Math 113 (Spring 2009) Yum-Tong Siu 1
Homework Assigned on April 2, 2009
due April 7, 2009
(numbering of problems continued from
the last assignment with the same due date)
Problem 6. (a) Find a bounded harmonic function uon the upper half-plane
so that
(i) its boundary value on −∞ < x < 0 is 1,
(ii) its boundary value on 0 < x < 1 is 2, and
(iii) its boundary value on 1 < x < ∞is 0.
Hint: Use the superposition of two harmonic functions, each of which assume
a constant value at each of only two intervals.
(b) Find a bounded harmonic function ϕon the strip {0< y < π }such that
(i) its boundary value on {y=π}is 1,
(ii) its boundary value on {x < 0, y = 0 }is 2, and
(iii) its boundary value on {x > 0, y = 0 }is 0.
Hint: Find ϕas the function uin (a) composed with a linear fractional
transformation and the exponential function.
Problem 7 (Poisson’s Integral Formula for the Exterior of a Circle). Suppose
uis a bounded twice continuously differentiable harmonic function on the
exterior {x2+y2> a2}of the circle Caof radius a > 0 centered at the origin.
Assume that uis continuous up to the circle Caand let h(θ) = u¡aeiθ¢for
0≤θ≤2π. Derive the following Poisson’s integral formula for the exterior
of the circle Ca.
u¡reiθ ¢=¡r2−a2¢Z2π
ϕ=0
h(ϕ)
a2−2ar cos (θ−ϕ) + r2
dϕ
2π
for r > a.
