Problems

In Problems 1-6, put the complex number in the form a + ib.

In Problems 7-12, find the roots of the given equation.

In Problems 13-18, put the complex number into the polar form r cis θ.

In Problems 19-24, use the polar form to simplify the given expression.

In Problems 25-28, compute both square roots of the given complex number using the polar form.

In Problems 29-32, put the given exponent in the form a + ib.

In Problems 33-36, find the derivative.

In Problems 37-40, find the general solution of the complex differential equation.

In Problems 41-44, solve the given complex initial value problem.

46            Prove that the conjugate of a complex number r cis θ is r cis (-θ).

47            Prove that for any nonzero complex number z, , where θ is the argument of z.

48            Prove that for any two complex numbers u and z, the sum of the conjugates of u and z is equal to the conjugate of the sum of u and z, and similarly for products. In symbols,

49            Prove that for any two complex numbers u and z,

50            Use De Moivre's Formula with n = 2,

cos (2θ) + i sin (2θ) = (cos θ + i sin θ)², to obtain expressions for cos (2θ) and sin (2θ) in terms of cos θ and sin θ.

51            Use De Moivre's Formula with n = 3,

cos (3θ) + i sin (3θ) = (cos θ + i sin θ)³, to obtain expressions for cos (3θ) and sin (3θ) in terms of cos θ and sin θ.

52            Find the solution of the initial value problem

z' + (a + ib)z = 0, z(0) = e^c+id where a, b, c, and d are real numbers.

53            Show that every solution of the differential equation z' + ibz = 0 has constant absolute value (where b is a real number).