Franklin's Notes


Cauchy-Riemann Equations

A holomorphic function $f:D\to \mathbb C$ is such that it has derivatives in a neighborhood of each point where it is defined. If we write $f(x+yi)=u(x,y)+v(x,y)i$ where $u,v:\mathbb R^2\to\mathbb R$ are real-valued functions, then we have that the limit exists for all $(x,y)$ in some neighborhood of any given point of $D$. In particular, the value that this expression converges to must not depend on the path via which $(h_x,h_y)$ approaches zero. If it approaches zero along the real axis, we obtain the limit and if it approaches zero along the imaginary axis, we obtain the limit Because these exrpressions are required to be equal in order for the derivative to be well-defined, we have by equating real and imaginary parts that These differential equations are called the Cauchy-Riemann Equations. It can similarly be shown that any pair of functions $u,v:\mathbb R^2\to\mathbb R$ satisfying these differential equations can be used as the real and imaginary part of a holomorphic function $f=u+vi$.

At a glance, it might not be obvious what this pair of differential equations actually means. Given a function $f$ defined in some neighborhood of a point $z_0\in\mathbb C$, its derivative should intuitively capture some information about how the output of $f$ changes with small perturbations of the input. If we express elements of $\mathbb C$ as points in a plane, we can see that any small perturbation is some linear combination of the perturbations in the real direction and in the imaginary direction. Each of these perturbations gives rise to a perturbation in the value of $f$, which can be expressed as a combination of a perturbation in the real direction and a perturbation in the complex direction.

The same can be said of any function $\mathbb R^2\to\mathbb R^2$ which is differentiable in both of its components with respect to both input variables. The Cauchy-Riemann equations, however, express that not just any differentiable function $\mathbb R^2\to\mathbb R^2$ gives rise to a holomorphic function. Rather, what makes holomorphic functions special is that the perturbations to the output resulting from a perturbation in the real direction and a perturbation in the imaginary direction form the same angle as the perturbations in the real/imaginary directions, and have proportional magnitudes. More generally, rotating and scaling some perturbation to the input results in a perturbation to the output that is rotated and scaled in the same way! The Cauchy-Riemann equations encapsulate how complex differentiable functions are more "nice" than just differentiable functions $\mathbb R^2\to\mathbb R^2$ - they also preserve local information about angles.

complex-analysis

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