User:Uffishbongo/Schwarz integral formula

In complex analysis, the Schwarz integral formula allows one to recover a holomorphic function, up to an imaginary constant, from the boundary values of its real part.

Unit disc
Let $$f = u+iv$$ be a function which is holomorphic on the closed unit disc $$\{z \in \mathbb{C} \mid |z| \leq 1\}$$. Then

$$ f(z) = \frac{1}{2\pi i} \oint_{|z| = 1} \frac{\zeta + z}{\zeta - z} \text{Re}(f(\zeta)) \frac{d\zeta}{\zeta} + \text{Im}(f(0))$$

for all $$|z| < 1$$.

Upper half-plane
Let $$f = u+iv$$ be a function which is holomorphic on the closed upper half-plane $$\{z \in \mathbb{C} \mid \text{Im}(z) \geq 0\}$$ such that, for some $$\alpha > 0$$, $$|z^\alpha f(z)|$$ is bounded on the closed upper half-plane. Then

$$f(z) = \frac{1}{\pi i} \int_{-\infty}^\infty \frac{u(\zeta,0)}{\zeta - z} d\zeta$$

for all $$\text{Im}(z) > 0$$.