f( z ) = z gives u = x and v = y. The electric field from a potential is E = - — f, so these give uniform electric fields in the x- and y-directions respectively. The equipotentials are straight lines.
f( z ) = z2 gives u = x2 - y2 and v = 2 x y. The equipotentials are hyperbolas, and these respective hyperbolas intersect orthogonally. You can check this by taking the gradient of the two functions and evaluating their dot product.
A more complicated example is f( z ) = ln z. This gives u = ln r and v = q. The first one is the potential of a line charge, with an electric field going as 1/ r. The second is less familiar, but it too leads to some useful solutions. It has radial lines for equipotentials, so you achieve this physically with sheets of metal. Two flat conductors that are in the same plane and are almost touching is represented by the boundary value problem
A general question of how to solve problems involving the Laplacian: Can you start from a blank piece of paper and go all the way from Laplace's equation to the end of a boundary value problem in spherical or even cylindrical coordinates?
31 Mar Discuss the meaning of "residue at infinity" and how it may be defined.
What does it take to solve a partial differential equation such as
Laplace's equation in one of the standard coordinate systems? There are
many steps and a variety of procedures that you have to master in order
to understand everything.
1. What isthe Laplacian in this coordinate system?
2. Separate variables.
3. Analyze the singularities of the resulting ordinary differential
equations.
4. Determine the behavior of the solutions near those singularities.
5. Factor out that behavior.
6. Do a Frobenius series solution of the equation.
7. Analyze the behavior of these solutions.
8. How do the boundary conditions on the original problem affect the
choice of separation constants? (Up to this point, they are arbitrary
complex numbers.)
9. These will typically be Sturm-Liouville type of equations and
solutions, so understand their orthogonality properties.
10. Use these orthogonality properties to determine the coefficients in
an infinite series solution to the original problem.
11. Analyze the results.
How do you find the Laplacian in cylindrical coordinates? One way is to start in rectangular coordinates and use the chain rule a lot. Another way is to define the Laplacian of V as div curl V and to evaluate — · — V, where — is expressed in terms of basis vectors and partial differential operators. Be sure to understand how to differentiate the unit vectors when you do the second —.
Execute the separation of variables for the cylindrical Laplace equation, getting three ordinary differential equations with two (so far) arbitrary separation constants.
2 Apr Carry out the separation program for the potential probelem in cylindrical coordinates. Analyze the behavior of the radial equation near r = 0; it is a regular singular point. Find the Frobenius series solution to this radial differential equation. In this case, you get both solutions most of the time, except where one of the separation constants vanishes, then you get only one.
