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Solutions to
can be found by setting
Here
is an analytic function of the complex variable
.
Splitting the function
into real and imaginary parts we obtain
and both
and
satisfy Laplace's equation.
Example 5. .2Consider solutions to Laplace's equation with
In this case we may try
Expanding the square and expressing in terms of real and imaginary
parts we obtain
Thus, we have
and
Using the real part for the potential,
, the solution to Laplace's
equation is
It is straightforward to show that both Laplace's equation and the
boundary conditions are satisfied.
Next: Cylindrical Coordinates,
Up: Cartesian Coordinates,
Previous: Separable Solutions
Prof. Alan Hood
2000-11-06