Math 448 Homework 4 Due Fri., Sept. 21, 2007
Same instructions as last time. For infinite series problems, you may always use
formulas that have already been established.
(ungraded) §2.1 – 17, §2.2 – 3, 5, 7, 9, 17
1. (graded) §2.1 – 20 ad.
2. (graded) §2.2 – Problems 2 and 4. (Closed form not requested.)
3. (graded) §2.2 – Problems 8 and 10.
4. (graded) §2.2 – Problems 14 and 16.
5. (graded) (E) Let Cdenote that portion of the circle |z|= 2 running from z= 2 to
z= 2iin the usual counterclockwise fashion. Compute, from the definition,
ZC
(¯z+z2)dz.
6. (graded) (E) Verify that u(x, y) = xy + 7x+e−ysin xis a harmonic function, find all
harmonic conjugates v(x, y) and find an expression for f(z) = f(x+iy) = u(x, y) + iv(x, y)
which depends only on zand certain real constants.
7. (graded) (E) Suppose f(z) = f(x+iy) = u(x, y) + iv(x, y) is an entire function with
the property that v(x, y) = 2(u(x, y ))2for all (x, y). Use the Cauchy-Riemann equations
to prove that fmust be constant.
8. (bonus) Determine all entire functions f(z) = f(x, y) = u(x, y) + iv(x, y) with the
property that (x2−y2)u(x, y)−2xyv (x, y) = 0 for all (x, y). Hints: what can you say
about z2f(z)? What happens at z= 0?
9. (bonus) Determine all real homogeneous harmonic polynomials of degree 7; that is,
f(x, y) =
7
X
j=0
ajxjy7−j:∇f=∂2f
∂x2+∂2f
∂y2= 0
.
(Hint: it will be a vector space over Rof dimension two.)
10. (bonus) a. Verify from the definition the identity
sin 3z= 3 sin z−4 sin3z.
b. Use this identity and the formula at the bottom of p.100 (or any other correct and
explained method) to give a closed form for the power series for sin3(z) at z= 0.
c. Determine, by any correct method, a numerical expression for f(2007)(0), where f(z) =
sin3z. (Something like “723
·e34
37! ” is what I mean by a numerical expression.)