Laplace-like operator

Question

Help me please to apply a Laplace-like operator:$ \Delta f:= \frac{\partial^2 f}{\partial r^2} + \frac{\partial^2 f}{\partial z^2} + {1\over r}\,\frac{\partial f}{\partial r} - {f\over r^2} $ on the expression: $f:=\frac{r}{a}\rho^{-\alpha}\sin (\alpha\phi)$.

when $\rho=\sqrt{(r-a)^{2}+z^{2}} $

Thanks a lot!

Answer 1

You can use MATLAB Program :

   syms r a z alpha phi 
   rho=sqrt((r-a)^2+z^2);
   f=(r/a)*rho^(-alpha)*sin(alpha*phi);
   delta_f=diff(f,r,2)+diff(f,z,2)+(1/r)*diff(f,r)-f/(r^2);
   delta_f=simple(delta_f);
   pretty(delta_f)

$$ \Delta f:= \frac{\partial^2 f}{\partial r^2} + \frac{\partial^2 f}{\partial z^2} + {1\over r}\,\frac{\partial f}{\partial r} - {f\over r^2} $$ $$ \Delta f := \frac{\alpha \space sin (\alpha \phi) (3a -3r+r \alpha)}{\alpha (a^2-2ra+r^2+z^2)^{\frac{\alpha}{2}+1}} $$