 Holomorphic derivative of a quadrant Dirichlet solution

ID: holomorphic-derivative-of-a-quadrant-dirichlet-solution

u_{z} (z) = \frac{2 z}{πi} \int_{0}^{\infty} r [\frac{g _{2} ( r )}{( r ^{2} - z ^{2} ) ^{2}} + \frac{g _{1} ( r )}{( r ^{2} + z ^{2} ) ^{2}}] d r .

This follows by differentiating the Schwarz integral formula composed with

z^{2}

, remembering that for real boundary data

u_{z}

u

. For complex boundary data, apply the construction to the real and imaginary parts separately and use linearity. If the data and their derivatives decay sufficiently, integration by parts gives

u_{z} (z) = \frac{z}{πi} [\int_{0}^{\infty} \frac{g _{2}^{'} ( r )}{r ^{2} - z ^{2}} d r + \int_{0}^{\infty} \frac{g _{1}^{'} ( r )}{r ^{2} + z ^{2}} d r] .

The endpoint terms cancel exactly when

g_{1} (0) = g_{2} (0)

. Interior denominators do not vanish because

Im z^{2} > 0

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