Is Complex Differentiation Defined for Linear Transformations?

[r731]

Let z = [a b]^T be in the 2-dimensional vector space over real numbers, and T a linear transformation on the vector space.

Consider

$$\lim_{z'\rightarrow \mathbf{0}} \frac{T(z+z')-T(z)}{z'}$$

I argue this could be an alternative definition for complex derivative.

To illustrate this, z as a complex number $a+bi$ is a vector [a b]^T and z' is a tiny vector parallel to z. Taking the limit, z+z' approaches z and the limit approaches the rate of change of T at z.

<Moderator's note: please use the proper wrappers, ## ## and $$ $$, for LaTeX.>

 

[S.G. Janssens]

Is there a particular reason why you would need an alternative definition for the derivative of a complex function?

In your formula, it is not clear what you mean when you divide by $z'$ when $z'$ is regarded as an element of $\mathbb{R}^2$. For multivariable functions, the difference quotient definition of the total derivative is not useful.

In any case, any reasonable definition of the derivative of a linear transformation of $\mathbb{R}^2$ at a point will reproduce that linear transformation itself.

Incidentally, this is useful to find out how to embed $\LaTeX$ in your posts here.

 

[haushofer]

Let's calculate an explicit example with your new derivative.

 

[r731]

The reason is I want to get rid of the difficulties with the complex log multi-function.

An example: The derivative of $z^2 = (a + bi)^2$ equals $2z$. Let $z := [a\;\;\;b]^T$ and $T([a \;\;\; b]^T) := [a^2 \;\;\; b^2]^T$.

$\lim_{\delta z\rightarrow \mathbf{0}} \frac{T(z+\delta z)-T(z)}{\delta z}= \lim_{\delta z\rightarrow \mathbf{0}} \frac{[(a+\delta a)^2\;\; (b+\delta b)^2]^T}{[\delta a \;\;\; \delta b]^T}$

Simplifying and factoring out yields

$\lim_{\delta z\rightarrow \mathbf{0}} \frac{[2a\delta a\;\; 2b\delta b]^T}{[\delta a \;\;\; \delta b]^T} = 2[a\;\;\; b]^T.$

$2[a\;\;\; b]^2$ corresponds to $2z$.

 

[mfb]

Your example is not a linear transformation, violating the conditions of post 1. I don't understand why you wanted this to limit to linear transformations anyway, but you did.

The fraction of a vector and a vector is generally undefined.