Schwarz Integral Formula - Upper Half-plane

Upper Half-plane

Let ƒ = u + iv be a function that is holomorphic on the closed upper half-plane {z ∈ C | Im(z) ≥ 0} such that, for some α > 0, |zα ƒ(z)| is bounded on the closed upper half-plane. Then

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

for all Im(z) > 0.

Note that, as compared to the version on the unit disc, this formula does not have an arbitrary constant added to the integral; this is because the additional decay condition makes the conditions for this formula more stringent.

Read more about this topic: Schwarz Integral Formula

Famous quotes containing the word upper:

“The stately Homes of England,
How beautiful they stand,
To prove the upper classes
Have still the upper hand.”
—Noël Coward (1899–1973)