Physlib.Mathematics.Calculus.Wirtinger.Basic

Given a function $f: V \to \mathbb{C}$ defined on a complex vector space $V$, a complex weight $c \in \mathbb{C}$, and two directions $b_1, b_2 \in V$, this definition is the function mapping a base point $p \in V$ to the weighted combination of real Fréchet derivatives: $$p \mapsto \frac{1}{2} (d_{b_1} f(p) + c \cdot d_{b_2} f(p))$$ where $d_v f(p)$ denotes the real Fréchet derivative of $f$ at the point $p$ in the direction $v$.

Given a function $f: V \to \mathbb{C}$ defined on a complex vector space $V$, a direction vector $v \in V$, and a base point $u \in V$, the **holomorphic directional Wirtinger derivative** $\partial_v f(u)$ is defined as: $$\partial_v f(u) = \frac{1}{2} \left( d_v f(u) - i \cdot d_{i \cdot v} f(u) \right)$$ where $d_v f(u)$ denotes the real Fréchet derivative of $f$ at the point $u$ in the direction $v$, and $i \cdot v$ is the direction $v$ transformed by the complex structure of $V$. This corresponds to the weighted directional derivative field with weight $c = -i$ and directions $b_1 = v, b_2 = i \cdot v$.

For a function $f: V \to \mathbb{C}$ defined on a complex vector space $V$, the anti-holomorphic directional Wirtinger derivative at a base point $u \in V$ along a direction $v \in V$ is defined as: $$\bar{\partial}_v f(u) = \frac{1}{2} (d_v f(u) + i d_{iv} f(u))$$ where $d_v f(u)$ (also denoted as $fderiv \mathbb{R} f u v$) represents the real Fréchet derivative of $f$ at the point $u$ in the direction $v$.

For a function $g : V \to \mathbb{C}$ defined on a complex vector space $V$, the holomorphic directional Wirtinger derivative $\partial_v g$ at a base point $u \in V$ along a direction $v \in V$ is given by: $$\partial_v g(u) = \frac{1}{2} (d_v g(u) - i \cdot d_{iv} g(u))$$ where $d_v g(u)$ denotes the real Fréchet derivative of $g$ at $u$ in the direction $v$, and $i v$ is the direction $v$ transformed by the complex structure of $V$.

For a function $g: V \to \mathbb{C}$ defined on a complex vector space $V$, the anti-holomorphic directional Wirtinger derivative $\bar{\partial}_v g$ at a base point $u \in V$ along a direction $v \in V$ is given by the formula: $$\bar{\partial}_v g(u) = \frac{1}{2} \left( d_v g(u) + i d_{iv} g(u) \right)$$ where $d_v g(u)$ (the real Fréchet derivative $fderiv \mathbb{R} g u v$) denotes the derivative of $g$ at the point $u$ in the direction $v$, and $iv$ is the direction $v$ transformed by the complex structure of $V$.

Let $V$ be a complex vector space. For any constant $c \in \mathbb{C}$, the holomorphic directional Wirtinger derivative $\partial_v$ of the constant function $p \mapsto c$ at a point $u \in V$ along a direction $v \in V$ is zero: $$\partial_v c = 0$$ This property follows from the fact that the real Fréchet derivative of a constant function is zero in any direction.

For any constant $c \in \mathbb{C}$ and any vectors $v, u \in V$ in a complex vector space $V$, the anti-holomorphic directional Wirtinger derivative of the constant function $f(p) = c$ at the base point $u$ along the direction $v$ is zero: $$\bar{\partial}_v c = 0$$

Let $V$ be a complex vector space. For any function $g: V \to \mathbb{C}$, any direction vector $v \in V$, and any base point $u \in V$, the holomorphic directional Wirtinger derivative $\partial_v$ of the negated function $-g$ is the negation of the derivative of $g$: $$\partial_v (-g)(u) = -\partial_v g(u)$$ This property holds without requiring $g$ to be differentiable, as it inherits the linearity of the underlying real Fréchet derivative.

For any function $g: V \to \mathbb{C}$ and any vectors $v, u \in V$, the anti-holomorphic directional Wirtinger derivative of the negated function $-g$ at the base point $u$ along the direction $v$ satisfies $$\bar{\partial}_v (-g)(u) = -\bar{\partial}_v g(u)$$

Let $V$ be a complex vector space, $u, v \in V$ be a base point and a direction vector, and $g, h: V \to \mathbb{C}$ be functions. If $g$ and $h$ are real-differentiable at $u$, then the holomorphic directional Wirtinger derivative of their sum satisfies: $$\partial_v (g + h)(u) = \partial_v g(u) + \partial_v h(u)$$

Let $V$ be a complex vector space, and let $g, h: V \to \mathbb{C}$ be functions that are real-differentiable at a base point $u \in V$. For any direction $v \in V$, the anti-holomorphic directional Wirtinger derivative $\bar{\partial}_v$ is additive: $$\bar{\partial}_v (g + h)(u) = \bar{\partial}_v g(u) + \bar{\partial}_v h(u)$$ where $(g+h)$ denotes the pointwise sum of the functions.

Let $V$ be a complex vector space, $u, v \in V$ be a base point and a direction vector, and $c \in \mathbb{C}$ be a complex scalar. If $g: V \to \mathbb{C}$ is a function that is real-differentiable at $u$, then the holomorphic directional Wirtinger derivative of the scalar product $c \cdot g$ satisfies: $$\partial_v (c \cdot g)(u) = c \cdot \partial_v g(u)$$

Let $V$ be a complex vector space and $g: V \to \mathbb{C}$ be a function. If $g$ is real-differentiable at a point $u \in V$, then for any complex scalar $c \in \mathbb{C}$ and any direction $v \in V$, the anti-holomorphic directional Wirtinger derivative $\bar{\partial}_v$ satisfies: $$\bar{\partial}_v (c \cdot g)(u) = c \cdot \bar{\partial}_v g(u)$$ where $(c \cdot g)$ is the function defined by $(c \cdot g)(p) = c \cdot g(p)$.

Let $V$ be a complex vector space and $u, v \in V$ be a base point and a direction vector, respectively. If the functions $g, h : V \to \mathbb{C}$ are real-differentiable at $u$, then the holomorphic directional Wirtinger derivative of their product $g \cdot h$ satisfies the Leibniz rule: $$\partial_v (g \cdot h)(u) = \partial_v g(u) \cdot h(u) + g(u) \cdot \partial_v h(u)$$ where $\partial_v$ denotes the holomorphic directional Wirtinger derivative `dWirtingerDir`.

Let $V$ be a complex vector space and $g, h: V \to \mathbb{C}$ be two functions that are real-differentiable at a point $u \in V$. For any direction $v \in V$, the anti-holomorphic directional Wirtinger derivative $\bar{\partial}_v$ of the product $g \cdot h$ at $u$ satisfies the Leibniz rule: $$\bar{\partial}_v(g \cdot h)(u) = \bar{\partial}_v g(u) \cdot h(u) + g(u) \cdot \bar{\partial}_v h(u)$$

Let $V$ be a complex vector space and $s$ be a finite set of indices. Let $\{F_a\}_{a \in s}$ be a family of functions $F_a : V \to \mathbb{C}$ such that each $F_a$ is real-differentiable at a point $u \in V$. Then for any direction $v \in V$, the holomorphic directional Wirtinger derivative of the sum $\sum_{a \in s} F_a$ at $u$ is the sum of the holomorphic directional Wirtinger derivatives of the individual functions: $$\partial_v \left( \sum_{a \in s} F_a \right)(u) = \sum_{a \in s} \partial_v F_a(u)$$

Let $V$ be a complex vector space and $s$ be a finite set of indices. Let $\{F_a\}_{a \in s}$ be a family of functions $F_a : V \to \mathbb{C}$ such that each $F_a$ is real-differentiable at a point $u \in V$. Then for any direction $v \in V$, the anti-holomorphic directional Wirtinger derivative of the sum $\sum_{a \in s} F_a$ at $u$ is the sum of the anti-holomorphic directional Wirtinger derivatives of the individual functions: $$\bar{\partial}_v \left( \sum_{a \in s} F_a \right)(u) = \sum_{a \in s} \bar{\partial}_v F_a(u)$$

Let $E$ be a real normed vector space and $f: E \to \mathbb{C}$ be a function that is real-differentiable at a point $u \in E$. The real Fréchet derivative of the pointwise conjugate function $p \mapsto \overline{f(p)}$ at $u$ is equal to the complex conjugation map (denoted as `conjCLE`) composed with the real Fréchet derivative of $f$ at $u$. In physicists' notation, this is expressed as $d\bar{f} = \text{conj}(df)$.

Let $f: V \to \mathbb{C}$ be a function on a complex vector space $V$ that is real-differentiable at a point $u \in V$. For any direction $v \in V$, the holomorphic directional Wirtinger derivative of the pointwise conjugate function $\bar{f}$ (defined by $\bar{f}(p) = \overline{f(p)}$) at $u$ is equal to the complex conjugate of the anti-holomorphic directional Wirtinger derivative of $f$ at $u$: $$\partial_v \bar{f}(u) = \overline{\bar{\partial}_v f(u)}$$ where $\partial_v$ and $\bar{\partial}_v$ denote the holomorphic and anti-holomorphic directional Wirtinger derivatives, respectively.

Let $f: V \to \mathbb{C}$ be a function on a complex vector space $V$ that is real-differentiable at a point $u \in V$. For any direction $v \in V$, the anti-holomorphic directional Wirtinger derivative of the pointwise conjugate function $\bar{f}$ (defined by $\bar{f}(p) = \overline{f(p)}$) is equal to the complex conjugate of the holomorphic directional Wirtinger derivative of $f$: $$\bar{\partial}_v \bar{f}(u) = \overline{\partial_v f(u)}$$ where $\bar{\partial}_v$ and $\partial_v$ denote the anti-holomorphic and holomorphic directional Wirtinger derivatives, respectively.

Let $L: \mathbb{C} \to \mathbb{C}$ be a continuous real-linear map. For any complex number $w \in \mathbb{C}$, $L(w)$ can be split into a holomorphic and an anti-holomorphic part according to the formula: $$L(w) = \frac{1}{2}(L(1) - i L(i)) w + \frac{1}{2}(L(1) + i L(i)) \bar{w}$$ where $i$ is the imaginary unit and $\bar{w}$ denotes the complex conjugate of $w$. The coefficients $a = \frac{1}{2}(L(1) - i L(i))$ and $b = \frac{1}{2}(L(1) + i L(i))$ are known as the Wirtinger weights of the map $L$.

Let $V$ be a complex vector space. For any direction $v \in V$ and base point $u \in V$, let $f: V \to \mathbb{C}$ be a function real-differentiable at $u$, and let $g: \mathbb{C} \to \mathbb{C}$ be a function real-differentiable at $f(u)$. The holomorphic directional Wirtinger derivative of the composite function $g \circ f$ at $u$ is given by the two-term chain rule: $$\partial_v (g \circ f)(u) = \partial_1 g(f(u)) \cdot \partial_v f(u) + \bar{\partial}_1 g(f(u)) \cdot \partial_v \bar{f}(u)$$ where $\partial_1 g$ and $\bar{\partial}_1 g$ (also denoted as $\frac{\partial g}{\partial f}$ and $\frac{\partial g}{\partial \bar{f}}$) are the holomorphic and anti-holomorphic Wirtinger derivatives of $g$ evaluated at $f(u)$ in the unit direction $1 \in \mathbb{C}$, and $\bar{f}$ denotes the pointwise complex conjugate function $p \mapsto \overline{f(p)}$.

Let $V$ be a complex vector space. Suppose $f: V \to \mathbb{C}$ is a function real-differentiable at $u \in V$, and $g: \mathbb{C} \to \mathbb{C}$ is a function real-differentiable at $f(u)$. For any direction $v \in V$, the anti-holomorphic directional Wirtinger derivative of the composition $g \circ f$ at $u$ is given by the two-term chain rule: $$\bar{\partial}_v(g \circ f)(u) = \partial_1 g(f(u)) \cdot \bar{\partial}_v f(u) + \bar{\partial}_1 g(f(u)) \cdot \bar{\partial}_v \bar{f}(u)$$ where $\partial_1 g$ and $\bar{\partial}_1 g$ are the holomorphic and anti-holomorphic Wirtinger derivatives of $g$ in the direction $1 \in \mathbb{C}$, and $\bar{f}$ denotes the pointwise complex conjugate of $f$.

Let $V$ and $V'$ be vector spaces. Let $L: V \to V'$ be a continuous $\mathbb{R}$-linear map and $g: V' \to \mathbb{C}$ be a function. If $g$ is real-differentiable at $L(u)$ for some base point $u \in V$, then for any direction $x \in V$, the real Fréchet derivative of the composition $g \circ L$ at $u$ along the direction $x$ is equal to the real Fréchet derivative of $g$ at $L(u)$ along the direction $L(x)$: $$d_x (g \circ L)(u) = d_{L x} g(L u)$$ where $d_v f(u)$ denotes the real Fréchet derivative of $f$ at $u$ in direction $v$ (given by `fderiv ℝ f u v`).

Let $V$ and $V'$ be complex vector spaces. Suppose $L: V \to V'$ is a continuous $\mathbb{R}$-linear map that is conjugate-linear, satisfying $L(i \cdot x) = -i \cdot L(x)$ for all $x \in V$. Let $g: V' \to \mathbb{C}$ be a function that is real-differentiable at $L(u)$ for some base point $u \in V$. Then for any direction $v \in V$, the holomorphic directional Wirtinger derivative of the composition $g \circ L$ at $u$ along $v$ is equal to the anti-holomorphic directional Wirtinger derivative of $g$ at $L(u)$ along the direction $L(v)$: $$\partial_v (g \circ L)(u) = \bar{\partial}_{L v} g(L u)$$

Let $V$ and $V'$ be complex vector spaces. Let $g : V' \to \mathbb{C}$ be a function and $L : V \to V'$ be a continuous $\mathbb{R}$-linear map. Suppose $L$ is conjugate-$\mathbb{C}$-linear, meaning $L(i \cdot x) = -i \cdot L(x)$ for all $x \in V$. If $g$ is real-differentiable at $L(u)$ for some base point $u \in V$, then for any direction $v \in V$, the anti-holomorphic directional Wirtinger derivative of the composition $g \circ L$ at $u$ along $v$ equals the holomorphic directional Wirtinger derivative of $g$ at $L(u)$ along $L(v)$: $$\bar{\partial}_v (g \circ L)(u) = \partial_{L v} g (L u)$$

For a function $f: V \to \mathbb{C}$ on a complex vector space $V$, along a direction $v \in V$ where the real Fréchet derivative at a point $u \in V$ is $\mathbb{C}$-linear (that is, $d_{i \cdot v} f(u) = i \cdot d_v f(u)$), a **holomorphic collapse** occurs where the holomorphic directional Wirtinger derivative $\partial_v f(u)$ is equal to the real Fréchet derivative $d_v f(u)$.

Let $f: V \to \mathbb{C}$ be a function on a complex vector space $V$. If the real Fréchet derivative of $f$ at a point $u \in V$ is $\mathbb{C}$-linear in the direction $v \in V$, satisfying $d_{iv} f(u) = i d_v f(u)$, then the anti-holomorphic directional Wirtinger derivative of $f$ at $u$ along $v$ vanishes: $$\bar{\partial}_v f(u) = 0$$

Let $f: V \to \mathbb{C}$ be a function defined on a complex vector space $V$. If the real Fréchet derivative of $f$ at a point $u \in V$ is conjugate-linear in the direction $v \in V$, satisfying the condition $d_{i \cdot v} f(u) = -i \cdot d_v f(u)$, then the holomorphic directional Wirtinger derivative of $f$ at $u$ along $v$ vanishes: $$\partial_v f(u) = 0$$

Let $V$ be a complex vector space and $f: V \to \mathbb{C}$ a function. For any direction $v \in V$, if the real Fréchet derivative of $f$ at a point $u$ is conjugate-linear along $v$, meaning it satisfies the condition $d_{iv} f(u) = -i d_v f(u)$, then the anti-holomorphic directional Wirtinger derivative $\bar{\partial}_v f(u)$ is equal to the real derivative $d_v f(u)$.

Let $V$ be a complex vector space and $f: V \to \mathbb{C}$ a function. For any point $u \in V$ and direction $v \in V$, if the anti-holomorphic directional Wirtinger derivative vanishes, $\bar{\partial}_v f(u) = 0$, then the real Fréchet derivative of $f$ at $u$ is complex-linear along $v$, satisfying: $$d_{i \cdot v} f(u) = i \cdot d_v f(u)$$ where $d_v f(u)$ denotes the real Fréchet derivative of $f$ at the point $u$ in the direction $v$.

Let $V$ be a complex vector space and $f: V \to \mathbb{C}$ be a function. Then $f$ is complex-differentiable at a point $u \in V$ if and only if $f$ is real-differentiable at $u$ and for every direction $v \in V$, the anti-holomorphic directional Wirtinger derivative vanishes: $$\bar{\partial}_v f(u) = 0.$$ Here, the anti-holomorphic directional Wirtinger derivative is defined as $\bar{\partial}_v f(u) = \frac{1}{2} (d_v f(u) + i d_{iv} f(u))$, where $d_v f(u)$ denotes the real Fréchet derivative of $f$ at $u$ in the direction $v$.

Let $V$ be a complex vector space and $f: V \to \mathbb{C}$ be a function. If $f$ is twice continuously differentiable (in the real sense) at a point $u \in V$ (i.e., $f \in C^2$ at $u$), then for any direction $v \in V$, the holomorphic directional Wirtinger derivative field $p \mapsto \partial_v f(p)$ is real-differentiable at $u$.

Let $V$ be a complex vector space and $f: V \to \mathbb{C}$ be a function. If $f$ is twice continuously differentiable (in the real sense) at a point $u \in V$ (i.e., $f \in C^2$ at $u$), then for any direction $w \in V$, the anti-holomorphic directional Wirtinger derivative field $p \mapsto \bar{\partial}_w f(p)$ is real-differentiable at $u$.

Let $V$ be a complex vector space and $u \in V$ be a base point. If two functions $f_1, f_2: V \to \mathbb{C}$ are equal on some neighborhood of $u$, then for any direction $v \in V$, their holomorphic directional Wirtinger derivatives at $u$ are equal: $$\partial_v f_1(u) = \partial_v f_2(u)$$

Let $V$ be a complex vector space and $u \in V$ be a base point. If two functions $f_1, f_2: V \to \mathbb{C}$ are equal on some neighborhood of $u$, then for any direction $v \in V$, their anti-holomorphic directional Wirtinger derivatives at $u$ are equal: $$\bar{\partial}_v f_1(u) = \bar{\partial}_v f_2(u)$$

Let $V$ be a complex vector space and $f: V \to \mathbb{C}$ be a function. If $f$ is twice continuously real-differentiable (of class $C^2$) at a base point $u \in V$, then for any two direction vectors $v, w \in V$, the holomorphic directional Wirtinger derivative along $v$ and the anti-holomorphic directional Wirtinger derivative along $w$ commute at $u$: $$\partial_v (\bar{\partial}_w f)(u) = \bar{\partial}_w (\partial_v f)(u)$$ Here, $\partial_v$ denotes the holomorphic directional Wirtinger derivative defined by $\partial_v f = \frac{1}{2}(d_v f - i d_{iv} f)$, and $\bar{\partial}_w$ denotes the anti-holomorphic directional Wirtinger derivative defined by $\bar{\partial}_w f = \frac{1}{2}(d_w f + i d_{iw} f)$, where $d$ is the real Fréchet derivative.