Theorems · Theorem · potential theory
AnalyticAt.harmonicAt_im
∀ {x : ℂ} {f : ℂ → ℂ}, AnalyticAt ℂ f x → InnerProductSpace.HarmonicAt (fun z => (f z).im) xIf f : ℂ → ℂ is complex-analytic, then its imaginary part is harmonic.
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- Foundations
- Depth 248 from the axioms · uses propext, Classical.choice, Quot.sound
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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement · cited by 25,697
- Complexstatement and proof · cited by 5,565
- Complex.imstatement · cited by 591
- AnalyticAtstatement and proof · cited by 321
- InnerProductSpace.HarmonicAtstatement · cited by 23
- Complex.imCLMproof · cited by 17
- InnerProductSpace.HarmonicAt.comp_CLMproof · cited by 4
- AnalyticAt.harmonicAtproof · cited by 3
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