Statement: Suppose that \( f(z)=u(x, y)+i v(x, y) \) and \( f^{\prime}\left(z_{0}\right) \) exists at a point \( z_{0}=x_{0}+i y_{0} \) then the first order partial derivatives of \( u \) and \( v \) must exist at \( \left(x_{0}, y_{0}\right) \) and they must satisfy the Cauchy- Riemann equation \( u_{x}=v_{y}, u_{y}=-v_{x} \). Also, \( f^{\prime}\left(z_{0}\right)=u_{x}+i v_{x} \) where these partial derivatives are to be

The Cauchy-Riemann equations form a cornerstone of complex analysis, linking real and imaginary parts of a complex function to establish conditions for differentiability in the complex plane. Dating back to mathematicians like Augustin-Louis Cauchy and Bernhard Riemann, these equations demonstrate that if a function is analytic at a point, its derivatives not only exist but are also closely interrelated. This elegant interplay allows for the powerful application of complex functions in various fields, including engineering and physics. In practical terms, if you’re computationally evaluating complex functions, always check for the Cauchy-Riemann equations after establishing that the first-order partial derivatives exist. A common mistake is overlooking the requirement that these equations must hold for differentiability. Remember, violating these equations means the function isn’t differentiable at that point, which can lead to erroneous conclusions in calculations or when applying theorems like the Residue Theorem for evaluating integrals. So, keep an eye out for those derivatives working together!