## laplace's equation in two dimensions_clyindrical coordinates

1. The problem statement, all variables and given/known data
an infinitely long thin conducting cylindrical shell(radius R) of surface charge density

$$\sigma$$=$$\sigma_{1}$$sin(2$$\Phi$$)+$$\sigma_{2}$$cos($$\Phi$$).
what are the four boundary conditions for this problem?
using the four boundary conditions and the identification of the coefficients of sin(n$$\Phi$$)and cos(n$$\Phi$$)find the expression of the potential inside and outside the cylindrical shell.
2. Relevant equations
the general solution of laplace's equation in this case can be written:
V(r,$$\Phi$$)=$$\sum^{n=1}_{\infty}$$[$$A_{n}$$cos(n$$\Phi$$)+
$$B_{n}$$sin(n$$\Phi$$))r$$^{n}$$+(C$$_{}n$$cos(n$$\Phi$$)+D$$_{}n$$sin(n$$\Phi$$)r$$^{-n}$$]+A$$_{}0$$ln(r)+C$$_{}0$$.
take C$$_{0}$$=0 inside the cylinder
3. The attempt at a solution

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