1. Suppose f is a function on the complex plane C, so for each complex number z the function f outputs some (possibly complex) number f(z). We can view f as a function of two real variables x, y in the following way: if z=x+iy with x and y real numbers then f(z) = f(x+iy). Now suppose for each z = x+iy we let u(x, y) be the real part of f(x+iy) and we let u(x, y) be the imaginary part of f(x + iy), so that f(z) = f(x+iy) = u(x, y) + iv(x, y). (1) Suppose that u = vy and y = -vr. Show that u and v are both harmonic functions, meaning Au == = 0. (2) Suppose z=x+iy as above. Define the complex exponential e* by = e+iy = eeiy Also define the complex sine function by sin(z) = eiz - e-iz 2i (recall that if x is real we know this formula is valid). Find two real-valued functions u(x, y) and v(x, y) such that sin(z) = u(x, y) + iv(x, y). (3) Continuing from part (2), show that we have u=vy and uy = -vr. So by part (1) the real and imaginary parts of sin(2) are harmonic functions.