SlowGuess/NeurIPS_2023_OCR huggingface dataset:
For an open set  $U \subseteq \mathbb{C}^n$ , a function  $f \in C^k(U; \mathbb{C})$  and a multi-index  $\ell \in \mathbb{N}_0^n$  with  $|\ell| \leq k$  we write  $\partial_{\mathrm{wirt}}^\ell f$  and  $\overline{\partial}_{\mathrm{wirt}}^\ell f$  for the iterated Wirtinger derivatives according to the multi-index  $\ell$ .
Proposition A.1. Let  $U \subseteq \mathbb{C}^n$  be an open set and  $f \in C^k(U; \mathbb{C})$ . Then, identifying  $f$  with the function  $f \circ \varphi_n$  with  $\varphi_n$  as in (A.1), for any multi-indices  $m, \ell \in \mathbb{N}_0^n$  with  $|m + \ell| \leq k$  we have the representation
$$
\partial_{\mathrm{wirt}}^{\boldsymbol{m}}\overline{\partial}_{\mathrm{wirt}}^{\boldsymbol{\ell}}f(a) = \sum_{\substack{\boldsymbol {p} = (\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime \prime})\in \mathbb{N}_{0}^{2n}\\ \boldsymbol{p}^{\prime} + \boldsymbol{p}^{\prime \prime} = \boldsymbol {m} + \boldsymbol {\ell}}}b_{\boldsymbol{p}}\left(\partial^{\boldsymbol{p}}f\right)(a)\quad \forall a\in U
$$
with complex numbers  $b_{p} \in \mathbb{C}$  only depending on  $p, m$  and  $\ell$  and writing  $\pmb{p} = (p', p'')$  for the concatenation of the multi-indices  $\pmb{p'}$  and  $\pmb{p}'' \in \mathbb{N}_0^n$ . In particular, the coefficients do not depend on  $f$ .
Proof. The proof is by induction over  $\mathbf{m}$  and  $\ell$ . We first assume  $\mathbf{m} = 0$  and show the claim for all  $\ell \in \mathbb{N}_0^n$  with  $|\ell| \leq k$ . In the case  $\ell = 0$  there is nothing to show, so we assume the claim to be true for fixed  $\ell \in \mathbb{N}_0^n$  with  $|\ell| < k$  and take  $j \in \{1, \dots, n\}$  arbitrarily. Then we get
$$
\begin{array}{l} \overline{\partial}_{\mathrm{wirt}}^{\ell +e_{j}}f(a) = \overline{\partial}_{\mathrm{wirt}}^{e_{j}}\overline{\partial}_{\mathrm{wirt}}^{\ell}f(a)\stackrel {\mathrm{IH}}{=}\sum_{\substack{\mathbf{p} = (\mathbf{p}^{\prime},\mathbf{p}^{\prime \prime})\in \mathbb{N}_{0}^{2n}\\ \mathbf{p}^{\prime} + \mathbf{p}^{\prime \prime} = \boldsymbol {\ell}}}b_{\mathbf{p}}\overline{\partial}_{\mathrm{wirt}}^{e_{j}}\left(\partial^{\mathbf{p}}f\right)(a) \\ = \sum_ {\substack {\mathbf {p} = (\mathbf {p} ^ {\prime}, \mathbf {p} ^ {\prime \prime}) \in \mathbb {N} _ {0} ^ {2 n} \\ \mathbf {p} ^ {\prime} + \mathbf {p} ^ {\prime \prime} = \ell}} \frac {b _ {\mathbf {p}}}{2} \left(\partial^ {(\mathbf {p} ^ {\prime} + e _ {j}, \mathbf {p} ^ {\prime \prime})} f\right) (a) + \frac {i b _ {\mathbf {p}}}{2} \left(\partial^ {(\mathbf {p} ^ {\prime}, \mathbf {p} ^ {\prime \prime} + e _ {j})} f\right) (a) \\ =: \sum_{\substack{\mathbf{p} = (\mathbf{p}^{\prime},\mathbf{p}^{\prime \prime})\in \mathbb{N}_{0}^{2n}\\ \mathbf{p}^{\prime} + \mathbf{p}^{\prime \prime} = \boldsymbol {\ell} + e_{j}}}b_{\mathbf{p}}\left(\partial^{\mathbf{p}}f\right)(a), \\ \end{array}
$$
Since we have shown the case  $\mathbf{m} = 0$  we may assume the claim to be true for fixed  $\mathbf{m},\ell \in \mathbb{N}_0^n$  with  $|\mathbf{m} + \boldsymbol {\ell}| < k$ . Then we deduce
$$
\begin{array}{l} \partial_{\mathrm{wirt}}^{\mathbf{m} + e_{j}}\overline{\partial}_{\mathrm{wirt}}^{\ell}f(a) = \partial_{\mathrm{wirt}}^{e_{j}}\partial_{\mathrm{wirt}}^{\mathbf{m}}\overline{\partial}_{\mathrm{wirt}}^{\ell}f(a)\stackrel {\mathrm{IH}}{=}\sum_{\substack{\mathbf{p} = (\mathbf{p}^{\prime},\mathbf{p}^{\prime \prime})\in \mathbb{N}_{0}^{2n}\\ \mathbf{p}^{\prime} + \mathbf{p}^{\prime \prime} = \mathbf{m} + \ell}}b_{\mathbf{p}}\partial_{\mathrm{wirt}}^{e_{j}}\left(\partial^{\mathbf{p}}f\right)(a) \\ = \sum_ {\substack {\mathbf {p} = (\mathbf {p} ^ {\prime}, \mathbf {p} ^ {\prime \prime}) \in \mathbb {N} _ {0} ^ {2 n} \\ \mathbf {p} ^ {\prime} + \mathbf {p} ^ {\prime \prime} = \mathbf {m} + \ell}} \frac {b _ {\mathbf {p}}}{2} \left(\partial^ {\left(\mathbf {p} ^ {\prime} + e _ {j} , \mathbf {p} ^ {\prime \prime}\right)} f\right) (a) - \frac {i b _ {\mathbf {p}}}{2} \left(\partial^ {\left(\mathbf {p} ^ {\prime} , \mathbf {p} ^ {\prime \prime} + e _ {j}\right)} f\right) (a) \\ =: \sum_{\substack{\mathbf{p} = (\mathbf{p}^{\prime},\mathbf{p}^{\prime \prime})\in \mathbb{N}_{0}^{2n}\\ \mathbf{p}^{\prime} + \mathbf{p}^{\prime \prime} = \mathbf{m} + \boldsymbol {\ell} + e_{j}}}b_{\mathbf{p}}\left(\partial^{\mathbf{p}}f\right)\left(a\right). \\ \end{array}
$$
Hence the claim follows using the principle of mathematical induction.
Proposition A.1 is technical but crucial: In the course of the paper we will need to approximate Wirtinger derivatives of certain functions. In fact, however, we will approximate real derivatives and take advantage of the fact that Wirtinger derivatives are linear combinations of these.
A.2 Concrete examples of activation functions
In this section we analyze concrete activation functions that are commonly used in practical applications of complex-valued neural networks. We are going to show that those activation functions are smooth and non-polyharmonic on some non-empty open subset of  $\mathbb{C}$ . Our first result analyzes a family of "real activation functions interpreted as complex activation functions".
Proposition A.2. Let  $\rho \in C^{\infty}(\mathbb{R};\mathbb{R})$  be non-polynomial and let  $\psi :\mathbb{C}\to \mathbb{C}$  be defined as
$$
\psi (z) := \rho (\operatorname {R e} (z)) \quad r e s p. \quad \psi (z) := \rho (\operatorname {I m} (z)).
$$
Then  $\psi$  is smooth and non-polyharmonic.
Proof. Since  $\psi$  depends only on the real-, resp. imaginary part of the input, we see directly from the definition of the Wirtinger derivatives that
$$
\partial_ {\mathrm {w i r t}} \psi (z) = \overline {{\partial}} _ {\mathrm {w i r t}} \psi (z) = \frac {1}{2} \rho^ {\prime} (\mathrm {R e} (z)) \quad \mathrm {r e s p .} \quad \partial_ {\mathrm {w i r t}} \psi (z) = - \overline {{\partial}} _ {\mathrm {w i r t}} \psi (z) = - \frac {i}{2} \rho^ {\prime} (\mathrm {I m} (z)).
$$
Hence we see for arbitrary  $m,\ell \in \mathbb{N}_0$  that
$$
\left| \partial_ {\mathrm {w i r t}} ^ {m} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {\ell} \psi (z) \right| = \frac {1}{2 ^ {m + \ell}} \left| \rho^ {(m + \ell)} (\mathrm {R e} (z)) \right| \quad \mathrm {r e s p .} \quad \left| \partial_ {\mathrm {w i r t}} ^ {m} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {\ell} \psi (z) \right| = \frac {1}{2 ^ {m + \ell}} \left| \rho^ {(m + \ell)} (\mathrm {I m} (z)) \right|.
$$
Since  $\rho$  is non-polynomial we can choose a real number  $x$ , such that  $\rho^{(n)}(x) \neq 0$  for all  $n \in \mathbb{N}_0$  (cf. for instance [20, p. 53]). Thus,  $\partial_{\mathrm{wirt}}^m \overline{\partial}_{\mathrm{wirt}}^\ell \psi(z) \neq 0$  for all  $m, \ell \in \mathbb{N}_0$ , whenever  $z \in \mathbb{C}$  with  $\operatorname{Re}(z) = x$ , or  $\operatorname{Im}(z) = y$ , respectively.
Next, we consider the modReLU which was defined in Section 2.
Theorem A.3. Let  $b \in (-\infty, 0)$ . Writing  $\sigma = \sigma_{\mathrm{modReLU},b}$ , one has for every  $z \in \mathbb{C}$  with  $|z| + b > 0$  the identity
$$
\left(\partial_ {\mathrm {w i r t}} ^ {m} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {\ell} \sigma\right) (z) = \left\{ \begin{array}{l l} z + b \frac {z}{| z |}, & m = \ell = 0, \\ 1 + \frac {b}{2} \cdot \frac {1}{| z |}, & m = 1, \ell = 0, \\ b \cdot q _ {m, \ell} \cdot \frac {z ^ {\ell - m + 1}}{| z | ^ {2 \ell + 1}}, & m \leq \ell + 1, \ell \geq 1, \\ b \cdot q _ {m, \ell} \cdot \frac {\overline {{z}} ^ {m - \ell - 1}}{| z | ^ {2 m - 1}}, & m \geq \ell + 1, m \geq 2 \end{array} \right.
$$
for every  $m \in \mathbb{N}_0$  and  $\ell \in \mathbb{N}_0$ . Here, the numbers  $q_{m,\ell}$  are non-zero and rational. Furthermore, note that all cases for choices of  $m$  and  $\ell$  are covered, by observing that we can either have the case  $m \geq \ell + 1$  (where either  $m \geq 2$  or  $m = 1$ ,  $\ell = 0$ ) or  $m \leq \ell + 1$ , where the latter case is again split into  $\ell = 0$  and  $\ell \geq 1$ .
Proof. We first calculate certain Wirtinger derivatives for  $z \neq 0$ . First note
$$
\begin{array}{l} \bar {\partial} _ {\text {w i r t}} \left(\frac {1}{| z | ^ {m}}\right) = \frac {1}{2} \left(\partial^ {(1, 0)} \left(\frac {1}{| z | ^ {m}}\right) + i \cdot \partial^ {(0, 1)} \left(\frac {1}{| z | ^ {m}}\right)\right) \\ = \frac {1}{2} \left(\left(- \frac {m}{2}\right) \frac {2 \operatorname {R e} (z) + i \cdot 2 \operatorname {I m} (z)}{| z | ^ {m + 2}}\right) \\ = - \frac {m}{2} \cdot \frac {z}{| z | ^ {m + 2}} \\ \end{array}
$$
$$
\partial_ {\mathrm {w i r t}} \left(\frac {1}{| z | ^ {m}}\right) = - \frac {m}{2} \cdot \frac {\bar {z}}{| z | ^ {m + 2}}
$$
for any  $m \in \mathbb{N}$ . Using the product rule for Wirtinger derivatives (see Appendix A.1), we see
$$
\bar {\partial} _ {\text {w i r t}} \left(\frac {z ^ {\ell}}{| z | ^ {m}}\right) = \underbrace {\bar {\partial} _ {\text {w i r t}} \left(z ^ {\ell}\right)} _ {= 0} \cdot \frac {1}{| z | ^ {m}} + z ^ {\ell} \cdot \bar {\partial} _ {\text {w i r t}} \left(\frac {1}{| z | ^ {m}}\right) = - \frac {m}{2} \cdot \frac {z ^ {\ell + 1}}{| z | ^ {m + 2}} \tag {A.2}
$$
for any  $m\in \mathbb{N}$  and  $\ell \in \mathbb{N}_0$  and furthermore
$$
\begin{array}{l} \partial_ {\text {w i r t}} \left(\frac {z ^ {\ell}}{| z | ^ {m}}\right) = \partial_ {\text {w i r t}} \left(z ^ {\ell}\right) \cdot \frac {1}{| z | ^ {m}} + z ^ {\ell} \cdot \partial_ {\text {w i r t}} \left(\frac {1}{| z | ^ {m}}\right) \\ = \ell \cdot z ^ {\ell - 1} \cdot \frac {1}{| z | ^ {m}} - z ^ {\ell} \cdot \frac {m}{2} \cdot \frac {\overline {{z}}}{| z | ^ {m + 2}} \\ = \left(\ell - \frac {m}{2}\right) \cdot \frac {z ^ {\ell - 1}}{| z | ^ {m}} \tag {A.3} \\ \end{array}
$$
for  $m,\ell \in \mathbb{N}$  , and finally
$$
\partial_ {\text {w i r t}} \left(\frac {\bar {z} ^ {\ell}}{| z | ^ {m}}\right) = \underbrace {\partial_ {\text {w i r t}} \left(\bar {z} ^ {\ell}\right)} _ {= 0} \cdot \frac {1}{| z | ^ {m}} + \bar {z} ^ {\ell} \cdot \partial_ {\text {w i r t}} \left(\frac {1}{| z | ^ {m}}\right) = - \frac {m}{2} \cdot \frac {\bar {z} ^ {\ell + 1}}{| z | ^ {m + 2}} \tag {A.4}
$$
for  $m\in \mathbb{N}$  and  $\ell \in \mathbb{N}_0$
Having proven those three identities, we can start with the actual computation. We first fix  $m = 0$  and prove the claimed identity by induction over  $\ell$ . The case  $\ell = 0$  follows directly from the definition of the modReLU and for  $\ell = 1$ , we note
$$
\overline {{\partial}} _ {\mathrm {w i r t}} \sigma (z) = \underbrace {\overline {{\partial}} _ {\mathrm {w i r t}} (z)} _ {= 0} + b \cdot \overline {{\partial}} _ {\mathrm {w i r t}} \left(\frac {z}{| z |}\right) \stackrel {(\mathrm {A}, 2)} {=} b \cdot \left(- \frac {1}{2}\right) \frac {z ^ {2}}{| z | ^ {3}},
$$
which is the claimed form. Then, using induction, we compute
$$
\overline {{\partial}} _ {\mathrm {w i r t}} ^ {\ell + 1} \sigma (z) = \overline {{\partial}} _ {\mathrm {w i r t}} \left(b \cdot q _ {0, \ell} \cdot \frac {z ^ {\ell + 1}}{| z | ^ {2 \ell + 1}}\right) \stackrel {{(\mathtt {A}, 2)}} {=} b \cdot \underbrace {q _ {0 , \ell} \cdot \left(- \frac {2 \ell + 1}{2}\right)} _ {=: q _ {0, \ell + 1}} \cdot \frac {z ^ {\ell + 2}}{| z | ^ {2 \ell + 3}},
$$
so that the case  $m = 0$  is complete.
Now we deal with the case  $m \leq \ell + 1$ . The case  $\ell = 0$  and  $m = 1$  is proven by computing
$$
\partial_ {\mathrm {w i r t}} \sigma (z) = \partial_ {\mathrm {w i r t}} (z) + b \cdot \partial_ {\mathrm {w i r t}} \left(\frac {z}{| z |}\right) \stackrel {{\text {(A . 3)}}} {=} 1 + b \cdot \frac {1}{2} \cdot \frac {1}{| z |}
$$
so we can assume  $\ell > 0$ . Since we already dealt with the case  $m = 0$ , we can inductively assume the claim to be true for a fixed  $m \leq \ell$ . Then we compute
$$
\left(\partial_ {\mathrm {w i r t}} ^ {m + 1} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {\ell} \sigma\right) (z) = \partial_ {\mathrm {w i r t}} \left(b \cdot q _ {m, \ell} \cdot \frac {z ^ {\ell - m + 1}}{| z | ^ {2 \ell + 1}}\right) \stackrel {(\mathrm {A}. 3)} {=} b \cdot \underbrace {q _ {m , \ell} \cdot \left(- m + \frac {1}{2}\right)} _ {=: q _ {m + 1, \ell}} \cdot \frac {z ^ {\ell - m}}{| z | ^ {2 \ell + 1}},
$$
which is of the desired form. Note that (A.3) is indeed applicable because  $\ell - m + 1 \geq 1$ .
Finally, we consider the case where  $m \geq \ell + 1$  and  $m \geq 2$ . The case  $m = \ell + 1$  has already been shown. Using induction, we see
$$
\begin{array}{l} \left(\partial_ {\mathrm {w i r t}} ^ {m + 1} \bar {\partial} _ {\mathrm {w i r t}} ^ {\ell} \sigma\right) (z) = \partial_ {\mathrm {w i r t}} \left(\delta_ {(m, \ell) = (1, 0)} + b \cdot q _ {m, \ell} \cdot \frac {\bar {z} ^ {m - \ell - 1}}{| z | ^ {2 m - 1}}\right) \\ \stackrel {{(\mathrm {A}. 4)}} {{=}} b \cdot \underbrace {q _ {m , \ell} \cdot \left(- m + \frac {1}{2}\right)} _ {=: q _ {m + 1, \ell}} \cdot \frac {\overline {{z}} ^ {m - \ell}}{| z | ^ {2 m + 1}}, \\ \end{array}
$$
so the proof is complete.
From Theorem A.3 we can now deduce that the modReLU is smooth and non-polyharmonic on some non-empty open subset of  $\mathbb{C}$ .
Corollary A.4. Let  $b \in (-\infty, 0)$  and  $z \in \mathbb{C}$  with  $|z| > -b$ . Then we have
$$
\partial_ {\mathrm {w i r t}} ^ {m} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {\ell} \sigma_ {\mathrm {m o d R e L U}, b} (z) \neq 0
$$
for all  $m, \ell \in \mathbb{N}_0$ . In particular,  $\sigma_{\mathrm{modReLU},b}$  is smooth and non-polyharmonic on the set
$$
\{z \in \mathbb {C}: | z | > - b \}.
$$
Furthermore,  $\sigma_{\mathrm{modReLU},b}$  is continuous on the entire complex plane.
Proof. The first part follows from Theorem A.3 by noting that if  $|z| > -b$  we have in particular  $|z| > -b/2$  and  $|z| > 0$ . For the second part, we first note that  $\sigma_{\mathrm{modReLU}, b}$  is trivially continuous in  $z \in \mathbb{C}$  if  $|z| \neq -b$ . Hence, we assume  $|z| = -b$ . Take any sequence  $(z_j)_{j \in \mathbb{N}}$  with  $z_j \to z$  as  $j \to \infty$  and note
$$
| \sigma_ {\mathrm {m o d R e L U}, b} (z _ {j}) - \sigma_ {\mathrm {m o d R e L U}, b} (z) | = | \sigma_ {\mathrm {m o d R e L U}, b} (z _ {j}) | \to \left\{ \begin{array}{l l} 0 \to 0, & \mathrm {i f} | z _ {j} | <   b, \\ | z _ {j} | + b \to | z | + b = 0, & \mathrm {i f} | z _ {j} | \geq b. \end{array} \right.
$$
This shows the continuity of  $\sigma_{\mathrm{modReLU},b}$ .
Next we analyze the complex cardioid, which was defined in Section 2.
Theorem A.5. For any  $z \in \mathbb{C}$  with  $z \neq 0$  and any  $m, \ell \in \mathbb{N}_0$  we have
$$
\partial_ {\mathrm {w i r t}} ^ {m} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {\ell} \operatorname {c a r d} (z) = \left\{ \begin{array}{l l} \frac {1}{2} z + \frac {1}{4} \frac {z ^ {2}}{| z |} + \frac {| z |}{4}, & m = \ell = 0, \\ a _ {m, \ell} \frac {z ^ {\ell - m}}{| z | ^ {2 \ell - 1}} + b _ {m, \ell} \frac {z ^ {\ell + 2 - m}}{| z | ^ {2 \ell + 1}}, & m \leq \ell \neq 0, \\ \frac {1}{2} + \frac {1}{8} \cdot \frac {\overline {{z}}}{| z |} + \frac {3}{8} \cdot \frac {z}{| z |}, & m = \ell + 1 = 1, \\ a _ {m, \ell} \frac {\overline {{z}}}{| z | ^ {2 \ell + 1}} + b _ {m, \ell} \frac {z}{| z | ^ {2 \ell + 1}}, & m = \ell + 1 > 1, \\ a _ {m, \ell} \frac {\overline {{z}} ^ {m - \ell}}{| z | ^ {2 m - 1}} + b _ {m, \ell} \frac {\overline {{z}} ^ {m - \ell - 2}}{| z | ^ {2 m - 3}}, & m \geq \ell + 2. \end{array} \right.
$$
Here, the numbers  $a_{m,\ell}$  and  $b_{m,\ell}$  are non-zero and rational. Furthermore, note that all cases for possible choices of  $m$  and  $\ell$  are covered: The case  $m \leq \ell$  is split into  $\ell = 0$  and  $\ell \neq 0$ . The case  $m = \ell + 1$  is split into  $m = 1$  and  $m > 1$ . Then, the case  $m \geq \ell + 2$  remains.
Proof. For the following we always assume  $z \in \mathbb{C}$  with  $z \neq 0$ . Then we can apply the identity  $\cos(\triangleleft z) = \frac{\operatorname{Re}(z)}{|z|}$ , so we can rewrite
$$
\operatorname {c a r d} (z) = \frac {1}{2} \left(1 + \frac {\operatorname {R e} (z)}{| z |}\right) z = \frac {1}{2} z + \frac {1}{4} \frac {(z + \bar {z}) z}{| z |} = \frac {1}{2} z + \frac {1}{4} \frac {z ^ {2}}{| z |} + \frac {| z |}{4}. \tag {A.5}
$$
This establishes the case  $m = \ell = 0$ . Next, we compute
$$
\bar {\partial} _ {\mathrm {w i r t}} (| z |) = \frac {1}{2} \left(\frac {1}{2} \frac {2 \operatorname {R e} (z)}{| z |} + \frac {i}{2} \frac {2 \operatorname {I m} (z)}{| z |}\right) = \frac {1}{2} \frac {z}{| z |} \tag {A.6}
$$
$$
\partial_ {\mathrm {w i r t}} (| z |) = \frac {1}{2} \frac {\bar {z}}{| z |}. \tag {A.7}
$$
We deduce
$$
\overline {{\partial}} _ {\mathrm {w i r t}} \operatorname {c a r d} (z) \overset {(\mathrm {A}. 2), (\mathrm {A}. 6)} {=} \underbrace {\frac {1}{4} \cdot \left(- \frac {1}{2}\right)} _ {=: b _ {0, 1}} \cdot \frac {z ^ {3}}{| z | ^ {3}} + \underbrace {\frac {1}{8}} _ {=: a _ {0, 1}} \cdot \frac {z}{| z |}.
$$
Using induction, we derive
$$
\begin{array}{l} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {\ell + 1} \operatorname {c a r d} (z) = a _ {0, \ell} \overline {{\partial}} _ {\mathrm {w i r t}} \left(\frac {z ^ {\ell}}{| z | ^ {2 \ell - 1}}\right) + b _ {0, \ell} \overline {{\partial}} _ {\mathrm {w i r t}} \left(\frac {z ^ {\ell + 2}}{| z | ^ {2 \ell + 1}}\right) \\ \stackrel {{(\mathrm {A}. 2)}} {{=}} \underbrace {a _ {0 , \ell} \cdot \left(- \frac {2 \ell - 1}{2}\right)} _ {=: a _ {0, \ell + 1}} \cdot \frac {z ^ {\ell + 1}}{| z | ^ {2 \ell + 1}} + \underbrace {b _ {0 , \ell} \cdot \left(- \frac {2 \ell + 1}{2}\right)} _ {=: b _ {0, \ell + 1}} \cdot \frac {z ^ {\ell + 3}}{| z | ^ {2 \ell + 3}}, \\ \end{array}
$$
so the claim has been shown if  $m = 0$ . If we now fix any  $\ell \in \mathbb{N}$  and assume that the claim holds true for some  $m \in \mathbb{N}_0$  with  $m < \ell$ , we get
$$
\begin{array}{l} \partial_ {\mathrm {w i r t}} ^ {m + 1} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {\ell} \operatorname {c a r d} (z) = a _ {m, \ell} \partial_ {\mathrm {w i r t}} \left(\frac {z ^ {\ell - m}}{| z | ^ {2 \ell - 1}}\right) + b _ {m, \ell} \partial_ {\mathrm {w i r t}} \left(\frac {z ^ {\ell + 2 - m}}{| z | ^ {2 \ell + 1}}\right) \\ \stackrel {{(\mathrm {A . 3})}} {{=}} \underbrace {a _ {m , \ell} \cdot \left(\frac {1}{2} - m\right)} _ {=: a _ {m + 1, \ell}} \cdot \frac {z ^ {\ell - m - 1}}{| z | ^ {2 \ell - 1}} + \underbrace {b _ {m , \ell} \cdot \left(\frac {3}{2} - m\right)} _ {=: b _ {m + 1, \ell}} \cdot \frac {z ^ {\ell + 2 - m - 1}}{| z | ^ {2 \ell + 1}}, \\ \end{array}
$$
so the claim holds true if  $m \leq \ell$ .
The case  $m = \ell + 1$  is split into the case  $m = 1$  and  $m > 1$ . If  $m = 1$ , then  $\ell = 0$  and we compute
$$
\partial_ {\mathrm {w i r t}} \operatorname {c a r d} (z) \stackrel {{(\mathrm {A . 3}), (\mathrm {A . 5}), (\mathrm {A . 7})}} {=} \frac {1}{2} + \frac {1}{4} \left(2 - \frac {1}{2}\right) \cdot \frac {z}{| z |} + \frac {1}{8} \cdot \frac {\overline {{z}}}{| z |}.
$$
If  $m > 1$ , we get
$$
\begin{array}{l} \partial_ {\mathrm {w i r t}} ^ {\ell + 1} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {\ell} \operatorname {c a r d} (z) = a _ {\ell , \ell} \partial_ {\mathrm {w i r t}} \left(\frac {1}{| z | ^ {2 \ell - 1}}\right) + b _ {\ell , \ell} \cdot \partial_ {\mathrm {w i r t}} \left(\frac {z ^ {2}}{| z | ^ {2 \ell + 1}}\right) \\ \stackrel {{(\mathrm {A}. 3), (\mathrm {A}. 4)}} {{=}} \underbrace {a _ {\ell , \ell} \cdot \left(- \frac {2 \ell - 1}{2}\right)} _ {=: a _ {\ell + 1, \ell}} \cdot \frac {\overline {{z}}}{| z | ^ {2 \ell + 1}} + \underbrace {b _ {\ell , \ell} \cdot \left(2 - \frac {2 \ell + 1}{2}\right)} _ {=: b _ {\ell + 1, \ell}} \cdot \frac {z}{| z | ^ {2 \ell + 1}}. \\ \end{array}
$$
Next, we deal with the case  $m = \ell + 2$ : Here, we see
$$
\begin{array}{l} \partial_ {\mathrm {w i r t}} ^ {\ell + 2} \bar {\partial} _ {\mathrm {w i r t}} ^ {\ell} \operatorname {c a r d} (z) = \partial_ {\mathrm {w i r t}} \left(\frac {1}{2} \delta_ {\ell = 0} + a _ {\ell + 1, \ell} \frac {\bar {z}}{| z | ^ {2 \ell + 1}} + b _ {\ell + 1, \ell} \frac {z}{| z | ^ {2 \ell + 1}}\right) \\ \stackrel {{(\mathrm {A}. 3), (\mathrm {A}. 4)}} {{=}} \underbrace {a _ {\ell + 1 , \ell} \cdot \left(- \frac {2 \ell + 1}{2}\right)} _ {=: a _ {\ell + 2, \ell}} \cdot \frac {\overline {{z}} ^ {2}}{| z | ^ {2 \ell + 3}} + \underbrace {b _ {\ell + 1 , \ell} \cdot \left(1 - \frac {2 \ell + 1}{2}\right)} _ {=: b _ {\ell + 2, \ell}} \cdot \frac {1}{| z | ^ {2 \ell + 1}}. \\ \end{array}
$$
If we assume the claim to be true for a fixed  $m \geq \ell + 2$ , we get
$$
\begin{array}{l} \partial_ {\mathrm {w i r t}} ^ {m + 1} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {\ell} \operatorname {c a r d} (z) = a _ {m, \ell} \cdot \partial_ {\mathrm {w i r t}} \left(\frac {\bar {z} ^ {m - \ell}}{| z | ^ {2 m - 1}}\right) + b _ {m, \ell} \cdot \partial_ {\mathrm {w i r t}} \left(\frac {\bar {z} ^ {m - \ell - 2}}{| z | ^ {2 m - 3}}\right) \\ \stackrel {{(\mathrm {A}, 4)}} {{=}} \underbrace {a _ {m , \ell} \cdot \left(- \frac {2 m - 1}{2}\right)} _ {=: a _ {m + 1, \ell}} \cdot \frac {\overline {{z}} ^ {m + 1 - \ell}}{| z | ^ {2 m + 1}} + \underbrace {b _ {m , \ell} \cdot \left(- \frac {2 m - 3}{2}\right)} _ {=: b _ {m + 1, \ell}} \cdot \frac {\overline {{z}} ^ {m - \ell - 1}}{| z | ^ {2 m - 1}}. \\ \end{array}
$$
Hence, using induction, we have proven the claimed identity.
The statement regarding the non-polyharmonicity of the complex cardioid is formulated in the following corollary.
Corollary A.6. For every  $z \in \mathbb{C}$  with  $z \notin \mathbb{R} \cup i\mathbb{R}$  and every  $m, \ell \in \mathbb{N}_0$  we have
$$
\partial_ {\mathrm {w i r t}} ^ {m} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {\ell} \operatorname {c a r d} (z) \neq 0.
$$
In particular, card is smooth and non-polyharmonic on  $\mathbb{C} \setminus (\mathbb{R} \cup i\mathbb{R})$ . Furthermore, card is continuous on the entire complex plane.
Proof. We start with the first part of the corollary. For the following, let  $z \in \mathbb{C}$  with  $z \notin \mathbb{R} \cup i\mathbb{R}$ , i.e.,  $\operatorname{Re}(z) \neq 0$  and  $\operatorname{Im}(z) \neq 0$ . Using the definition of card, we see
$$
\operatorname {c a r d} (z) = 0 \quad \Longleftrightarrow \quad z = 0 \quad \text {o r} \quad \cos (\triangleleft z) = - 1 \quad \Longleftrightarrow \quad z \in \mathbb {R} _ {\leq 0},
$$
and thus,  $\operatorname{card}(z) \neq 0$  since  $\operatorname{Im}(z) \neq 0$ . In the case  $m \leq \ell \neq 0$  we use Theorem A.5 to get the following chain of implications:
$$
\partial_ {\mathrm {w i r t}} ^ {m} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {\ell} \operatorname {c a r d} (z) = 0 \Leftrightarrow a _ {m, \ell} + b _ {m, \ell} \frac {z ^ {2}}{| z | ^ {2}} = 0 \Leftrightarrow z ^ {2} = - | z | ^ {2} \cdot \frac {a _ {m , \ell}}{b _ {m , \ell}} \Rightarrow z ^ {2} \in \mathbb {R} \Leftrightarrow z \in \mathbb {R} \cup i \mathbb {R}
$$
holds, and thus  $\partial_{\mathrm{wirt}}^m\overline{\partial}_{\mathrm{wirt}}^\ell$  card  $(z)\neq 0$  for  $z\notin \mathbb{R}\cup i\mathbb{R}$
For the case  $m = \ell + 1 = 1$ , note
$$
\partial_ {\mathrm {w i r t}} \operatorname {c a r d} (z) = 0 \Leftrightarrow \frac {1}{8} \cdot \bar {z} + \frac {3}{8} \cdot z = - \frac {| z |}{2} \Rightarrow \frac {3}{8} \operatorname {I m} (z) - \frac {1}{8} \operatorname {I m} (z) = 0 \Leftrightarrow \operatorname {I m} (z) = 0,
$$
and thus  $\partial_{\mathrm{wirt}}\operatorname {card}(z)\neq 0$  since  $z\notin \mathbb{R}$
For  $m = \ell + 1 > 1$ , we see by considering the real- and imaginary parts that
$$
\begin{array}{l} \partial_ {\mathrm {w i r t}} ^ {\ell + 1} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {\ell} \operatorname {c a r d} (z) = 0 \Leftrightarrow a _ {m, \ell} \bar {z} + b _ {m, \ell} z = 0 \\ \begin{array}{c} \operatorname {R e} (z), \operatorname {I m} (z) \neq 0 \\ \Leftrightarrow \end{array} a _ {m, \ell} + b _ {m, \ell} = 0 \text {a n d} - a _ {m, \ell} + b _ {m, \ell} = 0 \\ \Leftrightarrow a _ {m, \ell} = b _ {m, \ell} = 0, \\ \end{array}
$$
and hence  $\partial_{\mathrm{wirt}}^{\ell +1}\overline{\partial}_{\mathrm{wirt}}^{\ell}\operatorname {card}(z)\neq 0$  , since  $a_{m,\ell}\neq 0\neq b_{m,\ell}$  by Theorem A.5.
Therefore, it remains to consider the case  $m \geq \ell + 2$ . But here we easily see