CauchyRiemanni

The Cauchy-Riemann equations are a pair of differential equations that are fundamental in complex analysis. They provide a necessary and sufficient condition for a complex function to be differentiable. A complex function $f(z) = u(x, y) + iv(x, y)$, where $z = x + iy$, $u(x, y)$ is the real part, and $v(x, y)$ is the imaginary part, is said to be complex differentiable at a point $z_0$ if the limit $\lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}$ exists.

The Cauchy-Riemann equations are expressed as: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$. If these equations hold and the partial derivatives are continuous in a neighborhood of a point $z_0$, then the function $f(z)$ is differentiable at $z_0$. Furthermore, if a function is differentiable in a region, it is called analytic or holomorphic in that region.

These equations are crucial because they connect the concept of differentiability of a complex function to the behavior of its real and imaginary parts as functions of two real variables. If a function satisfies the Cauchy-Riemann equations, its derivative can be computed in a particularly simple form: $f'(z) = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x}$, or equivalently $f'(z) = \frac{\partial v}{\partial y} - i\frac{\partial u}{\partial y}$. The existence of analytic functions is what distinguishes complex analysis from the calculus of real variables, leading to powerful theorems like Cauchy's integral theorem and the residue theorem.

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