# Solution theory for Cauchy-Riemann Equations in the spirit of transport equation

This is just a peculiar thought I had while I was brushing up on my complex analysis notes and coincidentally having the (homog.) transport equation in mind:

Comparing the two set of equations

$$\partial_t u = \partial_x u \text{ in } \mathbb{R}^2$$ $$u_0 = u(0, x)$$

and

$$\partial_t u = i * \partial_x u \text{ in } \mathbb{R}^2$$ $$\text{(Here, I identify \mathbb{C} with \mathbb{R}^2 by setting z=x+i*t)}$$ $$u_0 = u(0, x)$$

look awfully similar with just a factor $$"i"$$ in front of one of the partials. I was wondering whether this would give us an opportunity to characterize what it means for a function to be holomorphic, since for the transport equation, we characterized with the initial data all of the solutions (namely they are of the form $$u_0 = u(x+t)$$).

Is there any more knowledge gained from this or is there no substantial merit in comparing?

The difference is striking: $$\partial_t f -\partial_x f = 0$$ means the solution depends on $$x-t$$ only and is constant in directions $$x+t$$. The same is true for Cauchy-Riemann, but here the solutions depend on $$z = x + i t$$ only (holomorphic) and are independent on $$z^\star = x- i y$$, generating by this shortest known condition a complete theory of complex differentiable functions, that constitute most of what is known about real computable functions in any dimensions.