Function of a differentiable complex variable to the real direction

This article constitutes primarily an introduction to the article on the equations of Cauchy-Riemann which it makes it possible to approach directly. It defines, for the functions of a complex variable and with complex values: the derivative partial (compared to $\backslash \; X,\; y$ or $\backslash \; Z,\; \backslash \; bar\; \{Z\}$) and differentiability with the real direction.

One considers a function $\backslash \; F:\; U\; \backslash \; to\; \backslash \; mathbb$ of a complex variable, definite on a open subset U of the plane complex $\backslash \; mathbb$. The following notations will be used:

• the variable complexes $\backslash \; z$ will be noted $\backslash \; X\; +\; I\; \backslash ,\; y$, where X , is real there
• the parts real and imaginary of $\backslash \; F\; (Z)\; =\; F\; (X\; +\; I\; \backslash ,\; there)$ will be respectively noted $\backslash \; P\; (X,\; there)$ and $\backslash \; Q\; (X,\; there)$, i.e.: $\backslash \; F\; (Z)\; =\; P\; (X,\; there)\; +\; I\; \backslash ,\; Q\; (X,\; there)$, where $\backslash \; P,\; \backslash ,\; Q$ is two real functions of two real variables.

Derivative partial of a function of a complex variable

Derivative partial compared to X and there

Definition: that is to say $\backslash \; z\_0\; =\; x\_0\; +\; I\; \backslash ,\; y\_0\; \backslash \; in\; U$, where $\backslash \; x\_0,\; \backslash ,\; y\_0$ is real.

• one says that F admits a partial derivative (of order 1) at the point $\backslash \; z\_0$ compared to the variable X , noted $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; (z\_0)$ if limit (finished) $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; (z\_0)\; =\; \backslash \; lim\_\; \{U\; \backslash \; to\; 0,\; \backslash ,\; U\; \backslash ,\; \backslash \; in\; \backslash ,\; \backslash \; mathbb^*\}\; \backslash \; frac\; \{F\; (z\_0+u)\; -\; F\; (z\_0)\}\{U\}$ exists
• one says that F admits a partial derivative (of order 1) at the point $\backslash \; z\_0$ compared to the variable there , noted $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\}\; (z\_0)$ if limit (finished) $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\}\; (z\_0)\; =\; \backslash \; lim\_\; \{v\; \backslash \; to\; 0,\; \backslash ,\; v\; \backslash ,\; \backslash \; in\; \backslash ,\; \backslash \; mathbb^*\}\; \backslash \; frac\; \{F\; (z\_0+i\; \backslash ,\; v)\; -\; F\; (z\_0)\}\{v\}$ exists

Property :

• the partial derivative $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; (z\_0)$ exists if and only if the derivative partial $\backslash \; frac\; \{\backslash \; partial\; P\}\; \{\backslash \; partial\; X\}\; (x\_0,\; y\_0)$, $\backslash \; frac\; \{\backslash \; partial\; Q\}\; \{\backslash \; partial\; X\}\; (x\_0,\; y\_0)$ exists, and then $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; (z\_0)\; =\; \backslash \; frac\; \{\backslash \; partial\; P\}\; \{\backslash \; partial\; X\}\; (x\_0,\; y\_0)\; +\; I\; \backslash ,\; \backslash \; frac\; \{\backslash \; partial\; Q\}\; \{\backslash partial\; X\}\; (x\_0,\; y\_0)$
• the partial derivative $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\}\; (z\_0)$ exists there if and only if the derivative partial $\backslash \; frac\; \{\backslash \; partial\; P\}\; \{\backslash \; partial\; there\}\; (x\_0,\; y\_0)$, $\backslash \; frac\; \{\backslash \; partial\; Q\}\; \{\backslash \; partial\}\; (x\_0,\; y\_0)$ exists there, and then $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\}\; (z\_0)\; =\; \backslash \; frac\; \{\backslash \; partial\; P\}\; \{\backslash \; partial\; there\}\; (x\_0,\; y\_0)\; +\; I\; \backslash ,\; \backslash \; frac\; \{\backslash \; partial\; Q\}\; \{\backslash \; partial\; there\}\; (x\_0,\; y\_0)$

Derivative partial of a higher nature :

• if, for example, $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; (z\_0)$ exists in any point $\backslash \; z\_0\; \backslash \; in\; U$, one defines the function $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}:\; U\; \backslash \; to\; \backslash \; mathbb,\; \backslash ,\; Z\; \backslash \; mapsto\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; (Z)$
• if, moreover, the function $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}$ admits a derivative partial of order 1 at the point $\backslash \; z\_0$ compared to the variable X , one notes it $\backslash \; frac\; \{\backslash \; partial^2\; F\}\; \{\backslash \; partial\; x^2\}\; (z\_0)$: $\backslash \; frac\; \{\backslash \; partial^2\; F\}\; \{\backslash \; partial\; x^2\}\; (z\_0)\; =\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; X\}\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; \backslash \; right)\; (z\_0)$. In a similar way, if $\backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; there\}\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; \backslash \; right)\; (z\_0)$ exists, it is noted $\backslash \; frac\; \{\backslash \; partial^2\; F\}\; \{\backslash \; partial\; there\; \backslash \; partial\; X\}\; (z\_0)$, etc

Derivative partial compared to $\backslash \; z$ and $\backslash \; \backslash \; bar\; \{Z\}$

Definition: it is supposed that F admits derivative partial of order 1 compared to X and there at the point $\backslash \; z\_0$. Then, one defines:

• $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; Z\}\; (z\_0)\; =\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash ,\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; (z\_0)\; -\; I\; \backslash ,\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\}\; (z\_0)\; \backslash \; right)$
• $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; \backslash \; bar\; \{Z\}\}\; (z\_0)\; =\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash ,\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; (z\_0)\; +\; I\; \backslash ,\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\}\; (z\_0)\; \backslash \; right)$

Property : by preserving the preceding assumptions

• $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; (z\_0)\; =\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; Z\}\; (z\_0)\; +\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; \backslash \; bar\; \{Z\}\}\; (z\_0)$
• $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\}\; (z\_0)\; =\; I\; \backslash ,\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; Z\}\; (z\_0)\; -\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; \backslash \; bar\; \{Z\}\}\; (z\_0)\backslash \; right)$

Differentiability with the real direction of the functions of a complex variable

It is said that a function of a complex variable is differentiable with the real direction, or $\backslash \; mathbb\; \{R\}$-différentiable in a point if one can locally approach it (in the vicinity of this point) by the sum of a constant and a function $\backslash \; mathbb\; \{R\}$-linéaire; the latter is then single, and is called differential of the function at the point considered.

More precisely, that wants to say that $\backslash \; f$, as a function of two real variables, admits in the vicinity of the point considered a Développement limited of order 1, of which the differential is the linear part.

• Definition: it is said that a $L\; application:\; \backslash \; mathbb\; \{C\}\; \backslash \; to\; \backslash \; mathbb\; \{C\}$ is $\backslash \; mathbb\; \{R\}$-linéaire if: $\backslash \; forall\; \backslash ,\; \backslash \; alpha\; \backslash \; in\; \backslash \; mathbb\; \{R\},\; \backslash \; forall\; \backslash ,\; \backslash \; beta\; \backslash \; in\; \backslash \; mathbb\; \{R\},\; \backslash \; forall\; \backslash ,\; Z\; \backslash \; in\; \backslash \; mathbb\; \{C\},\; \backslash \; forall\; \backslash ,\; W\; \backslash \; in\; \backslash \; mathbb\; \{C\},\; L\; (\backslash \; alpha\; \backslash ,\; Z\; +\; \backslash \; beta\; \backslash ,\; W)\; =\; \backslash \; alpha\; L\; (Z)\; +\; \backslash \; beta\; L\; (W)$.
• (then: $\backslash \; forall\; U\; \backslash \; in\; \backslash \; mathbb\; \{R\},\; \backslash ,\; \backslash \; forall\; v\; \backslash \; in\; \backslash \; mathbb\; \{R\},\; \backslash ,\; L\; (U\; +\; I\; \backslash ,\; v)\; =\; U\; L\; (1)\; +\; v\; L\; (I)$)

• Definition: it is said that the function $\backslash \; F:\; U\; \backslash \; to\; \backslash \; mathbb$ is $\backslash \; mathbb\; \{R\}$-différentiable in a point $z\_0\; \backslash \; in\; U$ if there exists an application $\backslash \; mathbb\; \{R\}$-linéaire $L:\; \backslash \; mathbb\; \{C\}\; \backslash \; to\; \backslash \; mathbb\; \{C\}$ and a function $\backslash \; \backslash \; epsilon$ of a variable complexes such as $\backslash \; epsilon\; (H)\; \backslash \; to\; 0$ when $h\; \backslash \; to\; 0$ and $f\; (z\_0+h)\; =\; F\; (z\_0)\; +\; L\; (H)\; +\; H\; \backslash ,\; \backslash \; epsilon\; (H)$ (by supposing that , where R is the ray of an open ball such as $\backslash \; B\; (z\_0,\; \backslash ,\; R)\; \backslash \; subset\; U$).
• When it exists, the application L is single (this results from the following property); it is called $\backslash \; mathbb\; \{R\}$-différentielle or differential of $\backslash \; f$ in $\backslash \; z\_0$ and one notes it usually $\backslash \; df\; (z\_0)$.
• One says that $\backslash \; f$ is $\backslash \; mathbb\; \{R\}$-différentiable on U if it is $\backslash \; mathbb\; \{R\}$-différentiable in any point of U .

• Property : if $\backslash \; f$ is $\backslash \; mathbb\; \{R\}$-différentiable in a point $\backslash \; z\_0\; \backslash \; in\; U$, then
• it is continuous in $\backslash \; z\_0$
• it admits derivative partial of order 1 in $\backslash \; z\_0$, and $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; (z\_0)\; =\; L\; (1)\; =\; df\; (z\_0)\; (1)$, $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\}\; (z\_0)\; =\; L\; (I)\; =\; df\; (z\_0)\; (I)$.

demonstration:

• continuity: $f\; (z\_0+h)\; =\; F\; (z\_0)\; +\; L\; (H)\; +\; H\; \backslash ,\; \backslash \; epsilon\; (H)\; \backslash \; to\; F\; (z\_0)$ when $h\; \backslash \; to\; 0$ because $L\; (H)\; \backslash \; to\; 0$ (the -différentielle $\backslash \; mathbb\; \{R\}$ L is a Endomorphisme of a vector Space of finished size, therefore it is continuous) and $H\; \backslash ,\; \backslash \; epsilon\; (H)\; \backslash \; to\; 0$.
• existence and expression of the derivative partial of order 1:
• for all U real such as , $f\; (z\_0+u)\; =\; F\; (z\_0)\; +\; L\; (U)\; +\; U\; \backslash ,\; \backslash \; epsilon\; (U)\; =\; F\; (z\_0)\; +\; U\; L\; (1)\; +\; U\; \backslash ,\; \backslash \; epsilon\; (U)$; therefore, if $u\; \backslash \; neq\; 0$, $\backslash \; frac\; \{F\; (z\_0+u)\; -\; F\; (z\_0)\}\{U\}\; =\; L\; (1)\; +\; \backslash \; epsilon\; (U)\; \backslash \; to\; L\; (1)$ when $u\; \backslash \; to\; 0$: this proves the existence of the derivative partial of the function $\backslash \; f$ in $\backslash \; z\_0$ compared to $\backslash \; x$, and the relation $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; (z\_0)\; =\; L\; (1)$
• for all v real such as , $f\; (z\_0+i\; \backslash ,\; v)\; =\; F\; (z\_0)\; +\; L\; (I\; \backslash ,\; v)\; +\; I\; \backslash ,\; v\; \backslash ,\; \backslash \; epsilon\; (I\; \backslash ,\; v)\; =\; F\; (z\_0)\; +\; v\; L\; (I)\; +\; I\; \backslash ,\; v\; \backslash ,\; \backslash \; epsilon\; (I\; \backslash ,\; v)$; therefore, if $v\; \backslash \; neq\; 0$, $\backslash \; frac\; \{F\; (z\_0+i\; \backslash ,\; v)\; -\; F\; (z\_0)\}\{v\}\; =\; L\; (I)\; +\; I\; \backslash ,\; \backslash \; epsilon\; (I\; \backslash ,\; v)\; \backslash \; to\; L\; (I)$ when $v\; \backslash \; to\; 0$: this proves the existence of the derivative partial of the function $\backslash \; f$ in $\backslash \; z\_0$ compared to $\backslash \; y$, and the relation $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\}\; (z\_0)\; =\; L\; (I)$.

• Theorem : a sufficient condition (nonnecessary) of $\backslash \; mathbb\; \{R\}$-differentiability in a point, or on open.
• Is $\backslash \; z\_0\; \backslash \; in\; U$. If $\backslash \; f$ admits derivative partial of order 1 compared to X and there (or with $\backslash \; z$ and $\backslash \; \backslash \; bar\; \{Z\}$) in any point of a Voisinage of $\backslash \; z\_0$, and if $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}$, $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\}$ (or $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; Z\}$, $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; \backslash \; bar\; \{Z\}\}$) are there continuous in $\backslash \; z\_0$, then $\backslash \; f$ is $\backslash \; mathbb\; \{R\}$-différentiable in $\backslash \; z\_0$
• In particular, if $\backslash \; f$ admits derivative partial of order 1 compared to X and there (or with $\backslash \; z$ and $\backslash \; \backslash \; bar\; \{Z\}$) definite and continuous in any point of open the U , the function $\backslash \; f$ is $\backslash \; mathbb\; \{R\}$-différentiable on U . In this case, one says that $\backslash \; f$ is $\backslash \; mathbb\; \{R\}$-continûment differentiable on U , or of class $\backslash \; C^1$ on U .

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