Cauchy-Riemann equations

The Cauchy-Riemann equations are conditions imposed on the form of a complex-valued analytic (holomorphic) function. which naturally arise from the properties of complex arithmetic and Complex function decomposition. Namely, if a function $𝑓 (𝑥 + 𝑦 𝑖) = 𝑢 (𝑥, 𝑦) + 𝑖 𝑣 (𝑥, 𝑦)$ is differentiable, then

𝜕 𝑢 𝜕 𝑥 = 𝜕 𝑣 𝜕 𝑦 \land 𝜕 𝑢 𝜕 𝑦 = - 𝜕 𝑣 𝜕 𝑥

which may be written in the equivalent complex form

𝑖 𝜕 𝑓 𝜕 𝑥 = 𝜕 𝑓 𝜕 𝑦

As an immediate consequence of this, any holomorphic function will satisfy Laplace's equation when considered as a vector field $ℝ 2 \to ℝ 2$ . It also follows that integrals over complex numbers are path-independent, i.e. as a vector field holomorphic functions are irrotational. This simplifies Complex integration.

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