KnowraCauchy–Riemann equationsLinked fromLinked fromThe 13 pages that link to Cauchy–Riemann equations, each with the reason it gives.All 13Broader topic 1Related 11Narrower topic 1Holomorphic functionRelated: They translate the complex derivative condition into relations between real and imaginary parts.Complex analysisRelated: Under suitable regularity conditions, they characterize complex differentiability.Augustin-Louis CauchyBroader topic: Cauchy helped establish these equations as a test for complex differentiability.Cauchy's integral formulaRelated: Under suitable smoothness conditions, these equations provide a differential test for holomorphicity.Cauchy problemRelated: They illustrate a system whose local solutions can be constrained by data along a curve.Conformal mapRelated: They enforce the local rotation-and-scaling structure behind planar conformality.Liouville's theoremRelated: They express the local structure behind the holomorphic functions constrained by the theorem.Complex differentiabilityRelated: They express a necessary local constraint for complex differentiability when the partial derivatives are continuous.Cauchy's integral theoremRelated: They offer a local analytic test for the differentiability behind the theorem.Complex derivativeRelated: They characterize complex differentiability when the relevant partial derivatives are continuous.Morera's theoremRelated: They offer a complementary route to holomorphy that Morera's integral condition can help establish.Analytic functionRelated: They express the local derivative constraints satisfied by complex analytic functions.Analyticity of holomorphic functionsNarrower topic: They characterize complex differentiability under suitable regularity, the hypothesis behind local analyticity.