MATH 4230 - Introduction to Complex Analysis Test 1 - Spring 2015 1. Suppose u and v are harmonic in a domain containing the closed disc |z| 1. True or false? (a) If u(z) > v(z) whenever |z| = 1, then u(z) > v(z) for all z inside the disc as well. (b) If u(z) = v(z) whenever |z| = 1, then u(z) = v(z) for all z inside the disc as well. (c) If u(0) u(z) whenever |z| = 1, then u(z) is constant for |z| 1. (d) If = R2 and lim|z| u(z) exists (and is nite), then u is a constant function. (e) If = R2 and u(z) satises |u(z)| | sin(|z|)| for all z, then u is a constant function. 2. Let be a bounded domain and let u be a harmonic function on that is continuous on (the closure of ). Show that if a u(z) b on , then a u(z) b on . 3. Suppose the twice continuously dierentiable function u(z) is harmonic in the punctured disc 0 < |z| < . (a) Show that the expression 2 2 u(r, ) d log r r 0 0 u (r, ) d r is constant (i.e., its value is independent of the radius r, for 0 < r < ). Hint: Use Green's identity p q p q n n ds = (pq qp) dxdy with appropriately chosen functions p and q and take the domain to be the annulus r1 < |z| < r2 to conclude that 2 2 u(r1 , ) d log r1 0 r1 0 u (r1 , ) d = r 2 2 u(r2 , ) d log r2 0 r2 0 u (r2 , ) d. r (b) Also show that the integral 2 r 0 u (r, ) d r is constant (i.e., its value is independent of the radius r, for 0 < r < ). (c) Combine the results in parts (a) and (b) and conclude that the arithmetic mean of a harmonic function over concentric circles |z| = r is a linear function of log r, 1 2 2 u(r, ) d = log r + 0 where and are constants. (d) What are the values of the constants and if u(z) is harmonic in the disc |z| <