Cauchy-Riemann Equations

Cauchy-Riemann Equations

Let f(x+iy)=u(x,y)+iv(x,y). The Cauchy-Riemann equations are:

\begin{aligned}  u_x&=v_y\\  u_y&=-v_x.  \end{aligned}

Alternative Form (Wirtinger Derivative)

The Cauchy-Riemann equations can be written as a single equation \displaystyle \frac{\partial f}{\partial\bar z}=0 where \displaystyle \frac{\partial}{\partial\bar z}=\frac 12(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}) is the Wirtinger derivative with respect to the conjugate variable.

Goursat’s Theorem

Suppose f=u+iv is a complex-valued function which is differentiable as a function f:\mathbb{R}^2\to\mathbb{R}^2. Then f is analytic in an open complex domain \Omega iff it satisfies the Cauchy-Riemann equations in the domain.

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