Show that using Cauchy Residue Theorem. (p-over-q rule or phi-rule) Jo x4 +1 22 Jo x4 +1 22
2. More integrals! Evaluate each integral, using either a Cauchy Integral Formula or Cauchy's Residue Theorem. Take C to be the circle [2] = 3, oriented counter-clockwise. 1) Sota-1jad: 6) Se TH h) Sorºcos(1/2)da
Jo at 12. Residue theorem. Compute the following integral, 80 (1 +4232(2 points)
please 2 only, thanks Exercises dA (1) Use Cauchy's residue theorem to compute Jo 2+sin (2) Repeat the preceding exercise for 8" 131. (3) Let a be a complex number such that lal < 1. Prove that (2 27 Jo 1 - 2a cos 0 + a2d6 = 1 - 22 (4) What is the value of the integral in the preceding exercise when |al > 1? (Hint: Let b= 1.)
obtain the resukt of the following integrals by using complex numbers and by either the residual theorem or the cauchy euler theorem 2π cos2θ Jo 3-sino 7 ._ dx 8 J-oo (x2+1) (x2+2x+2) 2π cos2θ Jo 3-sino 7 ._ dx 8 J-oo (x2+1) (x2+2x+2)
4. Evaluate the following integrals using the Residue Theorem. Justify your calculations, show the work. (10 points each) a) 12 cos a + 13 2 da b) (x2 +6x + 10)2 x sin 2x 24 13 da: c) 4. Evaluate the following integrals using the Residue Theorem. Justify your calculations, show the work. (10 points each) a) 12 cos a + 13 2 da b) (x2 +6x + 10)2 x sin 2x 24 13 da: c)
8) (Problem 17 (a) on page 49) Let p and q be two distinct primes. Show that for any integer a, pq|(a p+q − a p+1 − a q+1 + a 2 ). Hint: Find the least residue of a p+q − a p+1 − a q+1 + a 2 modulo p, and then find the least residue of a p+q − a p+1 − a q+1 + a 2 modulo q. After that, use the following result: Suppose x,...
Show that the sequence is Cauchy using the definition of Cauchy se- quences. Sn 2n +1 n +4
Show that the irreducible polynomial x4 - 2 over Q, has roots a, b, c in its splitting field such that the fields Q(a, b) and Q(a, c) are not isomorphic over Q (Hint: The roots are (4√2, -4√2, 4√2i, -4√2i), and the splitting field is Q(4√2, i,).)
Let KQi, 2 (a) Show that K is a splitting field of X4- 2 over Q. (b) Find a Q-basis of K c) Find an automorphism of order four of K over i (d) Determine all the automorphisms of K over Q (e) The zeros of X4-2 form -(±Vitiy2). Describe the action of the set S Aut(K) on S (f) Find all subgroups of Aut (KQ). (g) Find all intermediate field extensions of C K. Let KQi, 2 (a) Show...
9. Using theorem 12.3, find the three angles of the triangle with vertices P (1,0,-1), Q = (3,-2,0), and R (1,3, 3). a b 2 cose 9. Using theorem 12.3, find the three angles of the triangle with vertices P (1,0,-1), Q = (3,-2,0), and R (1,3, 3). a b 2 cose