Abelian Ward splitting by Laurent series ($f_x(\zeta )={\sum }_{n\in \mathbb{Z}}{f}_n(x){\zeta }^n=h_1-h_0$)

On a twistor line use ${\pi }_{0^{\prime }}=\zeta ,{\pi }_{1^{\prime }}=1$ and ${\omega }^A=x^{A{0}^{\prime }}\zeta +x^{A{1}^{\prime }}$. For a degree-zero holomorphic patching logarithm, its Laurent series splits as $h_0=-{\sum }_{n\ge 0}{f}_n{\zeta }^n$, $h_1={\sum }_{n<0}{f}_n{\zeta }^n$. The twistor incidence relation implies ${\partial }_{A{0}^{\prime }}{f}_n={\partial }_{A{1}^{\prime }}{f}_{n-1}$. Hence $(-{\partial }_{A{0}^{\prime }}+\zeta {\partial }_{A{1}^{\prime }})h_{\alpha }={\partial }_{A{1}^{\prime }}{f}_{-1}$ on both patches. The gauge potential is $A_{A{0}^{\prime }}=-{\partial }_{A{1}^{\prime }}{f}_{-1}$, $A_{A{1}^{\prime }}=0$. Exponentiating the two additive factors produces the linewise holomorphic splitting of $e^f$.

The Abelian Ward splitting by Laurent series gives this gauge potential through the Cauchy integral formula, with $f_{,B}=\partial f/\partial {\omega }^B$ evaluated on the twistor line. The integrand has total projective degree zero. Holding the contour fixed locally and differentiating gives $F_{A{A}^{\prime }B{B}^{\prime }}={ϵ}_{A^{\prime }{B}^{\prime }}(2\pi i{)}^{-1}{\oint }_{\Gamma }{f}_{,AB}{\pi }_{D^{\prime }}d{\pi }^{D^{\prime }}$, using ${\pi }_{A^{\prime }}{\iota }_{B^{\prime }}-{\pi }_{B^{\prime }}{\iota }_{A^{\prime }}={ϵ}_{A^{\prime }{B}^{\prime }}({\iota }_{C^{\prime }}{\pi }^{C^{\prime }})$. Its unprimed two-component spinor is symmetric, so the primed gauge curvature vanishes and the field obeys the anti-self-dual Maxwell equations. These signs use ${ϵ}_{0^{\prime }{1}^{\prime }}=1$, ${ϵ}^{0^{\prime }{1}^{\prime }}=-1$ and raising by left multiplication; other consistent raising conventions can reverse both displayed overall signs.