# cv.complex variables – Bicomplex Conjugate Derivative

I have decided to first ask my question and second provide a list of steps I have already considered.

Thank you in advance.

Question: After reading the following paper on Bicomplex numbers, I am wondering if the derivative $$frac{partial F}{partial Z^{dagger}}$$ could be defined for a function $$F: mathbb{BC} to mathbb{BC}$$ as a means to generalize the notion of quasiconformality. Unfortunately, the fact that $$bar{z}$$ is not holomorphic appears to preclude the usage of the multivariate chain rule to calculate the derivative. I have written the four components of a bicomplex number as $$x$$, $$y$$, $$k$$, and $$l$$ as a 4-tuple in $$mathbb{R}^4$$. Could they be expressed in terms of $$Z$$ and $$Z^{dagger}$$ using an intermediate formula relying on some other notion than the complex conjugate? I wish that the dot product was defined for the underlying four tuples as one could trivially assert $$left<1, 0, 0, 0right> cdot Z = x$$.

Existing Thoughts: The notion of complex quasiconformality is defined as follows.

$${mu}$$ is a complex Lebesgue measure s.t. $$sup{leftlbrace{mu}{left({x}right)}rightrbrace}<{1}$$.

$$displaystyle frac{{partial{f}}}{{partialoverline{{{z}}}}}={mu}{left({z}right)}frac{{partial{f}}}{{partial{z}}}nonumber$$

$${mu}$$-quasiconformality may be ensured if $$frac{{partial{f}}}{{partial{bar{z}}}} in the strict sense. The existence of a measure $${mu}$$ is then guaranteed as no statement of continuity is attested. The above is justified using the operator $$frac{{partial}}{{partialoverline{{{z}}}}}=frac{{1}}{{2}}{left(frac{{partial}}{{partial{x}}}+{i}frac{{partial}}{{partial{y}}}right)}$$, $${x}=frac{{{z}+overline{{{z}}}}}{{2}}$$, and $${y}=frac{{{z}-overline{{{z}}}}}{{{2}{i}}}$$. By the multivariate chain rule, we define $$frac{{partial{f}}}{{partialoverline{{{z}}}}}$$ as follows.

$$displaystyle frac{{partial{f}}}{{partialoverline{{{z}}}}}=frac{{partial{f}}}{{partial{x}}}frac{{partial{x}}}{{partialoverline{{{z}}}}}+frac{{partial{f}}}{{partial{y}}}frac{{partial{y}}}{{partialoverline{{{z}}}}}=frac{{1}}{{2}}{left(frac{{partial{f}}}{{partial{x}}}+{i}frac{{partial{f}}}{{partial{y}}}right)}=frac{{1}}{{2}}{left({u}_{{x}}+{i}_{{x}}+{i}{u}_{{y}}-{v}_{{y}}right)}nonumber$$

Note that this equation will equal $$0$$ provided that $${u}_{{x}}-{v}_{{y}}={0}$$ and $${v}_{{x}}+{u}_{{y}}={0}$$, which guarantee the Cauchy-Riemann equations.

Naturally, we ask if $${mu}$$-quasiconformality can be generalized to a bicomplex Lebesgue measure $${mu}:{mathbb{{{B}}}}{mathbb{{{C}}}}to{mathbb{{{R}}}}$$ with $$sup{leftlbrace{mu}{left({Z}right)}rightrbrace}<{1}$$ in the following sense. $${F}:{mathbb{{{B}}}}{mathbb{{{C}}}}to{mathbb{{{B}}}}{mathbb{{{C}}}}$$ is a holomorphic function and $$frac{{partial}}{{partial{Z}^{dagger}}}$$ is defined by the same algebra as $$frac{{partial}}{{partialoverline{{{z}}}}}$$.

$$displaystyle frac{{partial{F}}}{{partial{Z}^{dagger}}}={mu}{left({Z}right)}frac{{partial{F}}}{{partial{Z}}}nonumber$$

We consider if such bicomplex quasiconformality guarantees complex quasiconformality on each of the complex components. A notion of a conformal mapping does not exist on $${mathbb{{{R}}}}$$, so no analogue of this property exists on $${mathbb{{{C}}}}$$.

Before proceeding, certain terminology will prove to be advantageous. $${F}{left({z}_{{1}},{z}_{{2}}right)}:{mathbb{{{B}}}}{mathbb{{{C}}}}to{mathbb{{{B}}}}{mathbb{{{C}}}}$$ can also be written as $${F}{left({x},{y},{k},{l}right)}={left({U}+{i}{V}+{j}{K}+{i}{j}{L}right)}:{mathbb{{{R}}}}^{{{4}}}to{mathbb{BC}}$$. Our notion of the components of a complex number from linear combinations of the conjugate generalizes to $${mathbb{{{B}}}}{mathbb{{{C}}}}$$ as $${z}_{{{1}}}=frac{{{Z}^{dagger}+{Z}}}{{2}}$$, $${z}_{{{2}}}=frac{{-{Z}^{dagger}+{Z}}}{{{2}{j}}}$$, $${x}=frac{{{z}_{{{1}}}+overline{{{z}_{{{1}}}}}}}{{2}}$$, $${y}=frac{{{z}_{{{1}}}-overline{{{z}_{{{1}}}}}}}{{{2}{i}}}$$, $${k}=frac{{{z}_{{{2}}}+overline{{{z}_{{{2}}}}}}}{{2}}$$, and $${l}=frac{{{z}_{{{2}}}-overline{{{z}_{{{2}}}}}}}{{{2}{i}}}$$.

Where Things Break Down: The fact that the complex conjugate $$overline{{{z}}}$$ is nowhere differentiable presents a particular difficulty when attempting to apply the multivariate chain rule.

$$displaystyle frac{{partial{F}}}{{partial{Z}^{dagger}}}=frac{{partial{F}}}{{partial{x}}}{left(frac{{partial{x}}}{{partial{z}_{{1}}}}frac{{partial{z}_{{1}}}}{{partial{Z}^{dagger}}}+frac{{partial{x}}}{{partialoverline{{{z}_{{1}}}}}}frac{{partialoverline{{{z}_{{1}}}}}}{{partial{Z}^{dagger}}}right)}+frac{{partial{F}}}{{partial{y}}}{left(frac{{partial{y}}}{{partial{z}_{{1}}}}frac{{partial{z}_{{1}}}}{{partial{Z}^{dagger}}}+frac{{partial{y}}}{{partialoverline{{{z}_{{1}}}}}}frac{{partialoverline{{{z}_{{1}}}}}}{{partial{Z}^{dagger}}}right)}+ \ frac{{partial{F}}}{{partial{k}}}{left(frac{{partial{k}}}{{partial{z}_{{2}}}}frac{{partial{z}_{{2}}}}{{partial{Z}^{dagger}}}+ frac{{partial{k}}}{{partialoverline{{{z}_{{2}}}}}}frac{{partialoverline{{{z}_{{2}}}}}}{{partial{Z}^{dagger}}}right)}+frac{{partial{F}}}{{partial{l}}}{left(frac{{partial{l}}}{{partial{z}_{{2}}}}frac{{partial{z}_{{2}}}}{{partial{Z}^{dagger}}}+frac{{partial{l}}}{{partialoverline{{{z}_{{2}}}}}}frac{{partialoverline{{{z}_{{2}}}}}}{{partial{Z}^{dagger}}}right)}$$

For example, I cannot possibly see how to define the derivative of $$bar{{z}_{{{1}}}}=overline{frac{{{Z}^{dagger}+{Z}}}{{2}}}$$ considering that the complex conjugate is neither analytic nor holomorphic. (On $$mathbb{BC}$$, holomorphicity and analyticity are not logically equivalent.) Does anyone have an alternate idea to define quasiconformality for $$mathbb{BC}$$ in a meaningful fashion? Could the relationship to $$bar{z}$$ be avoided?