# ap.evaluation of pdes – Beltrami equation with harmonic coefficient Answer

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ap.evaluation of pdes – Beltrami equation with harmonic coefficient

Note that, in case you take $$phi=0$$, then the equation reduces to $$w_y =0$$, i.e., if $$Dsubset C$$ is the province of $$w$$ and $$x:Dtomathbb{R}$$ is the projection on the $$x$$-axis and has related fibers, then $$w= h(x)$$ for some $$C^1$$ duty $$h:x(D)tomathbb{C}$$, and that is the common resolution on such $$D$$.

Something related occurs in common: Write
$$mathrm{d}w = w_z,mathrm{d}z + w_{barrier z},mathrm{d}barrier z = w_z,(mathrm{d}z + mathrm{e}^{iphi(z)},mathrm{d}barrier z) = mathrm{e}^{iphi(z)/2}w_zleft(mathrm{e}^{-iphi(z)/2}mathrm{d}z + mathrm{e}^{iphi(z)/2},mathrm{d}barrier zright).$$
Then, setting $$alpha = mathrm{e}^{-iphi(z)/2}mathrm{d}z + mathrm{e}^{iphi(z)/2},mathrm{d}barrier z$$, we behold that $$alpha$$ is a real-valued $$1$$-form, and therefore all the time has a neighborhood integrating issue, i.e., it will probably breathe written domestically within the type $$alpha = f,mathrm{d}u$$ for some capabilities real-valued capabilities $$u$$ and $$f>0$$. Thus, if $$Dsubsetmathbb{C}$$ is a province such that $$alpha$$ can breathe written as $$alpha = f,mathrm{d}u$$ for some real-valued capabilities $$u$$ and $$f>0$$ on $$D$$ and the fibers of $$u:Dto u(D)subset mathbb{R}$$ are related, then any resolution of your equation on $$D$$ can breathe written within the type $$w = h(u)$$ for some $$C^1$$ duty $$h:u(D)tomathbb{C}$$, and each such $$h$$ that’s $$C^1$$ yields an answer. This is as a result of your equation for $$w:Dtomathbb{C}$$ reduces to $$mathrm{d}w = p,mathrm{d}u$$ for some duty $$p:Dtomathbb{C}$$.

The signficance of $$phi$$ being harmonic shouldn’t be actually limpid (aside from making certain that $$alpha$$ is real-analytic, in order that $$u$$ can breathe taken to breathe real-analytic too). Certainly, the conduct of $$phi$$ will learn which domains $$Dsubsetmathbb{C}$$ have the privilege configuration to uphold an integrating issue for $$alpha$$, however it isn’t limpid to me that simply requiring that $$phi$$ breathe harmonic provides you mighty simply accessible data alongside these strains.

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