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2 Problems on Homework for Complex Variables | MATH 303, Assignments of Mathematical Analysis

Material Type: Assignment; Class: Complex Variables W/ Apps; University: University of Hawaii at Hilo; Term: Unknown 1989;

Typology: Assignments

2009/2010

Uploaded on 04/12/2010

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Math 303 - Complex Variables
Homework due February 11
Recall that a real-valued function φ(x, y) : R2Ris 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) = x2y2and g(x, y ) = x33xy2are 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) = y4
(b) u(x, y) = x25xy2
(c) u(x, y) = excos y
(d) u(x, y) = xy x+y
1

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Math 303 - Complex Variables

Homework due February 11

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