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Material Type: Assignment; Class: Complex Variables W/ Apps; University: University of Hawaii at Hilo; Term: Unknown 1989;
Typology: Assignments
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Recall that a real-valued function φ(x, y) : R^2 → R is called harmonic if it satisfies Laplace’s equation:
∂^2 φ ∂x^2
∂^2 φ ∂y^2
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) = x^2 − y^2 and g(x, y) = x^3 − 3 xy^2 are 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 u is harmonic and (ii) find the harmonic conjugate of u.
(a) u(x, y) = y − 4
(b) u(x, y) = x^2 − 5 x − y^2
(c) u(x, y) = ex^ cos y
(d) u(x, y) = xy − x + y