Physlib.Mathematics.Calculus.Wirtinger.Coordinate

For a function $f: (\iota \to \mathbb{C}) \to \mathbb{C}$ and a coordinate index $I \in \iota$, the **holomorphic partial Wirtinger derivative** $\partial_I f$ is the function that maps each point $u$ to the holomorphic directional Wirtinger derivative of $f$ in the direction of the $I$-th standard basis vector $e_I$ (where $e_I$ is the vector with $1$ at index $I$ and $0$ elsewhere). At a point $u$, this is defined as: $$\partial_I f(u) = \partial_{e_I} f(u) = \frac{1}{2} \left( d_{e_I} f(u) - i \cdot d_{i e_I} f(u) \right)$$ where $d_v f(u)$ denotes the real Fréchet derivative of $f$ at $u$ in the direction $v$.

For a function $f: \mathbb{C}^\iota \to \mathbb{C}$ and a coordinate index $I \in \iota$, the anti-holomorphic partial Wirtinger derivative $\bar{\partial}_I f$ is the function from $\mathbb{C}^\iota$ to $\mathbb{C}$ defined at each point $u \in \mathbb{C}^\iota$ by: $$\bar{\partial}_I f(u) = \frac{1}{2} (D f(u)(e_I) + i D f(u)(i e_I))$$ where $D f(u)$ is the real Fréchet derivative of $f$ at $u$ and $e_I$ is the $I$-th standard basis vector (represented as `Pi.single I 1`).

For a function $f: \mathbb{C}^\iota \to \mathbb{C}$, a point $u \in \mathbb{C}^\iota$, and a coordinate index $I \in \iota$, the holomorphic partial Wirtinger derivative $\partial_I f(u)$ is expressed in terms of the real Fréchet derivative $Df(u)$ as: $$\partial_I f(u) = \frac{1}{2} \left( Df(u)(e_I) - i \cdot Df(u)(i e_I) \right)$$ where $e_I$ is the $I$-th standard basis vector (the vector with $1$ at index $I$ and $0$ elsewhere) and $i$ is the imaginary unit.

For a function $f: \mathbb{C}^\iota \to \mathbb{C}$ and a point $u \in \mathbb{C}^\iota$, the anti-holomorphic partial Wirtinger derivative with respect to the $I$-th coordinate, denoted $\bar{\partial}_I f(u)$, is given by the formula: $$\bar{\partial}_I f(u) = \frac{1}{2} \left( D_{\mathbb{R}} f(u)(e_I) + i D_{\mathbb{R}} f(u)(i e_I) \right)$$ where $D_{\mathbb{R}} f(u)$ is the real Fréchet derivative of $f$ at $u$, $e_I$ is the $I$-th standard basis vector (the vector with $1$ at index $I$ and $0$ elsewhere), and $i$ is the imaginary unit.

Let $f_1, f_2 : \mathbb{C}^\iota \to \mathbb{C}$ be functions and $u \in \mathbb{C}^\iota$ be a point. If $f_1$ and $f_2$ coincide on a neighborhood of $u$ (i.e., $f_1 = f_2$ eventually at the filter of neighborhoods of $u$), then for any coordinate index $I \in \iota$, their holomorphic partial Wirtinger derivatives at $u$ are equal: $$\partial_I f_1(u) = \partial_I f_2(u)$$

For any complex constant $c \in \mathbb{C}$ and any coordinate index $I \in \iota$, the holomorphic partial Wirtinger derivative of the constant function $f(u) = c$ with respect to the $I$-th coordinate is the zero function, denoted as $\partial_I c = 0$.

Let $f: \mathbb{C}^\iota \to \mathbb{C}$ be a function. For any point $u \in \mathbb{C}^\iota$ and any coordinate index $I \in \iota$, the holomorphic partial Wirtinger derivative of the negation of $f$ at $u$ satisfies: $$\partial_I (-f)(u) = -\partial_I f(u)$$

For any indices $I, J \in \iota$, the holomorphic partial Wirtinger derivative of the $J$-th coordinate projection function $z^J: \mathbb{C}^\iota \to \mathbb{C}$ (defined by $z^J(u) = u_J$) with respect to the $I$-th coordinate direction is the constant function equal to the Kronecker delta $\delta_{IJ}$: $$\partial_I z^J = \delta_{IJ}$$ where $\delta_{IJ} = 1$ if $I = J$ and $\delta_{IJ} = 0$ if $I \neq J$.

Let $f, g: \mathbb{C}^\iota \to \mathbb{C}$ be functions and $u \in \mathbb{C}^\iota$ be a point. If $f$ and $g$ are real-differentiable at $u$, then for any coordinate index $I \in \iota$, the holomorphic partial Wirtinger derivative of their sum satisfies: $$\partial_I (f + g)(u) = \partial_I f(u) + \partial_I g(u)$$ where $\partial_I f$ denotes the holomorphic partial Wirtinger derivative with respect to the $I$-th coordinate.

Let $f: \mathbb{C}^n \to \mathbb{C}$ be a function that is real-differentiable at a point $u \in \mathbb{C}^n$. For any complex scalar $c \in \mathbb{C}$ and any coordinate index $I \in \{1, \dots, n\}$, the holomorphic partial Wirtinger derivative satisfies $$\partial_I (c f)(u) = c \partial_I f(u)$$ where $\partial_I$ denotes the holomorphic partial Wirtinger derivative with respect to the $I$-th coordinate.

Let $f, g: (\iota \to \mathbb{C}) \to \mathbb{C}$ be functions. If $f$ and $g$ are real-differentiable at a point $u \in \iota \to \mathbb{C}$, then for any coordinate index $I \in \iota$, the holomorphic partial Wirtinger derivative of their product at $u$ satisfies the Leibniz rule: $$\partial_I (f \cdot g)(u) = \partial_I f(u) \cdot g(u) + f(u) \cdot \partial_I g(u)$$ where $\partial_I$ denotes the holomorphic partial Wirtinger derivative `dWirtingerCoord`.

Let $\iota$ be an index set and $\mathbb{C}^\iota$ be the space of complex coordinates. For any finite set $s$ and any family of functions $\{F_a\}_{a \in s}$ from $\mathbb{C}^\iota$ to $\mathbb{C}$, if each $F_a$ is real-differentiable at a point $u \in \mathbb{C}^\iota$, then the holomorphic partial Wirtinger derivative $\partial_I$ with respect to the $I$-th coordinate ($I \in \iota$) satisfies the finite-sum rule at $u$: $$\partial_I \left( \sum_{a \in s} F_a \right)(u) = \sum_{a \in s} \partial_I F_a(u)$$

Let $f: \mathbb{C}^\iota \to \mathbb{C}$ be a function and $\iota$ be a finite index set. If $f$ is complex-differentiable (holomorphic) at a point $u \in \mathbb{C}^\iota$, then for any coordinate index $I \in \iota$, the holomorphic partial Wirtinger derivative $\partial_I f(u)$ is equal to the complex Fréchet derivative of $f$ at $u$ evaluated in the direction of the $I$-th standard basis vector $e_I$. That is, $$\partial_I f(u) = D f(u)(e_I)$$ where $e_I$ is the vector in $\mathbb{C}^\iota$ defined by $(e_I)_J = \delta_{IJ}$.

Let $u \in \mathbb{C}^\iota$ be a point and $I \in \iota$ be a coordinate index. Suppose $g: \mathbb{C}^\iota \to \mathbb{C}$ is a function that is complex-differentiable (holomorphic) at $\bar{u}$, where $\bar{u}$ denotes the component-wise complex conjugate of $u$. Then the holomorphic partial Wirtinger derivative $\partial_I$ of the anti-holomorphic function $v \mapsto g(\bar{v})$ evaluated at the point $u$ is zero: $$\partial_I (g \circ \text{star})(u) = 0$$ where $\text{star}$ denotes the complex conjugation operator $v \mapsto \bar{v}$.

Let $\iota$ be an index set and $f_1, f_2 : \mathbb{C}^\iota \to \mathbb{C}$ be functions. If $f_1$ and $f_2$ are equal in a neighborhood of a point $u \in \mathbb{C}^\iota$, then for any coordinate index $I \in \iota$, their anti-holomorphic partial Wirtinger derivatives at $u$ are equal: $$\bar{\partial}_I f_1(u) = \bar{\partial}_I f_2(u)$$

For any complex constant $c \in \mathbb{C}$ and any coordinate index $I \in \iota$, the anti-holomorphic partial Wirtinger derivative $\bar{\partial}_I$ of the constant function $f(u) = c$ on $\mathbb{C}^\iota$ is zero. In notation, $\bar{\partial}_I c = 0$.

Let $f: \mathbb{C}^\iota \to \mathbb{C}$ be a function. For any point $u \in \mathbb{C}^\iota$ and coordinate index $I \in \iota$, the anti-holomorphic partial Wirtinger derivative of the negation of $f$ satisfies $$\bar{\partial}_I (-f)(u) = -\bar{\partial}_I f(u).$$

For any coordinate indices $I, J \in \iota$, the anti-holomorphic partial Wirtinger derivative of the $J$-th coordinate projection function $z^J: \mathbb{C}^\iota \to \mathbb{C}$ (defined by $u \mapsto u_J$) with respect to the $I$-th coordinate is zero: $$\bar{\partial}_I z^J = 0$$

For any indices $I, J \in \iota$, the holomorphic partial Wirtinger derivative of the conjugate coordinate function $\bar{z}^J$ (defined by $u \mapsto \overline{u_J}$) with respect to the $I$-th coordinate is zero: $$\partial_I \bar{z}^J = 0$$

Let $\iota$ be a finite index set. For any indices $I, J \in \iota$, let $\bar{z}^J: \mathbb{C}^\iota \to \mathbb{C}$ be the function that maps a vector $u$ to the complex conjugate of its $J$-th coordinate, $\bar{z}^J(u) = \overline{u_J}$. The anti-holomorphic partial Wirtinger derivative of $\bar{z}^J$ with respect to the $I$-th coordinate is the Kronecker delta $\delta_{IJ}$: $$\bar{\partial}_I \bar{z}^J = \delta_{IJ}$$ where $\delta_{IJ}$ is $1$ if $I = J$ and $0$ otherwise.

Let $\iota$ be an index set and let $f, g : \mathbb{C}^\iota \to \mathbb{C}$ be functions that are real-differentiable at a point $u \in \mathbb{C}^\iota$. For any coordinate index $I \in \iota$, the anti-holomorphic partial Wirtinger derivative $\bar{\partial}_I$ satisfies the following additivity property at $u$: $$\bar{\partial}_I (f + g)(u) = \bar{\partial}_I f(u) + \bar{\partial}_I g(u)$$

Let $\iota$ be a finite indexing set and $f: \mathbb{C}^\iota \to \mathbb{C}$ be a function. Suppose $f$ is real-differentiable at a point $u \in \mathbb{C}^\iota$. For any complex scalar $c \in \mathbb{C}$ and any coordinate index $I \in \iota$, the anti-holomorphic partial Wirtinger derivative $\bar{\partial}_I$ satisfies the following pointwise identity: $$\bar{\partial}_I (c f)(u) = c \bar{\partial}_I f(u)$$

Let $f, g: \mathbb{C}^\iota \to \mathbb{C}$ be functions and $u \in \mathbb{C}^\iota$ be a point. If $f$ and $g$ are real-differentiable at $u$, then for any coordinate index $I \in \iota$, the anti-holomorphic partial Wirtinger derivative $\bar{\partial}_I$ of their product at $u$ satisfies the Leibniz rule: $$\bar{\partial}_I (f \cdot g)(u) = \bar{\partial}_I f(u) \cdot g(u) + f(u) \cdot \bar{\partial}_I g(u)$$ where $\bar{\partial}_I$ denotes the directional Wirtinger derivative along the $I$-th standard basis vector $e_I$.

Let $s$ be a finite set and $\{F_a\}_{a \in s}$ be a collection of functions from $\mathbb{C}^\iota$ to $\mathbb{C}$. For any point $u \in \mathbb{C}^\iota$ and coordinate index $I \in \iota$, if each $F_a$ is real-differentiable at $u$, then the anti-holomorphic partial Wirtinger derivative $\bar{\partial}_I$ of their sum is equal to the sum of their individual anti-holomorphic partial Wirtinger derivatives: $$\bar{\partial}_I \left( \sum_{a \in s} F_a \right)(u) = \sum_{a \in s} \bar{\partial}_I F_a(u).$$

Let $u \in \mathbb{C}^\iota$ be a point and $I \in \iota$ be a coordinate index. If the function $g: \mathbb{C}^\iota \to \mathbb{C}$ is complex-differentiable at the pointwise complex conjugate $\bar{u}$, then the anti-holomorphic partial Wirtinger derivative of the function $v \mapsto g(\bar{v})$ at $u$ is equal to the complex Fréchet derivative of $g$ at $\bar{u}$ evaluated in the direction of the $I$-th standard basis vector $e_I$: $$\bar{\partial}_I (g \circ \text{conj})(u) = D_{\mathbb{C}} g(\bar{u})(e_I)$$ where $\bar{\partial}_I$ is the anti-holomorphic partial Wirtinger derivative, $\text{conj}$ denotes pointwise complex conjugation, and $e_I$ is the basis vector with $1$ at index $I$ and $0$ elsewhere.

Let $f: \mathbb{C}^\iota \to \mathbb{C}$ be a function. If $f$ is complex-differentiable at a point $u \in \mathbb{C}^\iota$, then its anti-holomorphic partial Wirtinger derivative with respect to the $I$-th coordinate is zero at that point: $$\bar{\partial}_I f(u) = 0$$

For any coordinate indices $I, J \in \iota$, let $z^J$ be the $J$-th coordinate projection function $v \mapsto v_J$ and $\bar{z}^J$ be its pointwise complex conjugate $v \mapsto \overline{v_J}$. The holomorphic partial Wirtinger derivative $\partial_I$ of their difference is given by the Kronecker delta $\delta_{IJ}$: $$\partial_I (z^J - \bar{z}^J) = \delta_{IJ}$$ where $\delta_{IJ} = 1$ if $I = J$ and $0$ otherwise.

For any coordinate indices $I, J \in \iota$, the anti-holomorphic partial Wirtinger derivative $\bar{\partial}_I$ of the function mapping a vector $z \in \mathbb{C}^\iota$ to the difference between its $J$-th coordinate $z^J$ and its conjugate $\bar{z}^J$ is equal to the negative of the Kronecker delta: $$\bar{\partial}_I (z^J - \bar{z}^J) = -\delta_{IJ}$$ where $\delta_{IJ} = 1$ if $I = J$ and $0$ otherwise.

Let $f: \mathbb{C}^\iota \to \mathbb{C}$ and $g: \mathbb{C} \to \mathbb{C}$ be functions. If $f$ is real-differentiable at a point $u \in \mathbb{C}^\iota$ and $g$ is real-differentiable at the point $f(u) \in \mathbb{C}$, then for any coordinate index $I \in \iota$, the holomorphic partial Wirtinger derivative of the composition $g \circ f$ at $u$ satisfies: $$\partial_I (g \circ f)(u) = \frac{\partial g}{\partial z}(f(u)) \cdot \partial_I f(u) + \frac{\partial g}{\partial \bar{z}}(f(u)) \cdot \partial_I \bar{f}(u)$$ where $\partial_I$ denotes the holomorphic partial Wirtinger derivative with respect to the $I$-th coordinate, $\frac{\partial g}{\partial z}$ and $\frac{\partial g}{\partial \bar{z}}$ are the one-variable holomorphic and anti-holomorphic Wirtinger derivatives of $g$ respectively, and $\bar{f}$ denotes the pointwise complex conjugate of $f$.

Let $u \in \mathbb{C}^n$ (represented as $\iota \to \mathbb{C}$) be a point and $I \in \iota$ be a coordinate index. If $g: \mathbb{C} \to \mathbb{C}$ is complex-differentiable at $f(u)$ and $f: \mathbb{C}^n \to \mathbb{C}$ is real-differentiable at $u$, then the holomorphic partial Wirtinger derivative of the composition $g \circ f$ at $u$ satisfies: $$\partial_I (g \circ f)(u) = g'(f(u)) \cdot \partial_I f(u)$$ where $\partial_I$ denotes the holomorphic partial Wirtinger derivative and $g'$ denotes the complex derivative of $g$.

Let $f: \mathbb{C}^\iota \to \mathbb{C}$ be a function that is real-differentiable at a point $u \in \mathbb{C}^\iota$. For any coordinate index $I \in \iota$, the holomorphic partial Wirtinger derivative $\partial_I$ of the complex conjugate function $\bar{f}$ (defined by $\bar{f}(v) = \overline{f(v)}$) evaluated at $u$ is equal to the complex conjugate of the anti-holomorphic partial Wirtinger derivative $\bar{\partial}_I f$ evaluated at $u$: $$\partial_I \bar{f}(u) = \overline{\bar{\partial}_I f(u)}$$

Let $f: \mathbb{C}^\iota \to \mathbb{C}$ and $g: \mathbb{C} \to \mathbb{C}$ be functions. If $f$ is real-differentiable at a point $u \in \mathbb{C}^\iota$ and $g$ is real-differentiable at $f(u)$, then for any coordinate index $I \in \iota$, the anti-holomorphic partial Wirtinger derivative of the composition $g \circ f$ at $u$ satisfies the chain rule: $$\bar{\partial}_I (g \circ f)(u) = \partial g(f(u)) \bar{\partial}_I f(u) + \bar{\partial} g(f(u)) \bar{\partial}_I \bar{f}(u)$$ where $\bar{\partial}_I$ is the anti-holomorphic partial Wirtinger derivative with respect to the $I$-th coordinate, $\partial g$ and $\bar{\partial} g$ are the holomorphic and anti-holomorphic directional Wirtinger derivatives of $g$ along the direction $1 \in \mathbb{C}$, and $\bar{f}$ denotes the pointwise complex conjugate of $f$.

Let $f: \mathbb{C}^\iota \to \mathbb{C}$ be a function that is real-differentiable at $u \in \mathbb{C}^\iota$, and let $g: \mathbb{C} \to \mathbb{C}$ be a holomorphic function (complex-differentiable) at $f(u)$. For any coordinate index $I \in \iota$, the anti-holomorphic partial Wirtinger derivative $\bar{\partial}_I$ of the composition $g \circ f$ at $u$ satisfies the single-term chain rule: $$\bar{\partial}_I (g \circ f)(u) = g'(f(u)) \cdot \bar{\partial}_I f(u)$$ where $g'$ denotes the standard complex derivative of $g$. This formula arises because the anti-holomorphic component of the derivative of the holomorphic function $g$ (often denoted $\frac{\partial g}{\partial \bar{z}}$) vanishes, leaving only the holomorphic contribution.

Let $f: \mathbb{C}^\iota \to \mathbb{C}$ be a function and $I \in \iota$ be a coordinate index. If $f$ is real-differentiable at $u \in \mathbb{C}^\iota$, then the anti-holomorphic partial Wirtinger derivative of the complex conjugate function $\bar{f}$ at $u$ is the complex conjugate of the holomorphic partial Wirtinger derivative of $f$ at $u$: $$\bar{\partial}_I \bar{f}(u) = \overline{\partial_I f(u)}$$ where $\partial_I$ and $\bar{\partial}_I$ denote the holomorphic and anti-holomorphic partial Wirtinger derivatives with respect to the $I$-th coordinate, and the overline denotes complex conjugation.

Let $f: (\iota \to \mathbb{C}) \to \mathbb{C}$ be a function. If $f$ is twice continuously differentiable (of class $C^2$) at a point $u$, then for any coordinate index $I \in \iota$, the holomorphic partial Wirtinger derivative $\partial_I f$ is real-differentiable at $u$.

Let $\iota$ be an index set and $f : \mathbb{C}^\iota \to \mathbb{C}$ be a function. If $f$ is twice continuously differentiable ($C^2$) at a point $u \in \mathbb{C}^\iota$ with respect to the real numbers, then for any coordinate index $J \in \iota$, the anti-holomorphic partial Wirtinger derivative $\bar{\partial}_J f$ is real-differentiable at $u$.

Let $f: \mathbb{C}^\iota \to \mathbb{C}$ be a function and $u \in \mathbb{C}^\iota$ be a point. If $f$ is twice continuously differentiable ($C^2$) at $u$ over the real numbers $\mathbb{R}$, then for any coordinate indices $I, J \in \iota$, the holomorphic partial Wirtinger derivative $\partial_I$ and the anti-holomorphic partial Wirtinger derivative $\bar{\partial}_J$ commute at $u$: $$\partial_I (\bar{\partial}_J f)(u) = \bar{\partial}_J (\partial_I f)(u)$$ Here, $\partial_I f$ and $\bar{\partial}_J f$ represent the partial Wirtinger derivatives along the $I$-th and $J$-th standard basis vectors of $\mathbb{C}^\iota$, respectively.