Question

Q.NO.3: Show that the function z=tan^-1(2xy/x^2-y^2)satisfies Laplace’s equation; then make the substitution
x=r cosθ, y= r sinθ
and show that the resulting function of satisfies the polar form of laplace’s equation

(∂^2z/∂r^2)+(1/r^2)(∂^2z/∂θ^2)+(1/r)(∂z/∂r)=0

Answer 1

well, you have

tanz = r^2 tan2?
z = arctan(r^2 tan2?)

?z/?r = (2r tan2?)/((r^2 tan2?)^2+1)
?z/?? = (2r^2 sec^2(2?))/((r^2 tan2?)^2+1)

the second partials get a bit messy, but you can get them here:

http://www.wolframalpha.com/input/?i=d%5E2%2Fdr%5E2+arctan(r%5E2+tan2%CE%B8)

http://www.wolframalpha.com/input/?i=d%5E2%2Fdy%5E2+arctan(r%5E2+tan(2y))
(where I used y instead of ? because it choked on ?

The rest is just algebra!