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**Unformatted text preview: **Remark : On a connected domain Ω ⊂ R 2 , all harmonic conjugates to a given function u ( x, y ) only differ by a constant: tildewide v ( x, y ) = v ( x, y ) + c ; see Exercise . Although most harmonic functions have harmonic conjugates, unfortunately this is not always the case. Interestingly, the existence or non-existence of a harmonic conjugate can depend on the underlying topology of its domain of definition. If the domain is simply connected, and so contains no holes, then one can always find a harmonic conjugate. On non-simply connected domains, there may not exist a single-valued harmonic conjugate to serve as the imaginary part of a complex function f ( z ). Example 7.12. The simplest example where the latter possibility occurs is the logarithmic potential u ( x, y ) = log r = 1 2 log( x 2 + y 2 ) . This function is harmonic on the non-simply connected domain Ω = C \ { } , but is not the real part of any single-valued complex function. Indeed, according to (7.15), the logarithmic potential is the real part of the multiply-valued complex logarithm log z , and so its harmonic conjugate † is ph z = θ , which cannot be consistently and continuously defined on all of Ω. On the other hand, on any simply connected subdomain tildewide Ω ⊂ Ω, one can select a continuous, single-valued branch of the angle θ = ph z , which is then a bona fide harmonic conjugate to log r restricted to this subdomain. The harmonic function u ( x, y ) = x x 2 + y 2 is also defined on the same non-simply connected domain Ω = C \ { } with a singularity at x = y = 0. In this case, there is a single valued harmonic conjugate, namely v ( x, y ) = − y x 2 + y 2 , which is defined on all of Ω. Indeed, according to (7.9), these functions define the real and imaginary parts of the complex function u + i v = 1 /z . Alternatively, one can directly check that they satisfy the Cauchy–Riemann equations (7.18).check that they satisfy the Cauchy–Riemann equations (7....

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