BRIEF ANSWERS TO TEST 1

• • False, the inequality is backwards. For example, 2 = |1|+|-1| is not less or equal to |1+(-1)| = 0.
  • False. Need to change v_y to v_x.
  • True. If u is harmonic in the plane, then it has a harmonic conjugate (since the plane is simply-connected). Therefore it is the real part of the analytic function u(x,y) + i v(x,y).
  • False. The left side is a multiple-valued function, and the right side is a single-valued function.
• By the quadratic formula, the roots are [-(1+square_root(3)i) +/- square_root(2+2*square_root(3)i)]/2 which equals [-(1+square_root(3)i) +/- (square_root(3)+i)]/2 which equals either (-1+square_root(3))/2 + (-square_root(3)+1)*i/2 or (-1-square_root(3))/2 + (-square_root(3)-1)*i/2.
• This is an old quiz problem! Answer: v(x,y) = (3/2)x^2y^2 - x^4/4 - y^4/4. [u(x,y) + i*v(x,y)]' = u_x + i*v_x = (3x^2y - y^3) + i*(3xy^2 - x^3).
• Using the procedure we learned in class, it is not hard to show that sinh^(-1)(z) = log(z + (z^2+1)^(1/2)). Using the usual differentiation rules from calculus, you find that [sinh^(-1)(z)]' = 1/(z^2+1)^(1/2). (This formula is valid on any branch.)
• One form of the answer is square_root((square_root(x^2+y^2)+x)/2) + i*square_root((square_root(x^2+y^2)-x)/2). (There are alternate forms that use trig and inverse trig functions.)

BRIEF ANSWERS TO TEST 2

• 2*Pi*i*z_0*r^2. (Use the parameterization z(t) = z_0 + r*e^(it)).
• 0. (Use the Cauchy Integral Formula.)
• (7/27)*Pi*i. (Use the Cauchy Integral Formula.)
• One way to show that the integral equals 2*Pi is to apply the Gauss Mean Value Theorem to the function e^z. (That is, the value of e^z at z=0 equals the average value of e^z over the unit circle.) Another way is to apply the Cauchy Integral Formula to the integral of (e^z)/z over the unit circle.
• The function G(z) = F(z)/e^z is entire, and |G(z)| is less or equal to 1 for all z. By Liouville's Theorem, G(z) = z_0 is a constant function. Necessarily the modulus of the constant is less or equal to 1. From F(z)/e^z = z_0 we get F(z) = z_0*e^z.