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can any1 explain some small stuff about harmonic and conjugate functions?

Mathematics
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i know that for a function to be harmonic it must satisfy the laplace equation in 2 dimension, and a conjugate function must satisfy the cauchy riemanns equation. so all conjugate must be harmonic also? or vice versa, or i cant make that relationship?
In mathematics, a function defined on some open domain is said to have as a conjugate a function if and only if they are respectively real and imaginary part of a holomorphic function of the complex variable That is, is conjugate to if is holomorphic on As a first consequence of the definition, they are both harmonic real-valued functions on . Moreover, the conjugate of if it exists, is unique up to an additive constant. Also, is conjugate to if and only if is conjugate to .
so a conjugate function is always harmonic, ie. called a conjugate harmonic function? since conjugate function need to satisfy the CR equation, and by the 2nd CR equation it satisfies the laplace equation. so can i conclude like this?

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