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The standard type of regularity in Clifford Calculus is the one introduced by Feuter, namely:

a function is Feuter regular iff it is in the zero set of the Clifford-Dirac

 

operator $D= \partial x_0 + \sum e_i \partial x_i$

 

where the $e_i$ are the orthonormal basis of vectors, bivectors, $\ldots$ of the algebra

Now, onto my question. In the case of the Clifford algebra $Cl(0, 1)$, aka the complex numbers, Feuter regularity at a point plus suitable differentiability in its neighborhood implies the existence and uniqueness of the derivative in all directions.

Is this true in a general Clifford algebra? I suspect not (*).

In that case, what goes wrong in generalizing the standard proof there (by that I mean the usual proof in complex analysis books, such as the one reported by the wiki using the decomposition via the Wirtinger derivatives)? What else can be done to ensure that the derivative in all directions exists and it is unique?

(*) actually, I am pretty certain. In the quaternions, aka $Cl(0, 2)$, the usual naive version of derivative is satisfied only by a trivial class of functions (namely the linear ones), whereas there are tons of Feuter regular functions there.

The standard type of regularity in Clifford Calculus is the one introduced by Feuter, namely:

a function is Feuter regular iff it is in the zero set of the Clifford-Dirac

operator $D= \partial x_0 + \sum e_i \partial x_i$

where the $e_i$ are the orthonormal basis of vectors, bivectors, $\ldots$ of the algebra

Now, onto my question. In the case of the Clifford algebra $Cl(0, 1)$, aka the complex numbers, Feuter regularity at a point plus suitable differentiability in its neighborhood implies the existence and uniqueness of the derivative in all directions.

Is this true in a general Clifford algebra? I suspect not (*).

In that case, what goes wrong in generalizing the standard proof there (by that I mean the usual proof in complex analysis books, such as the one reported by the wiki using the decomposition via the Wirtinger derivatives)? What else can be done to ensure that the derivative in all directions exists and it is unique?

(*) actually, I am pretty certain. In the quaternions, aka $Cl(0, 2)$, the usual naive version of derivative is satisfied only by a trivial class of functions (namely the linear ones), whereas there are tons of Feuter regular functions there.

The standard type of regularity in Clifford Calculus is the one introduced by Feuter, namely:

a function is Feuter regular iff it is in the zero set of the Clifford-Dirac

operator $D= \partial x_0 + \sum e_i \partial x_i$

where the $e_i$ are the orthonormal basis of vectors, bivectors, $\ldots$ of the algebra

Now, onto my question. In the case of the Clifford algebra $Cl(0, 1)$, aka the complex numbers, Feuter regularity at a point plus suitable differentiability in its neighborhood implies the existence and uniqueness of the derivative in all directions.

Is this true in a general Clifford algebra? I suspect not (*).

In that case, what goes wrong in generalizing the standard proof there (by that I mean the usual proof in complex analysis books, such as the one reported by the wiki using the decomposition via the Wirtinger derivatives)? What else can be done to ensure that the derivative in all directions exists and it is unique?

(*) actually, I am pretty certain. In the quaternions, aka $Cl(0, 2)$, the usual naive version of derivatives is satisfied only by a trivial class of functions, whereas there are tons of Feuter regular functions there.

The standard type of regularity in Clifford Calculus is the one introduced by Feuter, namely:

a function is Feuter regular iff it is in the zero set of the Clifford-Dirac

operator $D= \partial x_0 + \sum e_i \partial x_i$

where the $e_i$ are the orthonormal basis of vectors, bivectors, $\ldots$ of the algebra

Now, onto my question. In the case of the Clifford algebra $Cl(0, 1)$, aka the complex numbers, Feuter regularity at a point plus suitable differentiability in its neighborhood implies the existence and uniqueness of the derivative in all directions.

Is this true in a general Clifford algebra? I suspect not (*).

In that case, what goes wrong in generalizing the standard proof there (by that I mean the usual proof in complex analysis books, such as the one reported by the wiki using the decomposition via the Wirtinger derivatives)? What else can be done to ensure that the derivative in all directions exists and it is unique?

(*) actually, I am pretty certain. In the quaternions, aka $Cl(0, 2)$, the usual naive version of derivative is satisfied only by a trivial class of functions (namely the linear ones), whereas there are tons of Feuter regular functions there.

The standard type of regularity in Clifford Calculus is the one introduced by Feuter, namely:

a function is Feuter regular iff it is in the zero set of the Clifford-Dirac

operator $D= \partial x_0 + \sum e_i \partial x_i$

where the $e_i$ are the orthonormal basis of vectors, bivectors, $\ldots$ of the algebra

Now, onto my question. In the case of the Clifford algebra $Cl(0, 1)$, aka the complex numbers, Feuter regularity at a point plus suitable differentiability in its neighborhood implies the existence and uniqueness of the derivative in all directions.

Is this true in a general Clifford algebra? I suspect not.

In that case, what goes wrong in generalizing the standard proof there (by that I mean the usual proof in complex analysis books, such as the one reported by the wiki using the decomposition via the Wirtinger derivatives)? What else can be done to ensure that the derivative in all directions exists and it is unique?

The standard type of regularity in Clifford Calculus is the one introduced by Feuter, namely:

a function is Feuter regular iff it is in the zero set of the Clifford-Dirac

operator $D= \partial x_0 + \sum e_i \partial x_i$

where the $e_i$ are the orthonormal basis of vectors, bivectors, $\ldots$ of the algebra

Now, onto my question. In the case of the Clifford algebra $Cl(0, 1)$, aka the complex numbers, Feuter regularity at a point plus suitable differentiability in its neighborhood implies the existence and uniqueness of the derivative in all directions.

Is this true in a general Clifford algebra? I suspect not (*).

In that case, what goes wrong in generalizing the standard proof there (by that I mean the usual proof in complex analysis books, such as the one reported by the wiki using the decomposition via the Wirtinger derivatives)? What else can be done to ensure that the derivative in all directions exists and it is unique?

(*) actually, I am pretty certain. In the quaternions, aka $Cl(0, 2)$, the usual naive version of derivatives is satisfied only by a trivial class of functions, whereas there are tons of Feuter regular functions there.