(z)| < M for all z in the image of y. Prove that < ||, f(z)dz| ≤ ML(y). (3 marks) (c) State and prove Liouville's theorem. You may use Cauchy's integral formula without proof. (d) Let R0. Let w € C. Let (10 marks) U = { z Є C : | z − w| < R} . Let f UC be a holomorphic function such that 0 < |ƒ(w)| < |f(z)| for all z Є U. Show, using the local maximum modulus principle, that f is constant. (6 marks)
3. (a) Let A be an algebra. Define the notion of an A-module M. When is a module M a simple module? (b) State and prove Schur's Lemma for simple modules. (c) Let AM(K) and M = K" the natural A-module. (i) Show that M is a simple K-module. (ii) Prove that if ƒ € Endд(M) then ƒ can be written as f(m) = am, where a is a matrix in the centre of M, (K). [Recall that the centre, Z(M,(K)) == {a Mn(K) | ab M,,(K)}.] = ba for all bЄ (iii) Explain briefly why this means End₁(M) K, assuming that Z(M,,(K))~ K as K-algebras. Is this consistent with Schur's lemma?

Chapter 4 Solutions

Advanced Engineering Mathematics

Ch. 4.1 - Find a general solution of the given ODE (a) by...Ch. 4.1 - Find a general solution of the given ODE (a) by...Ch. 4.1 - Find a general solution of the given ODE (a) by...Ch. 4.1 - Prob. 14PCh. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - Find a real general solution of the following...Ch. 4.3 - Prob. 9PCh. 4.3 - Solve the following initial value problems. Ch. 4.3 - 10–15 IVPs Solve the following initial value...Ch. 4.3 - Prob. 12PCh. 4.3 - Solve the following initial value problems. Ch. 4.3 - Solve the following initial value problems. Ch. 4.3 - Solve the following initial value problems. Ch. 4.3 - Prob. 16PCh. 4.3 - Prob. 17PCh. 4.3 - Prob. 18PCh. 4.3 - Prob. 19PCh. 4.4 - Prob. 1PCh. 4.4 - Prob. 2PCh. 4.4 - Prob. 3PCh. 4.4 - Prob. 4PCh. 4.4 - Prob. 5PCh. 4.4 - Prob. 6PCh. 4.4 - Prob. 7PCh. 4.4 - Prob. 8PCh. 4.4 - Prob. 9PCh. 4.4 - Prob. 10PCh. 4.4 - Prob. 11PCh. 4.4 - Prob. 12PCh. 4.4 - Prob. 13PCh. 4.4 - Prob. 14PCh. 4.4 - Prob. 15PCh. 4.4 - Prob. 16PCh. 4.4 - Prob. 17PCh. 4.5 - Prob. 1PCh. 4.5 - Prob. 2PCh. 4.5 - Prob. 4PCh. 4.5 - Prob. 5PCh. 4.5 - Prob. 6PCh. 4.5 - Prob. 7PCh. 4.5 - Prob. 8PCh. 4.5 - Prob. 9PCh. 4.5 - Prob. 10PCh. 4.5 - Prob. 11PCh. 4.5 - Prob. 12PCh. 4.5 - Prob. 13PCh. 4.6 - Prob. 1PCh. 4.6 - Prob. 2PCh. 4.6 - Prob. 3PCh. 4.6 - Prob. 4PCh. 4.6 - Prob. 5PCh. 4.6 - Prob. 6PCh. 4.6 - Prob. 7PCh. 4.6 - Prob. 9PCh. 4.6 - Prob. 10PCh. 4.6 - Prob. 11PCh. 4.6 - Prob. 12PCh. 4.6 - Prob. 13PCh. 4.6 - Prob. 14PCh. 4.6 - Prob. 15PCh. 4.6 - Prob. 16PCh. 4.6 - Prob. 17PCh. 4.6 - Prob. 19PCh. 4 - Prob. 1RQCh. 4 - Prob. 2RQCh. 4 - How can you transform an ODE into a system of...Ch. 4 - What are qualitative methods for systems? Why are...Ch. 4 - Prob. 5RQCh. 4 - Prob. 6RQCh. 4 - What are eigenvalues? What role did they play in...Ch. 4 - Prob. 8RQCh. 4 - Prob. 9RQCh. 4 - Prob. 10RQCh. 4 - Find a general solution. Determine the kind and...Ch. 4 - Find a general solution. Determine the kind and...Ch. 4 - Find a general solution. Determine the kind and...Ch. 4 - Find a general solution. Determine the kind and...Ch. 4 - Prob. 15RQCh. 4 - Prob. 16RQCh. 4 - Prob. 17RQCh. 4 - Prob. 18RQCh. 4 - Prob. 19RQCh. 4 - Prob. 20RQCh. 4 - Prob. 21RQCh. 4 - Prob. 22RQCh. 4 - Prob. 23RQCh. 4 - Prob. 24RQCh. 4 - Prob. 25RQCh. 4 - Network. Find the currents in Fig. 103 when R = 1...Ch. 4 - Prob. 27RQCh. 4 - Prob. 28RQCh. 4 - Find the location and kind of all critical points...Ch. 4 - Find the location and kind of all critical points...
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