# quadrature formula for a lune

I have been reading about quadrature formulas in the complex plane.

On the set $\mathbb{D} = \{|z| < 1\}$ we have $\int_{\mathbb{D}} f(z) dA = \pi f(0)$. Standard result in Harmonic functions.

The sample paper shows for the limaçon, $\Omega = \{ z + \frac{1}{2}z^2: |z| < 1\}$ we have $\int_{\mathbb{D}} f(z) dA = \frac{3\pi}{2} f(0) + \frac{\pi}{2} f'(0)$.

Does a quadrature formula exist for the lune? Let $f(z)$ be an holomorphic function (on a large enough area). Given a lune shape (such as) $L=\big\{z\in\mathbb{C}:|z-1|<\frac{3}{4},\;|z|>1\big\}$ is there a formula for $\int_L f(z) \, dA$ that is a linear combination of $f(z)$ and it's derivatives evaluated at specific points?

For more information on quadrature formulas, here is an article Topology of Quadrature Domains.

• are you certain that the picture matches with the formula ? – mercio Dec 8 '17 at 17:38
• @mercio I have re-written the equation. Look any better? – cactus314 Dec 8 '17 at 17:46