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I face a problem asking for CR equation for anti-holomorphic function. They ask for three forms: in rectangle coordinate, polar coordinate and complex coordinate.

My approach is that : Let $f = u + iv$ be a function which is anti-holomorphic. Let $ \bar{f} = g = s + it$. So $g$ satisfies normal CR in rectangle coordinate, that is, $s_x = v_y, s_y = -v_x$. So $u_x = -v_y, u_y = v_x$ is CR for anti-holomorphic $f$. Also, $rU_r = -V_\theta, rV_r = U_\theta$ is a polar CR for anti-holomirphic. But I do not know what is CR for complex coordinate. Actaully, I am not sure if CR for rectangle coordinate is the same as CR for complex coordinate since complex coordinate can be expressed as xy-coordinate.

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  • $\begingroup$ Any suggestion, anyone ? 😑😕 $\endgroup$ – Both Htob Feb 3 '15 at 12:26
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Suppose $f = u + iv$ is an anti-holomorphic function. Then $\bar{f} = u - iv$ is holomorphic, so $\frac{d\bar{f}}{d\bar{z}} = 0$. That is, $$\frac{1}{2}\left(\frac{\partial \bar{f}}{\partial x} + i \frac{\partial \bar{f}}{\partial y}\right) = 0.$$ Taking the conjugate of the equation yields $$\frac{1}{2}\left(\frac{\partial f}{\partial x} - i\frac{\partial f}{\partial y}\right) = 0.$$ In other words, $$\frac{df}{dz} = 0.$$ This gives the CR in complex coordinates. The CR in rectangular coordinates are $u_x = -v_y$, $u_y = v_x$, and in polar coordinates, we have $ru_r = -v_\theta$, $rv_r = u_\theta$.

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