##  [Cauchy–Riemann Equations](/cauchy-riemann-equations) 

  ##  [Cauchy–Riemann Equations](https://natural.quantumdictionary.io/cauchy-riemann-equations-0) 

  

 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)



**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A pair of first-order partial differential relations on the real and imaginary parts of a complex-valued function defined on an open subset of the plane; they are the local necessary and sufficient conditions for the function to be complex-differentiable (holomorphic).