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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?

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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.

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