Complex Derivatives of Functions of Several Variables

Introduction

Frequently, in the field of signal processing, the most appropriate interpretation of the data is given by writing it in terms of complex numbers ($\mathbb{C}$). The common operations with different signals can be simplified with the rules of addition and multiplication in $\mathbb{C}$, namely:

$$[a_1,\;a_2] + [b_1,\;b_2] = [a_1+b_1,\;a_2+b_2]$$

and

$$[a_1,\;a_2] \cdot [b_1,\;b_2] = [a_1b_1-a_2b_2,\;a_1b_2+a_2b_1]$$

so that you can simply write $a+b$ and $a\cdot b$ instead, where $a=[a_1,\;a_2]$ and $b=[b_1,\;b_2]$.

Many textbooks on the field of signal processing present the algorithms separately into their real and complex forms, while some don't even present the complex form of some algorithms (e.g. [1]). The conversion of an algorithm from the complex form to the real one is usually trivial, but the other way around usually isn't. That shows the importance of having an adequate definition of derivatives of functions $f:\mathbb{C}^N\to\mathbb{C}^M$ that is the closest possible analogue of the real derivative.

In the literature of complex analysis, it's hard to find a good reference for this problem, because most textbooks don't develop the study of Wirtinger derivatives of non-analytical functions (e.g. [2]), even though there are many applications of such functions, such as complex least squares problems. In Wikipedia, we find the following quote:

Despite their ubiquitous use, it seems that there is no text listing all the properties of Wirtinger derivatives: however, fairly complete references are the short course on multidimensional complex analysis by Andreotti (1976, pp. 3–5), the monograph of Gunning & Rossi (1965, pp. 3–6), and the monograph of Kaup & Kaup (1983, p. 2,4) which are used as general references in this and the following sections.

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https://en.wikipedia.org/w/index.php?title=Wirtinger_derivatives&oldid=692448620&#Formal_definition (Permalink)

In this post, we present rigorous definitions of derivatives of functions $f:\mathbb{C}^N\to\mathbb{C}^M$ that apply to certain non-analytical functions, with countless practical applications. Our definitions, apart from transposition operations, are, in essence, the same as the ones given in [3], with the advantage of simplifying the analogy between the real and the complex cases.

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