Math 303 - Complex Variables
Homework due February 11
Recall that a real-valued function φ(x, y) : R2→Ris called harmonic if it satisfies Laplace’s equation:
∂2φ
∂x2+∂2φ
∂y2= 0.
Question 1. Assume that two functions f(x, y) and g(x, y) are both harmonic.
(a) Prove that the function αf(x, y ) is also harmonic.
(b) Prove that the sum f(x, y) + g(x, y) is also harmonic.
(c) Show that f(x, y) = x2−y2and g(x, y ) = x3−3xy2are both harmonic functions, but their
product f(x, y)·g(x, y ) is not harmonic. Thus, in general, the product of two harmonic functions
is not harmonic.
Recall that if a function u(x, y) is harmonic, then its harmonic conjugate v(x, y) is another harmonic
function such that u(x, y) + iv(x, y) is analytic.
Question 2. For the following u(x, y), (i) verify that uis harmonic and (ii) find the harmonic conjugate
of u.
(a) u(x, y) = y−4
(b) u(x, y) = x2−5x−y2
(c) u(x, y) = excos y
(d) u(x, y) = xy −x+y
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