A real function can have directional derivatives, partial derivatives, and even a smooth real Jacobian without behaving like multiplication by a complex number. Complex differentiability isolates exactly this missing constraint. It asks whether, near a point $z_0$, the change in $f$ is controlled to first order by one complex scalar multiplying the increment $z-z_0$. The first useful failure is conjugation. The map $z \mapsto \bar z$ is perfectly regular as a real map of the plane, but it reverses orientation, while multiplication by a complex number can only rotate and scale. The complex difference quotient detects this mismatch by approaching the same point along different directions. [example: Directional Failure of Conjugation] Let $f:\mathbb C\to\mathbb C$ be $f(z)=\bar z$. At $z_0=0$, we have $f(0)=\bar 0=0$, so for every nonzero $h\in\mathbb C$, \begin{align*} \frac{f(0+h)-f(0)}{h} &=\frac{\overline{h}-0}{h}\\ &=\frac{\bar h}{h}. \end{align*} Along the real approach $h=t$ with $t\in\mathbb R\setminus\{0\}$ and $t\to0$, \begin{align*} \frac{\bar h}{h} &=\frac{\bar t}{t}\\ &=\frac{t}{t}\\ &=1. \end{align*} Along the imaginary approach $h=it$ with $t\in\mathbb R\setminus\{0\}$ and $t\to0$, \begin{align*} \frac{\bar h}{h} &=\frac{\overline{it}}{it}\\ &=\frac{\bar i\,\bar t}{it}\\ &=\frac{(-i)t}{it}\\ &=-1. \end{align*} The same difference quotient has limiting value $1$ along the real axis and limiting value $-1$ along the imaginary axis, so the limit as $h\to0$ does not exist. Therefore $f$ is not complex differentiable at $0$. The failure is not special to the origin. If $z_0\in\mathbb C$ and $h\ne0$, then \begin{align*} \frac{f(z_0+h)-f(z_0)}{h} &=\frac{\overline{z_0+h}-\bar z_0}{h}\\ &=\frac{\bar z_0+\bar h-\bar z_0}{h}\\ &=\frac{\bar h}{h}, \end{align*} so the same two approaches give the two different limiting values $1$ and $-1$. Thus conjugation is not complex differentiable at any point of $\mathbb C$. [/example] This example is not a curiosity; it is the basic warning. Complex differentiability is a two-dimensional limit with a one-dimensional algebraic answer. The rest of the chapter explains how that tension becomes a usable theory. ## Definition The derivative at $z_0$ should test all nearby complex increments. For that reason, the domain must contain a small disk around $z_0$; otherwise missing directions could make the quotient misleading. This leads to the pointwise definition on open subsets of $\mathbb C$. [definition: Complex Derivative] Let $\Omega\subset\mathbb C$ be open, let $f:\Omega\to\mathbb C$ be a function, and let $z_0\in\Omega$. The complex derivative of $f$ at $z_0$ is the limit \begin{align*} f'(z_0)=\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0} \end{align*} when this limit exists in $\mathbb C$. [/definition] Writing $h=z-z_0$, the same condition is \begin{align*} f'(z_0)=\lim_{h\to0}\frac{f(z_0+h)-f(z_0)}{h}. \end{align*} A derivative at one point supports a local computation, but complex analysis is built from functions whose derivative exists as the point moves through a region; this regional condition will be named holomorphicity below. At this stage the definition asks only for existence of the limit at a point, not continuity, boundedness, or a [power series](/page/Power%20Series). The elementary examples show that the definition agrees with algebraic differentiation when the function respects complex multiplication. [example: Constants, Affine Maps, and Powers] Let $\Omega=\mathbb C$ and fix $z_0\in\mathbb C$. For a constant function $f(z)=c$, every nonzero $h\in\mathbb C$ gives \begin{align*} \frac{f(z_0+h)-f(z_0)}{h} &=\frac{c-c}{h}\\ &=\frac{0}{h}\\ &=0, \end{align*} so the difference quotient is constantly $0$ as $h\to0$, and therefore $f'(z_0)=0$. For an affine function $f(z)=az+b$ with $a,b\in\mathbb C$, every nonzero $h$ gives \begin{align*} \frac{f(z_0+h)-f(z_0)}{h} &=\frac{a(z_0+h)+b-(az_0+b)}{h}\\ &=\frac{az_0+ah+b-az_0-b}{h}\\ &=\frac{ah}{h}\\ &=a. \end{align*} Thus the difference quotient is constantly $a$, so $f'(z_0)=a$. For $f(z)=z^2$, every nonzero $h$ gives \begin{align*} \frac{f(z_0+h)-f(z_0)}{h} &=\frac{(z_0+h)^2-z_0^2}{h}\\ &=\frac{z_0^2+2z_0h+h^2-z_0^2}{h}\\ &=\frac{2z_0h+h^2}{h}\\ &=2z_0+h. \end{align*} Since $2z_0+h\to2z_0$ as $h\to0$, we get $f'(z_0)=2z_0$. These three computations show that the complex derivative agrees with the expected algebraic rules for constants, affine maps, and the first non-linear power. [/example] These computations explain why polynomials behave as expected. They do not explain how to test a function written in real coordinates. For that we need to see what complex differentiability demands of the two real component functions. ## The Cauchy-Riemann Equations A complex-valued function is also a real map from the plane to the plane. Writing its real and imaginary parts makes ordinary partial derivatives available. The price is that complex differentiability becomes a structural condition on the real Jacobian. [definition: Real and Imaginary Parts of a Complex Function] Let $\Omega\subset\mathbb C$ be open, let $\Omega_{\mathbb R}=\{(x,y)\in\mathbb R^2:x+iy\in\Omega\}$, and let $f:\Omega\to\mathbb C$. The real and imaginary parts of $f$ are the functions $u,v:\Omega_{\mathbb R}\to\mathbb R$ defined by \begin{align*} f(x+iy)=u(x,y)+iv(x,y) \end{align*} for $x+iy\in\Omega$. [/definition] ### The Complex-Linearity Test A general real [linear map](/page/Linear%20Map) $\mathbb R^2\to\mathbb R^2$ has four independent entries. Multiplication by a complex number has only two: a scale-rotation pattern. The equations below express exactly the condition that the real Jacobian has this complex-linear form. [definition: Cauchy-Riemann Equations] Let $\Omega\subset\mathbb C$ be open, let $\Omega_{\mathbb R}=\{(x,y)\in\mathbb R^2:x+iy\in\Omega\}$, let $f:\Omega\to\mathbb C$, and write $f(x+iy)=u(x,y)+iv(x,y)$ with $u,v:\Omega_{\mathbb R}\to\mathbb R$. The Cauchy-Riemann equations at $z_0=x_0+iy_0\in\Omega$ are \begin{align*} \partial_xu(x_0,y_0)&=\partial_yv(x_0,y_0),\\ \partial_yu(x_0,y_0)&=-\partial_xv(x_0,y_0). \end{align*} [/definition] ### Directional Tests for Differentiability The equations are forced by comparing the complex difference quotient along the real and imaginary directions. If the two directional limits disagree, then the full two-dimensional limit cannot exist. This turns the Cauchy-Riemann equations into a necessary test for complex differentiability: before asking for a full two-dimensional limit, one should first check whether the real and imaginary directional derivatives can even agree with a single complex number. The next formal result records that obstruction precisely and identifies the only possible value of the derivative when the obstruction vanishes. [quotetheorem:333] The theorem rejects conjugation immediately: $u(x,y)=x$ and $v(x,y)=-y$ give $\partial_xu=1$ and $\partial_yv=-1$. For a positive test, however, the equations need enough real regularity to control all approach directions. Continuous first partial derivatives provide the standard sufficient hypothesis. [quotetheorem:327] Combining this differentiability criterion with the Cauchy-Riemann characterisation above gives the usual $C^1$ sufficient test for complex differentiability. ### Wirtinger Notation The same condition is often packaged as a first-order differential equation. This is useful in analysis and PDE because it separates the complex-linear part of the derivative from the conjugate-linear obstruction. The notation for that separation is Wirtinger notation. In the next definition, $C^1(\Omega_{\mathbb R};\mathbb C)$ denotes the complex-valued functions on $\Omega_{\mathbb R}$ with continuous first partial derivatives, and $C^0(\Omega;\mathbb C)$ denotes the continuous complex-valued functions on $\Omega$. [definition: Wirtinger Derivatives] Let $\Omega\subset\mathbb C$ be open and let $\Omega_{\mathbb R}=\{(x,y)\in\mathbb R^2:x+iy\in\Omega\}$. The Wirtinger derivative operators are the maps \begin{align*} \partial_z:C^1(\Omega_{\mathbb R};\mathbb C)&\to C^0(\Omega;\mathbb C),\\ \partial_{\bar z}:C^1(\Omega_{\mathbb R};\mathbb C)&\to C^0(\Omega;\mathbb C) \end{align*} defined for $f\in C^1(\Omega_{\mathbb R};\mathbb C)$, viewed as a function on $\Omega$ by $z=x+iy$, by \begin{align*} \partial_zf(z)&=\frac{1}{2}\left(\partial_xf(x,y)-i\partial_yf(x,y)\right),\\ \partial_{\bar z}f(z)&=\frac{1}{2}\left(\partial_xf(x,y)+i\partial_yf(x,y)\right). \end{align*} [/definition] For functions with continuous first partial derivatives, complex differentiability is equivalent to the vanishing of the $\bar z$ derivative. That reformulation makes holomorphicity look like the differential equation $\partial_{\bar z}f=0$ in the $C^1$ setting. It also prepares the notation used in applied complex analysis and several complex variables. This is exactly the Cauchy-Riemann characterisation rewritten in Wirtinger notation: the $\partial_{\bar z}$ term measures the conjugate-linear part of the real derivative, so its vanishing leaves a complex-linear derivative. The Cauchy-Riemann viewpoint turns a limit into a first-order system. The next step is to study functions satisfying that system throughout a domain. ## Holomorphic Functions and Local Rigidity A derivative at a point is a local test. A derivative at every point of an [open set](/page/Open%20Set) is a structure strong enough to support global theorems. The standard name for this regional form of complex differentiability is holomorphicity. [definition: Holomorphic Function] Let $\Omega\subset\mathbb C$ be open. A function $f:\Omega\to\mathbb C$ is holomorphic on $\Omega$ if $f$ is complex differentiable at every point of $\Omega$. [/definition] Some arguments require functions with no finite singularities at all. This happens for polynomials, exponentials, and global rigidity results on the whole plane. The special name for holomorphicity on all of $\mathbb C$ is entire. [definition: Entire Function] A function $f:\mathbb C\to\mathbb C$ is entire if it is holomorphic on $\mathbb C$. [/definition] Examples become useful only if we can combine them. Since addition, multiplication, quotient formation away from zeros, and composition are the basic ways functions are built, holomorphicity must be stable under these operations. The next two theorems give that stability. [quotetheorem:4949] [Algebraic closure](/page/Algebraic%20Closure) builds rational functions from polynomials and explains why singularities occur at zeros of denominators. The next missing operation is substitution: if a holomorphic map is used as a coordinate change or as a parametrisation, differentiability should survive after another [holomorphic function](/page/Holomorphic%20Function) is applied. The chain rule is the mechanism that makes holomorphic functions closed under this kind of change of variables. [quotetheorem:323] The chain rule explains how derivatives behave after substitution, but it also raises the reverse question: when can a holomorphic change of variables be undone holomorphically? A nonzero derivative means that the first-order complex multiplication is invertible, so the local geometry has no infinitesimal collapse. The [holomorphic inverse function theorem](/theorems/4950) turns that first-order condition into a genuine local inverse. [quotetheorem:4950] Local invertibility does not imply global injectivity. The exponential map has nonzero derivative everywhere but repeats its values with period $2\pi i$. This example separates local holomorphic geometry from global topology. [example: Exponential Is Locally Invertible but Not Globally Injective] Let $f:\mathbb C\to\mathbb C$ be $f(z)=e^z$. From the power-series computation of the complex exponential, \begin{align*} f'(z) &=(e^z)'\\ &=e^z. \end{align*} Writing $z=x+iy$ with $x,y\in\mathbb R$, [Euler's formula](/theorems/2014) gives \begin{align*} e^z &=e^{x+iy}\\ &=e^x(\cos y+i\sin y), \end{align*} so \begin{align*} |e^z| &=|e^x|\,|\cos y+i\sin y|\\ &=e^x\sqrt{\cos^2y+\sin^2y}\\ &=e^x\\ &>0. \end{align*} Thus $f'(z)=e^z\ne0$ for every $z\in\mathbb C$. By the *Holomorphic [Inverse Function Theorem](/page/Inverse%20Function%20Theorem)*, for each $z_0\in\mathbb C$ there are open neighbourhoods $U$ of $z_0$ and $V$ of $e^{z_0}$ such that $f|_U:U\to V$ has a holomorphic inverse; these local inverses are local holomorphic branches of the logarithm. The same function is not globally injective. For every $z=x+iy\in\mathbb C$, \begin{align*} e^{z+2\pi i} &=e^{x+i(y+2\pi)}\\ &=e^x(\cos(y+2\pi)+i\sin(y+2\pi))\\ &=e^x(\cos y+i\sin y)\\ &=e^{x+iy}\\ &=e^z. \end{align*} Since $z+2\pi i\ne z$, the two distinct inputs $z$ and $z+2\pi i$ have the same image under $f$. Hence the exponential map is locally invertible everywhere but not globally injective on $\mathbb C$. [/example] Local invertibility asks whether values near one point determine nearby preimages, but complex analysis also needs a stronger question: when does agreement on a small set determine an entire function on a domain? For real smooth functions, matching on a convergent sequence says little, because a bump function can vanish on a large set without being zero nearby. Holomorphic functions are more rigid, and the identity theorem records how a small accumulation of agreement determines the function on a connected domain. [quotetheorem:3357] The identity theorem is the first major sign that holomorphic functions behave like power series rather than arbitrary differentiable functions. The next section makes that statement exact. ## Power Series and Analyticity A power series is the natural local model for a holomorphic function because it records all orders of approximation at one centre. Before proving that holomorphic functions have such expansions, we need to specify what kind of series is being used. The centre and coefficients are part of the data. [definition: Complex Power Series] Let $z_0\in\mathbb C$ and let $(a_n)_{n=0}^{\infty}$ be a sequence in $\mathbb C$. The complex power series with centre $z_0$ and coefficients $(a_n)$ is the formal series \begin{align*} \sum_{n=0}^{\infty}a_n(z-z_0)^n. \end{align*} [/definition] The formal expression by itself is not yet a function. If the series converges only at the centre, it carries almost no local analytic information; if it converges in a neighbourhood, it becomes a genuine candidate for a holomorphic function. We therefore need a single piece of data that records the largest disk on which the infinite sum is available for differentiation, comparison, and later [analytic continuation](/page/Analytic%20Continuation). That data is the [radius of convergence](/theorems/265). [definition: Radius of Convergence] Let \begin{align*} \sum_{n=0}^{\infty}a_n(z-z_0)^n \end{align*} be a complex power series. Its [radius of convergence](/theorems/262) is the number $R\in[0,\infty]$ such that the series converges absolutely for $|z-z_0|<R$ and diverges for $|z-z_0|>R$. [/definition] The radius tells us where the series defines a function. The next question is whether this function is complex differentiable and whether differentiation respects the series. Inside the convergence disk, termwise differentiation is valid. [quotetheorem:335] A first test for the radius concept is the exponential series, where the convergence disk is the whole plane. The example matters because it gives an entire function directly from a series, and its derivative computation shows how termwise differentiation turns coefficient data into analytic behaviour. [example: The Complex Exponential] Define $\exp:\mathbb C\to\mathbb C$ by \begin{align*} \exp z=\sum_{n=0}^{\infty}\frac{z^n}{n!}. \end{align*} For a fixed $z\in\mathbb C$, the absolute-value series has successive ratio \begin{align*} \frac{|z|^{n+1}/(n+1)!}{|z|^n/n!} &=\frac{|z|^{n+1}n!}{(n+1)!|z|^n}\\ &=\frac{|z|}{n+1}\\ &\to0 \end{align*} as $n\to\infty$, so the series converges absolutely for every $z\in\mathbb C$. Hence its [radius of convergence](/theorems/273) is infinite. By *[Holomorphicity of Power Series](/theorems/335)*, $\exp$ is holomorphic on all of $\mathbb C$, and for every $z\in\mathbb C$, \begin{align*} (\exp z)' &=\left(\sum_{n=0}^{\infty}\frac{z^n}{n!}\right)'\\ &=\sum_{n=1}^{\infty}n\frac{z^{n-1}}{n!}\\ &=\sum_{n=1}^{\infty}\frac{nz^{n-1}}{n(n-1)!}\\ &=\sum_{n=1}^{\infty}\frac{z^{n-1}}{(n-1)!}. \end{align*} Putting $m=n-1$, the index $n=1$ becomes $m=0$, and therefore \begin{align*} (\exp z)' &=\sum_{m=0}^{\infty}\frac{z^m}{m!}\\ &=\exp z. \end{align*} Thus the exponential function is entire, and it is its own complex derivative. [/example] The exponential shows one direction: a convergent power series produces a holomorphic function. The deeper question is whether holomorphic functions ever escape this series world. The full analyticity theorem answers that question using [Cauchy's integral formula](/page/Cauchy's%20Integral%20Formula), so it is recorded later after contour integration has been introduced. Analyticity turns the next local question into a coefficient question: if $f(z_0)=0$, how strongly does the function vanish at $z_0$? The first nonzero coefficient separates a simple crossing from a higher-order contact. The definition of order records that first nonzero power in an invariant way. [definition: Zero of Order $m$] Let $\Omega\subset\mathbb C$ be open, let $f:\Omega\to\mathbb C$ be holomorphic, and let $z_0\in\Omega$. The point $z_0$ is a zero of order $m\in\mathbb N$ with $m\ge1$ of $f$ if there exists a holomorphic function $g:\Omega\to\mathbb C$ such that $g(z_0)\ne0$ and \begin{align*} f(z)=(z-z_0)^mg(z) \end{align*} for every $z\in\Omega$. [/definition] The definition separates a zero into its exact vanishing factor and a remaining factor that does not vanish at the point. That separation should prevent zeros from clustering near $z_0$ unless every coefficient in the local power series has disappeared. This is the next rigidity phenomenon to isolate: for a holomorphic function on a connected domain, zeros are either locally separated points or the function has collapsed to the zero function everywhere. [quotetheorem:3357] Indeed, applying the [identity principle](/theorems/3357) to $f$ and the zero function shows that a nonzero holomorphic function of one complex variable cannot have a zero set with an accumulation point inside the domain. The order of a zero also controls geometry. A simple nonzero derivative preserves first-order angles, while a higher-order zero of $f(z)-f(z_0)$ changes angle behaviour. ## Conformal Meaning of the Derivative Multiplication by a nonzero complex number is a rotation followed by a scaling. Since the complex derivative is exactly such a multiplication at first order, a holomorphic function with nonzero derivative should preserve infinitesimal angles. This is the pointwise notion of conformality. [definition: Conformal at a Point] Let $\Omega\subset\mathbb C$ be open, let $f:\Omega\to\mathbb C$ be holomorphic, and let $z_0\in\Omega$. The function $f$ is conformal at $z_0$ if $f'(z_0)\ne0$. [/definition] For domain-level geometry, a pointwise condition is not enough. A useful change of variables should also be one-to-one and onto its target domain. This motivates the global notion of a conformal map. [definition: Conformal Map] Let $\Omega,\Lambda\subset\mathbb C$ be open. A function $f:\Omega\to\Lambda$ is a conformal map if $f$ is holomorphic, bijective, and $f'(z)\ne0$ for every $z\in\Omega$. [/definition] The definition is geometric, so it should imply an angle statement for curves. Tangent vectors to curves transform by multiplication by $f'(z_0)$, and a common nonzero complex factor preserves their oriented angle. The theorem gives the formal version. [quotetheorem:334] The nonzero derivative hypothesis cannot be removed. If the first nonzero term has degree greater than one, the map can multiply angles instead of preserving them. The squaring map is the standard local model. [example: Squaring Doubles Angles at the Origin] Let $f:\mathbb C\to\mathbb C$ be $f(z)=z^2$. For $h\ne0$, the difference quotient at $0$ is \begin{align*} \frac{f(0+h)-f(0)}{h} &=\frac{h^2-0^2}{h}\\ &=\frac{h^2}{h}\\ &=h, \end{align*} so as $h\to0$ this quotient tends to $0$. Hence $f'(0)=0$, and $f$ is not conformal at $0$. Now fix an angle $\theta\in\mathbb R$ and consider the ray $\gamma_\theta(t)=te^{i\theta}$ for $t\ge0$. Its image under $f$ is \begin{align*} f(\gamma_\theta(t)) &=f(te^{i\theta})\\ &=(te^{i\theta})^2\\ &=t^2e^{i\theta}e^{i\theta}\\ &=t^2e^{2i\theta}. \end{align*} Thus the ray with argument $\theta$ is sent to the ray with argument $2\theta$, with the radial parameter changed from $t$ to $t^2$. For two rays $\gamma_{\theta_1}$ and $\gamma_{\theta_2}$, the original oriented angle is $\theta_2-\theta_1$. Their images have arguments $2\theta_1$ and $2\theta_2$, so the image angle is \begin{align*} 2\theta_2-2\theta_1 &=2(\theta_2-\theta_1) \end{align*} modulo $2\pi$. Therefore the squaring map is holomorphic, but at the origin it doubles angles rather than preserving them, exactly because its derivative there is zero. [/example] [Conformal maps](/page/Conformal%20Maps) turn complex differentiability into a tool for geometry and PDE. They transport harmonic functions, boundary conditions, and analytic kernels between domains. This is why the same derivative condition appears in fluid flow, electrostatics, and planar elasticity. [illustration:angle-preserving-conformal-map] ## Complex Differentiability in Integral Theory The definition of $f'(z)$ is local, but holomorphicity controls contour integrals globally. The simplest reason an integral around a closed curve vanishes is that the integrand is the derivative of another holomorphic function. This motivates the complex analogue of an antiderivative. [definition: Primitive] Let $\Omega\subset\mathbb C$ be open and let $f:\Omega\to\mathbb C$. A primitive of $f$ on $\Omega$ is a holomorphic function $F:\Omega\to\mathbb C$ such that \begin{align*} F'(z)=f(z) \end{align*} for every $z\in\Omega$. [/definition] If $F'=f$, then integrating $f$ along a path should recover the change in $F$. The key obstruction is path dependence: without a primitive, the same endpoints can be joined by curves whose integrals differ. A primitive removes that obstruction by turning the integral into endpoint data, so every closed contour has zero integral. [quotetheorem:339] The converse can fail on domains with holes. A closed path can wind around a missing point and detect a singularity outside the domain of integration. The function $1/z$ on the punctured plane is the basic example. [example: A Holomorphic Function Without a Global Primitive] Let $\Omega=\mathbb C\setminus\{0\}$ and let $f:\Omega\to\mathbb C$ be $f(z)=1/z$. Since the identity function is holomorphic and does not vanish on $\Omega$, $f$ is holomorphic on $\Omega$ by *[Algebra of Holomorphic Functions](/theorems/4949)*. Consider the unit circle $\gamma:[0,2\pi]\to\Omega$ given by $\gamma(t)=e^{it}$. Then \begin{align*} \gamma'(t) &=ie^{it}. \end{align*} Using the definition of the contour integral along a parametrised path, \begin{align*} \oint_\gamma\frac{1}{z}\,dz &=\int_0^{2\pi}\frac{1}{\gamma(t)}\gamma'(t)\,dt\\ &=\int_0^{2\pi}\frac{1}{e^{it}}ie^{it}\,dt\\ &=\int_0^{2\pi}i\,dt\\ &=i\left[t\right]_{0}^{2\pi}\\ &=i(2\pi-0)\\ &=2\pi i. \end{align*} This value is nonzero. If $1/z$ had a primitive on $\mathbb C\setminus\{0\}$, then every closed contour integral of $1/z$ in that domain would vanish by the *[Fundamental Theorem of Contour Integration](/theorems/339)*. The unit circle is closed, but its integral is $2\pi i\ne0$, so $1/z$ has no global primitive on $\mathbb C\setminus\{0\}$. [/example] Simply connected domains remove this winding obstruction. On such domains, holomorphicity gives vanishing integrals around closed paths. This is the foundational integral theorem of complex analysis. [quotetheorem:344] Vanishing integrals explain why contour deformations are allowed, but the next problem is more precise: can the boundary values of a holomorphic function determine its value at an interior point? To recover $f(z_0)$, we insert the kernel $1/(\zeta-z_0)$, whose controlled singularity at $z_0$ extracts precisely the interior value. This produces Cauchy's integral formula. [quotetheorem:345] The integral formula does more than recover values: it forces local power series expansions. Here $\mathcal O(\Omega)$ denotes the set of holomorphic functions on $\Omega$. The following theorem is the structural answer promised earlier: complex differentiability is strong enough to make every holomorphic function locally analytic. [quotetheorem:3354] The integral formula extracts values from holomorphicity, but in applications we often face the reverse problem: a function is known through integrals, limits, or transforms, and direct difference quotients are hard to control. If a continuous function has the same integral cancellation that holomorphic functions have, differentiability should be recoverable from those integrals. [Morera's theorem](/theorems/349) confirms this under a continuity hypothesis. [quotetheorem:349] Morera's theorem is useful when functions are defined by limits, parameter integrals, or transforms. It lets holomorphicity be proved from integral cancellation rather than from direct difference quotient estimates. ## Beyond and Connected Topics Complex differentiability is the doorway into [Cambridge IB Complex Analysis](/page/Cambridge%20IB%20Complex%20Analysis), where this chapter continues into [Cauchy's theorem](/page/Cauchy's%20Theorem), contour integration, Taylor and [Laurent series](/page/Laurent%20Series), residues, and the [maximum modulus principle](/page/Maximum%20Modulus%20Principle). That course-level page is the natural next reference for the core theory. The higher-dimensional branch begins with [Androma Several Complex Variables I: Domains and Holomorphy](/page/Androma%20Several%20Complex%20Variables%20I%3A%20Domains%20and%20Holomorphy). There the one-variable condition of complex differentiability becomes holomorphy in several coordinates, and familiar local ideas meet new phenomena such as domains of holomorphy and stronger rigidity. Further several-variable directions include [Several Complex Variables III: $L^2$ Methods and Applications](/page/Several%20Complex%20Variables%20III%3A%20L%C2%B2%20Methods%20and%20Applications), where analytic questions are attacked using estimates and Hilbert-space methods, and [Several Complex Variables IV: Complex Geometry and Curvature](/page/Several%20Complex%20Variables%20IV%3A%20Complex%20Geometry%20and%20Curvature), where holomorphic functions interact with bundles, metrics, and curvature. Several standard one-variable topics refine the same ideas. Laurent series and residues study isolated singularities. The [maximum modulus principle](/theorems/491) turns local differentiability into global size control. The Riemann mapping theorem shows that conformal maps are abundant on simply connected proper domains, while several complex variables reveals even stronger rigidity in higher dimensions. ## References Androma, [Cambridge IB Complex Analysis](/page/Cambridge%20IB%20Complex%20Analysis). Androma, [Androma Several Complex Variables I: Domains and Holomorphy](/page/Androma%20Several%20Complex%20Variables%20I%3A%20Domains%20and%20Holomorphy). Androma, [Several Complex Variables III: $L^2$ Methods and Applications](/page/Several%20Complex%20Variables%20III%3A%20L%C2%B2%20Methods%20and%20Applications). Androma, [Several Complex Variables IV: Complex Geometry and Curvature](/page/Several%20Complex%20Variables%20IV%3A%20Complex%20Geometry%20and%20Curvature). Ahlfors, *Complex Analysis* (1979). Conway, *Functions of One Complex Variable I* (1978). Stein and Shakarchi, *Complex Analysis* (2003).