In each of 1 through 12, find u and v so that f = u + iv, determine all points (x, y) at

In each of Problems 1 through 10, carry out the indicated calculation.(3 4i)(6 + 2i)

In each of Problems 1 through 10, carry out the indicated calculation.i(6 2i) + |1 + i|

In each of Problems 1 through 10, carry out the indicated calculation.(2 + i)/(4 7i)

In each of Problems 1 through 10, carry out the indicated calculation.((2 + i) (3 4i))/(5 i)(3 + i)

In each of Problems 1 through 10, carry out the indicated calculation.(17 6i)(4 12i)

In each of Problems 1 through 10, carry out the indicated calculation.|3i/(4 + 8i)|

In each of Problems 1 through 10, carry out the indicated calculation.i 3 4i 2 + 2

In each of Problems 1 through 10, carry out the indicated calculation.(3 + i)3

In each of Problems 1 through 10, carry out the indicated calculation.((6 + 2i)/(1 8i))2

In each of Problems 1 through 10, carry out the indicated calculation.(3 8i)(2i)(4 i)

In each of Problems 11 through 16, determine the magnitude and all of the arguments of z.3i

In each of Problems 11 through 16, determine the magnitude and all of the arguments of z.2 + 2i

In each of Problems 11 through 16, determine the magnitude and all of the arguments of z.3 + 2i

In each of Problems 11 through 16, determine the magnitude and all of the arguments of z.8 + i

In each of Problems 11 through 16, determine the magnitude and all of the arguments of z.4

In each of Problems 11 through 16, determine the magnitude and all of the arguments of z.3 4i

In each of Problems 17 through 22, write the number in polar form2 + 2i

In each of Problems 17 through 22, write the number in polar form7i

In each of Problems 17 through 22, write the number in polar form5 2i

In each of Problems 17 through 22, write the number in polar form4 i

In each of Problems 17 through 22, write the number in polar form8 + i

In each of Problems 17 through 22, write the number in polar form12 + 3i

Show that, for any positive integer n, i 4n = 1,i 4n+1 = i,i 4n+2 = 1,i 4n+3 = i.

Let z = a + ib. Determine Re(z2 ) and Im(z2 )

Show that complex numbers z, w, and u form vertices of an equilateral triangle if and only if z2 + w2 + u2 = zw + zu + wu

Show that z2 = (z)2 if and only if z is either real or pure imaginary.

Let z and w be numbers with zw = 1. Suppose either z or w has magnitude 1. Prove that z w 1 zw = 1. Hint: Recall that |u| 2 = uu for every complex number u. 28. S

Show that, for any numbers z and w, |z + w| 2 + |z w| 2 = 2  |z| 2 + |w| 2  . Hint: Note the hint from Problem 27

In each of Problems 29 through 37, a set of complex numbers is specified. Determine whether the set is open, closed, open and closed, or not open and not closed. Specify all boundary points of the set (whether or not they belong to the set).M is the set of all z satisfying Im(z) < 7

In each of Problems 29 through 37, a set of complex numbers is specified. Determine whether the set is open, closed, open and closed, or not open and not closed. Specify all boundary points of the set (whether or not they belong to the set).S is the set of all z with |z| > 2

In each of Problems 29 through 37, a set of complex numbers is specified. Determine whether the set is open, closed, open and closed, or not open and not closed. Specify all boundary points of the set (whether or not they belong to the set).U is the set of all z with 1 < Re(z) 3

In each of Problems 29 through 37, a set of complex numbers is specified. Determine whether the set is open, closed, open and closed, or not open and not closed. Specify all boundary points of the set (whether or not they belong to the set).V is the set of all z with 2 < Re(z) 3 and 1 < Imz < 1

In each of Problems 29 through 37, a set of complex numbers is specified. Determine whether the set is open, closed, open and closed, or not open and not closed. Specify all boundary points of the set (whether or not they belong to the set).'W consists of all z with Re(z)>( Im(z))2

In each of Problems 29 through 37, a set of complex numbers is specified. Determine whether the set is open, closed, open and closed, or not open and not closed. Specify all boundary points of the set (whether or not they belong to the set).R is the set of all numbers 1/m + (1/n)i with m and n positive integers

In each of Problems 1 through 12, find u and v so that f = u + iv, determine all points (x, y) at which the Cauchy-Riemann equations hold, and determine all z at which f is differentiable. Familiar facts about continuity of real-valued functions of two real variables can be assumed.f (z) = z i

In each of Problems 1 through 12, find u and v so that f = u + iv, determine all points (x, y) at which the Cauchy-Riemann equations hold, and determine all z at which f is differentiable. Familiar facts about continuity of real-valued functions of two real variables can be assumed.f (z) = z2 i z

In each of Problems 1 through 12, find u and v so that f = u + iv, determine all points (x, y) at which the Cauchy-Riemann equations hold, and determine all z at which f is differentiable. Familiar facts about continuity of real-valued functions of two real variables can be assumed.f (z) = |z

In each of Problems 1 through 12, find u and v so that f = u + iv, determine all points (x, y) at which the Cauchy-Riemann equations hold, and determine all z at which f is differentiable. Familiar facts about continuity of real-valued functions of two real variables can be assumed.f (z) = (2z + 1)/z

In each of Problems 1 through 12, find u and v so that f = u + iv, determine all points (x, y) at which the Cauchy-Riemann equations hold, and determine all z at which f is differentiable. Familiar facts about continuity of real-valued functions of two real variables can be assumed.f (z) = i|z| 2

In each of Problems 1 through 12, find u and v so that f = u + iv, determine all points (x, y) at which the Cauchy-Riemann equations hold, and determine all z at which f is differentiable. Familiar facts about continuity of real-valued functions of two real variables can be assumed.f (z) = z + Im(z)

In each of Problems 1 through 12, find u and v so that f = u + iv, determine all points (x, y) at which the Cauchy-Riemann equations hold, and determine all z at which f is differentiable. Familiar facts about continuity of real-valued functions of two real variables can be assumed.f (z) = z/Re(z)

In each of Problems 1 through 12, find u and v so that f = u + iv, determine all points (x, y) at which the Cauchy-Riemann equations hold, and determine all z at which f is differentiable. Familiar facts about continuity of real-valued functions of two real variables can be assumed.f (z) = z3 8z + 2

In each of Problems 1 through 12, find u and v so that f = u + iv, determine all points (x, y) at which the Cauchy-Riemann equations hold, and determine all z at which f is differentiable. Familiar facts about continuity of real-valued functions of two real variables can be assumed.f (z) = (z)2

In each of Problems 1 through 12, find u and v so that f = u + iv, determine all points (x, y) at which the Cauchy-Riemann equations hold, and determine all z at which f is differentiable. Familiar facts about continuity of real-valued functions of two real variables can be assumed.f (z) = i z + |z

In each of Problems 1 through 12, find u and v so that f = u + iv, determine all points (x, y) at which the Cauchy-Riemann equations hold, and determine all z at which f is differentiable. Familiar facts about continuity of real-valued functions of two real variables can be assumed.f (z) = 4z + 1/z

In each of Problems 1 through 12, find u and v so that f = u + iv, determine all points (x, y) at which the Cauchy-Riemann equations hold, and determine all z at which f is differentiable. Familiar facts about continuity of real-valued functions of two real variables can be assumed.f (z) = (z i)/(z + i)

Let zn = an + ibn be a sequence of complex numbers. We say that this sequence converges to w = c + id if the real sequences an c and bn d. Show that, if f (z) is continuous at z0 and zn is a sequence converging to z0, f (zn ) converges to f (z0).

In each of Problems 1 through 10, write the function value in the form a + bi.ei

In each of Problems 1 through 10, write the function value in the form a + bi.sin(1 4i)

In each of Problems 1 through 10, write the function value in the form a + bi.cos(3 + 2i)

In each of Problems 1 through 10, write the function value in the form a + bi.tan(3i)

In each of Problems 1 through 10, write the function value in the form a + bi.e5+2i

In each of Problems 1 through 10, write the function value in the form a + bi.cot(1 i/4)

In each of Problems 1 through 10, write the function value in the form a + bi.sin2 (1 + i)

In each of Problems 1 through 10, write the function value in the form a + bi.cos(2 i) sin(2 i)

In each of Problems 1 through 10, write the function value in the form a + bi.ei/2

In each of Problems 1 through 10, write the function value in the form a + bi.sin(ei )

Determine u and v such that ez2 = u(x, y) + iv(x, y). Show that u and v satisfy the Cauchy-Riemann equations.

Find u and v such that e1/z =u(x, y)+iv(x, y). Show that u and v satisfy the Cauchy-Riemann equations.

Find u and v such that zez = u(x, y) +iv(x, y). Show that u and v satisfy the Cauchy-Riemann equations wherever they are defined.

Find u and v such that cos2 (z) = u(x, y) + iv(x, y). Show that u and v satisfy the Cauchy-Riemann equations wherever they are defined.

Find all solutions of ez = 2i.

Derive the following identities. (a) sin(z + w) = sin(z) cos(w) + cos(z)sin(w). (b) cos(z + w) = cos(z) cos(w) sin(z)sin(w)

Find all solutions of ez = 2.

Find all solutions of sin(z) = i.

In each of Problems 1 through 6, determine all values of the complex logarithm of z4i

In each of Problems 1 through 6, determine all values of the complex logarithm of z2 2i

In each of Problems 1 through 6, determine all values of the complex logarithm of z5

In each of Problems 1 through 6, determine all values of the complex logarithm of z1 + 5i

In each of Problems 1 through 6, determine all values of the complex logarithm of z9 + 2i

In each of Problems 1 through 6, determine all values of the complex logarithm of z5

Let z and w be nonzero complex numbers. Show that each value of log(zw) is equal to a value of log(z) plus a value of log(w).

Let z and w be nonzero complex numbers. Show that each value of log(z/w) is equal to a value of log(z) minus a value of log(w).

In each of Problems 1 through 14, determine all values of zw.i 1+i

In each of Problems 1 through 14, determine all values of zw.(1 + i)2

In each of Problems 1 through 14, determine all values of zw.ii

In each of Problems 1 through 14, determine all values of zw.. (1 + i)2i

In each of Problems 1 through 14, determine all values of zw.(1 + i)3i

In each of Problems 1 through 14, determine all values of zw.(1 i)1/3

In each of Problems 1 through 14, determine all values of zw.i 1/4

In each of Problems 1 through 14, determine all values of zw.161/4

In each of Problems 1 through 14, determine all values of zw.(4)2i

In each of Problems 1 through 14, determine all values of zw.623i

In each of Problems 1 through 14, determine all values of zw.(16)1/4

In each of Problems 1 through 14, determine all values of zw.][(1 + i)/(1 i)] 1/3

In each of Problems 1 through 14, determine all values of zw.11/6/2017

In each of Problems 1 through 14, determine all values of zw.(7i)3i

Let u1, , un be the nth roots of unity with n a positive integer and n 2. Prove that n j=1 u j = 0. Hint: Write each nth root of unity as a power of e2i/n . A vector argument can also be made based on plotting the nth roots as vertices of a polygon

Let n be a positive integer, and let =e2i/n  . Evaluate n1 j=0(1)j j .