1. Show that u (x, y) is harmonic in some domain and find a harmonic conjugate v (x, y) when
(a) u (x, y) = 2x (1 - y)
(b) u (x, y) = 2x - x3 + 3xy2
(c) u (x, y) = sinh x?sin y
(d) u (x, y) = y / (x2 + y2)
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This solution provides a detailed step by step explanation of the given calculus problem.