If u is harmonic function defined on (say) the open unit disc, then it can be continuously extended to the closed unit disc in such a way that it matches any continuous function f(θ) on unit circle, i.e. the boundary of the disc. But my understanding is the same cannot be said of holomorphic functions, and this bothers me, because it seems like I can easily prove that it can be. Since u is harmonic it is the real part of some holomorphic function H=u+iv defined on the open unit disc, and v is harmonic. Then if F(θ) is some continuous function defined on the unit circle, we can extend u continuously to the closed unit disc so that it matches Re(F(θ)) on the unit circle, and we can extend v continuously to the closed unit disc so that it matches Im(F(θ)) on the unit circle. Thus we have successfully extended H continuously to the closed unit disc so that it matches F(θ) on the boundary. Where am I going wrong?(adsbygoogle = window.adsbygoogle || []).push({});

Any help would be greatly appreciated.

Thank You in Advance.

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# Why can't holomorphic functions be extended to a closed disc?

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