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Holomorphic functions [ edit] Main article: Holomorphic function

Complex functions that are differentiable at every point of an open subset $\Omega$ of the complex plane are said to be holomorphic on $\Omega$. In the context of complex analysis, the derivative of $f$ at $z_0$ is defined to be ${ }^{[1]}$ $$ f^{\prime}\left(z_0 ight)=\lim _{z ightarrow z_0} rac{f(z)-f\left(z_0 ight)}{z-z_0} . $$

Superficially, this definition is formally analogous to that of the derivative of a real function. However, complex derivatives and differentiable functions behave in significantly different ways compared to their real counterparts. In particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we approach $z_0$ in the complex plane. Consequently, complex differentiability has much stronger implications than real differentiability. For instance, holomorphic functions are infinitely differentiable, whereas the existence of the $n$th derivative need not imply the existence of the $(n+1)$ th derivative for real functions. Furthermore, all holomorphic functions satisfy the stronger condition of analyticity, meaning that the function is, at every point in its domain, locally given by a convergent power series. In essence, this means that functions holomorphic on $\Omega$ can be approximated arbitrarily well by polynomials in some neighborhood of every point in $\Omega$. This stands in sharp contrast to differentiable real functions; there are infinitely differentiable real functions that are nowhere analytic; see Non-analytic smooth function $\S A$ smooth function which is nowhere real analytic.

Most elementary functions, including the exponential function, the trigonometric functions, and all polynomial functions, extended appropriately to complex arguments as functions $\mathbb{C} ightarrow \mathbb{C}$, are holomorphic over the entire complex plane, making them entire functions, while rational functions $p / q$, where $p$ and $q$ are polynomials, are holomorphic on domains that exclude points where $q$ is zero. Such functions that are holomorphic everywhere except a set of isolated points are known as meromorphic functions. On the other hand, the functions $z \mapsto \Re(z)$, $z \mapsto|z|$, and $z \mapsto ar{z}$ are not holomorphic anywhere on the complex plane, as can be shown by their failure to satisfy the Cauchy-Riemann conditions (see below).

An important property of holomorphic functions is the relationship between the partial derivatives of their real and imaginary components, known as the Cauchy-Riemann conditions. If $f: \mathbb{C} ightarrow \mathbb{C}$, defined by $f(z)=f(x+i y)=u(x, y)+i v(x, y)$, where $x, y, u(x, y), v(x, y) \in \mathbb{R}$, is holomorphic on a region $\Omega$, then for all $z_0 \in \Omega$, $$ rac{tial f}{tial ar{z}}\left(z_0 ight)=0, ext { where } rac{tial}{tial ar{z}}:=rac{1}{2}\left(rac{tial}{tial x}+i rac{tial}{tial y} ight) . $$ subscripts indicate partial differentiation. However, the Cauchy-Riemann conditions do not characterize holomorphic functions, without additional continuity conditions (see Looman-Menchoff theorem).

Holomorphic functions exhibit some remarkable features. For instance, Picard’s theorem asserts that the range of an entire function can take only three possible forms: $\mathbb{C}, \mathbb{C} ackslash\left\{z_0 ight\}$, or $\left\{z_0 ight\}$ for some $z_0 \in \mathbb{C}$. In other words, if two distinct complex numbers $z$ and $w$ are not in the range of an entire function $f$, then $f$ is a constant function. Moreover, a holomorphic function on a connected open set is determined by its restriction to any nonempty open subset. An exponential function $A^n$ of a discrete (integer) variable $n$, similar to geometric progression 图片描述 全纯函数 [ 编辑] 主文章:全纯函数

在复平面的开放子集 $\Omega$ 的每一点上都可微分的复函数被称为 $\Omega$ 上的全纯函数。在复分析中,$f$ 在 $z_0$ 处的导数定义为 ${ }^{[1]}$ $$ f^{\prime}\left(z_0 ight)=\lim _{z ightarrow z_0} rac{f(z)-f\left(z_0 ight)}{z-z_0} . $$

从表面上看,这个定义与实函数导数的形式类似。然而,复数导数和可微分函数的行为方式与实数导数和可微分函数的行为方式明显不同。特别是,无论我们以何种方式在复平面上接近 $z_0$,要使这一极限存在,差商的值必须接近相同的复数。因此,复可微性比实可微性具有更强的意义。例如,全形函数是无限可微的,而对于实函数来说,第 $n$ 次导数的存在并不一定意味着第 $(n+1)$ 次导数的存在。此外,所有全形函数都满足更强的解析性条件,即函数在其域中的每一点都由收敛幂级数局部给出。从本质上讲,这意味着$\Omega$上的全纯函数可以被$\Omega$上每一点的某个邻域中的多项式任意逼近。这与可微实函数形成了鲜明对比;有无穷可微的实函数是无处解析的;见非解析光滑函数 $\S A$ 无处实解析的光滑函数。

大多数初等函数,包括指数函数、三角函数和所有多项式函数,都可以适当地扩展为复数参数函数 $\mathbb{C} 。 ightarrow \mathbb{C}$,在整个复平面上是全态的,因此它们是全函数,而有理函数 $p / q$,其中 $p$ 和 $q$ 是多项式,在不包括 $q$ 为零的点的域上是全态的。这种除一组孤立点外在任何地方都是全形的函数被称作分形函数。另一方面,函数 $z \mapsto \Re(z)$、$z \mapsto|z|$ 和 $z \mapsto ar{z}$ 在复平面上的任何地方都不是全态的,这可以通过它们不满足考奇-黎曼条件(见下文)来证明。

全形函数的一个重要性质是其实部和虚部偏导数之间的关系,即所谓的 Cauchy-Riemann 条件。如果 $f:\mathbb{C} 由 $f(z)=f(x+i y)=u(x, y)+i v(x, y)$ 定义,其中 $x, y, u(x, y), v(x, y) \in \mathbb{R}$, 在区域 $\Omega$ 上是全态的,那么对于所有 $z_0 \in \Omega$、 $$ rac{(部分)f}{(部分)ar{z}}left(z_0 ight)=0, ext { 其中 } rac{partial}{partial ar{z}}:= rac{1}{2}\left( rac{partial}{partial x}+i rac{partial}{partial y} ight) . $$ 下标表示部分微分。然而,如果没有额外的连续性条件(见 Looman-Menchoff 定理),Cauchy-Riemann 条件并不能表征全形函数。

全形函数表现出一些显著特点。ackslash\left\{z_0 ight\}$ 或者 $\left\{z_0 ight\}$ 对于 \mathbb{C}$ 中的某个 $z_0 \。换句话说,如果两个不同的复数 $z$ 和 $w$ 不在一个全函数 $f$ 的范围内,那么 $f$ 是一个常数函数。此外,连通开集上的全形函数是由它对任何非空开集子集的限制决定的。 离散(整数)变量 $n$ 的指数函数 $A^n$,类似于几何级数 这里是文章内容。[内链关键词](你的内链URL)

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