To determine To express: The complex number − i in the form R ( cos θ + i sin θ ) = Re i θ . Explanation Consider the complex number − i . It is known that, for a complex number z = a + i b , the modulus R , is given R = | z | = a 2 + b 2 . Argument θ can be obtained from tan θ = b a . By comparing the given complex number − i with z = a + i b , the values of a = 0 and b = − 1 . Obtain the value of R as follows. R = | z | = ( 0 ) 2 + ( − 1 ) 2 = 1 R = 1 Obtain the value of θ as follows. tan θ = − 1 0 = ∞ θ = tan − 1 ( ∞ ) θ = π 2 It is known that tan is periodic with period π and the number − i is on the negative part of the imaginary line

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

See similar textbooks

Videos

Question

Chapter 4.2, Problem 4P

To determine

To express: The complex number −i in the form R(cosθ+isinθ)=Reiθ.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.2, Problem 3P

Chapter 4.2, Problem 5P

Students have asked these similar questions

= 1. Show (a) Let G = Z/nZ be a cyclic group, so G = {1, 9, 92,...,g" } with g": that the group algebra KG has a presentation KG = K(X)/(X” — 1). (b) Let A = K[X] be the algebra of polynomials in X. Let V be the A-module with vector space K2 and where the action of X is given by the matrix Compute End(V) in the cases (i) x = p, (ii) xμl. (67) · (c) If M and N are submodules of a module L, prove that there is an isomorphism M/MON (M+N)/N. (The Second Isomorphism Theorem for modules.) You may assume that MON is a submodule of M, M + N is a submodule of L and the First Isomorphism Theorem for modules.

(a) Define the notion of an ideal I in an algebra A. Define the product on the quotient algebra A/I, and show that it is well-defined. (b) If I is an ideal in A and S is a subalgebra of A, show that S + I is a subalgebra of A and that SnI is an ideal in S. (c) Let A be the subset of M3 (K) given by matrices of the form a b 0 a 0 00 d Show that A is a subalgebra of M3(K). Ꮖ Compute the ideal I of A generated by the element and show that A/I K as algebras, where 0 1 0 x = 0 0 0 001

(a) Let HI be the algebra of quaternions. Write out the multiplication table for 1, i, j, k. Define the notion of a pure quaternion, and the absolute value of a quaternion. Show that if p is a pure quaternion, then p² = -|p|². (b) Define the notion of an (associative) algebra. (c) Let A be a vector space with basis 1, a, b. Which (if any) of the following rules turn A into an algebra? (You may assume that 1 is a unit.) (i) a² = a, b²=ab = ba 0. (ii) a² (iii) a² = b, b² = abba = 0. = b, b² = b, ab = ba = 0. (d) Let u1, 2 and 3 be in the Temperley-Lieb algebra TL4(8). ገ 12 13 Compute (u3+ Augu2)² where A EK and hence find a non-zero x € TL4 (8) such that ² = 0.

Chapter 4 Solutions

Elementary Differential Equations

Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - Prob. 6P Ch. 4.1 - In each of Problems 7 through 10, determine...Ch. 4.1 - Prob. 8P Ch. 4.1 - In each of Problems 7 through 10, determine...Ch. 4.1 - Prob. 10P

Knowledge Booster

$Advanced Math$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.