Conducting spheroid in uniform electric field

[ShayanJ]

I want to calculate the field distortion caused by placing a conducting spheroid in a uniform electric field. The field direction is taken to be the z axis.
I'm using oblate spheroidal coordinates and the convention below:

$x=a \cosh\eta \sin\theta \cos\psi \\ y=a \cosh\eta \sin\theta \sin\psi \\ z=a \sinh\eta \cos\theta$

I calculated $\hat z$ to be the following:

$\hat z=\frac{\cosh\eta \cos\theta \hat \eta-\sinh\eta \sin\theta \hat\theta}{\sqrt{\cosh^2\eta \cos^2\theta+\sinh^2\eta \sin^2\theta}}$

But when I write the first few terms in the oblate spheroidal harmonics expansion as the electric potential, and take its gradient to get the electric field, as the gradient formula in oblate spheroidal coordinates dictates, there is only a $a\sqrt{\cosh^2\eta-\sin^2\theta}$ in the denominator but the initial uniform electric field is $E_0 \hat z$ which has a more complicated denominator and so it seems impossible to match them at infinity.
What should I do?

Thanks

 

[ShayanJ]

Sorry...I found the answer. Those two square roots are related through circular and hyperbolic identities.

 

[ShayanJ]

There is another problem.
The series expansion that should be used for finding the answer, by finding its coefficients, is:
$\sum_0^\infty [A_n P_n(i \sinh\eta)+B_n Q_n(i\sinh\eta)][C_n P_n(\cos\theta)+D_nQ_n(\cos\theta)]$
Where Ps and Qs are Legendre functions of first and second kind.
The problem is, nothing goes to zero at infinity and also no two of them can cancel each other at $\eta \rightarrow \infty$. So I'm confused and don't know what to do!