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Shackleford
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Homework Statement
Determine whether or not the following functions are harmonic:
[itex]u = z + \bar{z} [/itex]
[itex]u = 2z\bar{z} [/itex]
Homework Equations
[itex]z = u(x,y) + v(x,y)i [/itex]
[itex]\bar{z} = u(x,y) - v(x,y)i [/itex]
A function is harmonic if Δu = 0.
The Attempt at a Solution
[itex]Δu = Δz +Δ \bar{z} = u_{xx} + v_{xx} + u_{yy} + v_{yy} + u_{xx} - v_{xx} + u_{yy} + -v_{yy} = 2u_{xx} + 2u_{yy}
[/itex]
[itex]u = 2z\bar{z} = 2[u(x,y) + v(x,y)i][u(x,y) - v(x,y)i] = 2[u^2(x,y) - v^2(x,y)][/itex]
[itex]Δu = 2[2uu_{xx} + 2uu_{yy} - 2vv_{xx} - 2vv_{yy}]
[/itex]
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