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I'm unsure of which Cauchy-Riemann law to use when I'm given either a real or imaginary function. For instance. I might be given a real function and asked to work out the imaginary part.

For instance, if I'm given the real part: $-3xy^2-2y^2+x^3+2x$ and asked to work out the imaginary, then I'd need to use the $\frac{du}{dx}=\frac{dv}{dy}$ rule rather than the $-\frac{du}{dy}=\frac{dv}{dx}$ rule before finding the imaginary part. Why is this?

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    I assume you are talking about finding a *harmonic conjugate* (so you are given the real part of a holomorphic function and you're supposed to find the imaginary part)? It's not entirely clear. In that case, I'm pretty sure you would need both Cauchy-Riemann equations.2012-05-11
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    Hi, yes. Although from my solution here, I've only used one: u=−3xy^2 −2y^2 +x^3 +2x^2 ∂u/∂x = −3 y2 + 3 x2 + 4 x = ∂v/∂y by C-R Hence v = −y3 + 3 x2 y + 4 x y2012-05-11
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    Oh! I think I've got you now...thanks for the help!2012-05-11
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    Once you know $\partial v/\partial y$, you can find $v$ by integrating with respect to $y$ (I assume this is what you did). However, this gives a constant of integration that *still depends on $x$*. To find the integration constant, you need the other equation.2012-05-11
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    Are you sure this is the correct expression for $u$? It is not harmonic.2012-05-11

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