# Existence of complex function?

Motivated by a similar question Complex-doubly periodic function in two variables?, I would like to ask if there exists a non-zero function $$(z_1,z_2) \mapsto f(z_1,z_2)$$, where $$z_1,z_2 \in \mathbb C$$ are two complex variables, that satisfies $$(\partial_{z_2} + \partial_{z_1})f =0 \text{ and } (\partial_{\bar z_1} - \partial_{\bar z_2})f=0.$$

Notice that such a function $$f$$ is necessarily harmonic. $$(\partial_{\bar z_2} \partial_{z_2} + \partial_{\bar z_1} \partial_{z_1} )f=0.$$

In addition, I require for $$k_1,k_2 \in \mathbb R$$ the periodicity conditions $$\begin{gather*} f (z_1+1,z_2 ) = f(z_1,z_2 ), \quad f (z_1+i,z_2 ) = f(z_1,z_2 ), \\ \text{and}\quad f (z_1,z_2+1 ) = e^{ik_1} f(z_1,z_2), \quad f (z_1,z_2+i ) = e^{ik_2} f(z_1,z_2). \end{gather*}$$

Such a function $$f$$ must necessarily have poles, unless $$f$$ is constant, so what I am asking here is if there exists a function $$f$$ that satisfies the above differential equations up to a set of lower dimension where the functions exhibits poles and in addition all the periodicity conditions.

• which boundary conditions? Feb 1, 2021 at 22:46
• @FedorPetrov I now renamed them as periodicity conditions. The ones that link translates of $f$ to values of $f$. Feb 1, 2021 at 23:23
• I would use two new variables $x=\bar{z}_1+\bar{z}_2$, $y=z_1-z_2$, so $f$ is analytic in each of them. Feb 2, 2021 at 8:43
• You really should cite the earlier MO question, so that readers don't have to hunt for it: mathoverflow.net/questions/382480/… Feb 2, 2021 at 11:28
• @LSpice: Yes, indeed, the question is what I wanted to point to. I should have been more careful when I copied the link. Thanks. Feb 2, 2021 at 14:34

The answer is 'yes' there do exist such functions that are non-constant with singularities only along surfaces $$\Sigma\subset\mathbb{C}^2$$, and here is how one can understand them:
First, it helps to change coordinates, though, perhaps, a little more subtly than Fedor Petrov suggested: Let $$y_1 = \tfrac i2({\overline z}_1+{\overline z}_2 - z_1 + z_2) \quad\text{and}\quad y_2 = \tfrac12({\overline z}_1+{\overline z}_2 + z_1 - z_2)$$ Then $$(y_1,y_2):\mathbb{C}^2\to\mathbb{C}^2$$ is a diffeomorphism. Using the complex structure on $$\mathbb{C}^2$$ for which $$y_1$$ and $$y_2$$ are holomorphic coordinates, we find that $$\frac{\partial f}{\partial{\overline y}_1} = \frac{\partial f}{\partial{\overline y}_2} = 0,$$ so that $$f$$ is a holomorphic function of $$y_1$$ and $$y_2\,$$. Moreover, we have the periodicity relations $$f(y_1{+}1,y_2)= f(y_1,y_2{+}1) = f(y_1,y_2)$$ while $$f(y_1{+}i,y_2)= \mathrm{e}^{ik_1}f(y_1,y_2) \quad\text{and}\quad f(y_1,y_2{+}i) = \mathrm{e}^{-ik_2}f(y_1,y_2)$$
Now, one can construct the general meromorphic $$f$$ that satisfies these condition as follows:
First, let $$p_1$$ and $$p_2$$ be (not-identically-vanishing) meromorphic functions on $$\mathbb{C}$$ that satisfy the period relations $$p_1(y+1) = p_1(y)\quad\text{and}\quad p_1(y+i) = \mathrm{e}^{ik_1}\,p_1(y)$$ and $$p_2(y+1) = p_2(y)\quad\text{and}\quad p_2(y+i) = \mathrm{e}^{-ik_2}\,p_2(y).$$ Techniques for constructing meromorphic functions on $$\mathbb{C}$$ satisfying such double-periodicity relations are well-known, using $$\vartheta$$-series or the theory of elliptic curves. The point is that, if $$C = \mathbb{C}/\mathbb{Z}[i]$$ is the square torus (which is an elliptic curve), then the above relations on meromorphic functions on $$\mathbb{C}$$ essentially describe the meromorphic sections of two flat complex line bundles $$E_1$$ and $$E_2$$ over $$C$$, i.e., elements of $$\mathrm{Pic}_0(C)$$.
Given $$p_1$$ and $$p_2$$, then the general meromorphic $$f$$ that satisfies the above conditions can be written as a product $$f(y_1,y_2) = p_0(y_1,y_2)\,p_1(y_1)\,p_2(y_2),$$ where $$p_0$$ comes from any meromorphic function on $$C\times C$$, i.e., $$p_0$$ is a meromorphic function on $$\mathbb{C}^2$$ that satisfies $$p_0(y_1{+}m,y_2)= p_0(y_1,y_2{+}m) = p_0(y_1,y_2)$$ for any Gaussian integer $$m\in\mathbb{Z}[i]$$.
Note that the 'singularities' of $$f$$ occur along surfaces in $$\mathbb{C}^2$$ that are complex curves in the $$y$$-coordinates whose images in $$C\times C$$ are algebraic curves.