- #1

- 107

- 1

The author begins with Green's second identity:

[tex]

\int_V \alpha \nabla^2 \beta - \beta \nabla^2 \alpha \ dV =

\int_C \left( \alpha \frac{\partial \beta}{\partial n} - \beta \frac{\partial \alpha}{\partial n} \right) \ ds[/tex]

Here, C is a closed curve, s is the arc length for C and n is the outward unit normal. We then let [itex]\alpha[/itex] satisfy Laplace's equation and let [itex]g = 1/(4\pi)\log [(x-x^*)^2 + (y-y^*)^2][/itex], i.e. the free-space Green's function. Then he gets

[tex]

\alpha(x^*, y^*) = r\int_C \left( \alpha \frac{\partial \beta}{\partial n} - \beta \frac{\partial \alpha}{\partial n} \right) \ ds,[/tex]

where r = 1 if (x*,y*) is inside C, but r = 1/2 if (x*, y*) is on C.

I'm confused at this point. I thought that

[tex]

\int_V \alpha \nabla^2 \beta - \beta \nabla^2 \alpha \ dV =

\int_V \alpha \delta((x,y) - (x^*, y^*)) \ dV = \alpha(x^*, y^*)

[/tex]

Where is the factor of 1/2 or 1 coming in?

Moreover, the next equation, the factor of r = 1/2 has switched to the left-hand-side. I can't figure out how this is done (but perhaps if someone firsts helps me understand the above, this will be clear).