Question

Verify Laplace equation for U=(r+a^2/r) cos θ .Also find V and f(z)

I could verify,
Anyone knows how i can find V ?

Answer 1

If you have that f is an analytic function we could maybe use Cauchy-Riemann's equations.

Answer 2

elaborate ??

if i have real part of an analytic function, i can get the imaginary part using Milne-Thompson method, i know that...you're suggesting the same ?

Answer 3

I think that was a stupid assumption, if we had f analytic we wont even need to verify the Laplace equation.

Answer 4

hmm...so back to square one

Answer 5

** z= U+iV

Answer 6

laplace equation

\( ∇^2 U= 1/r (∂U/∂r)+(∂^2 U)/(∂r^2 )+(1/r^2) (∂^2 U)/(∂θ^2 ) = 0 \)
this I could verify

Answer 7

I think this is how we're supposed to find it : \[\frac{ \delta u }{ \delta r }=\frac{ 1 }{ r } \frac{ \delta v }{ \delta \theta }\] \[\frac{ \delta v }{ \delta r }=-\frac{ 1 }{ r } \frac{ \delta u }{ \delta \theta }\] We can use this since U is differentiable.

Answer 8

so i will get
\(\Large \frac{ \delta v }{ \delta \theta } and \frac{ \delta v }{ \delta r }\)

then integrate ? how ?

Answer 9

Hm.. I didn't really think about it.