A complex differentiable function is a severe object. At first glance it asks for the same kind of limit as real differentiability: compare the increment of a function to a linear approximation and let the increment tend to zero. The difference is that in the complex plane the increment can approach zero from infinitely many directions, and multiplication by a complex number is not an arbitrary real [linear map](/page/Linear%20Map). The Cauchy-Riemann equations are the test that detects when a real two-variable derivative is actually compatible with complex multiplication. The central tension is this: a function $f: \Omega \to \mathbb C$ can be viewed as a pair of real functions $u, v: \Omega \to \mathbb R$ by writing $f(z) = u(x,y) + i v(x,y)$ for $z = x + iy$. Real differentiability of $(u,v)$ allows any real $2 \times 2$ Jacobian matrix. [Complex differentiability](/page/Complex%20Differentiability) allows only matrices representing multiplication by one complex number. The Cauchy-Riemann equations are exactly the algebraic restriction that forces the real derivative to have that special form. [example: A Directional Limit That Fails] Let \begin{align*} f: \mathbb C &\to \mathbb C \\ z &\mapsto \bar z. \end{align*} We test complex differentiability at $0$ by examining the difference quotient from the definition of the complex derivative. Since $f(0)=\overline{0}=0$, for every $h \ne 0$ one has \begin{align*} \frac{f(0+h)-f(0)}{h} &= \frac{f(h)-0}{h} \\ &= \frac{\bar h}{h}. \end{align*} Along the real axis, write $h=t$ with $t \in \mathbb R$ and $t \ne 0$. Then $\bar h=\bar t=t$, so \begin{align*} \frac{\bar h}{h} &= \frac{t}{t} \\ &= 1. \end{align*} Thus the quotient tends to $1$ as $t \to 0$ through nonzero real values. Along the imaginary axis, write $h=it$ with $t \in \mathbb R$ and $t \ne 0$. Since $\overline{it}=-it$, we get \begin{align*} \frac{\bar h}{h} &= \frac{\overline{it}}{it} \\ &= \frac{-it}{it} \\ &= -1. \end{align*} Thus the quotient tends to $-1$ as $t \to 0$ through nonzero real values on this second path. A single complex limit cannot have two different path limits, so the complex derivative of $f$ at $0$ does not exist. [/example] This example is the basic warning. The map $z \mapsto \bar z$ is perfectly smooth as a real map from $\mathbb R^2$ to $\mathbb R^2$, but it reverses orientation and cannot be approximated to first order by complex multiplication. Holomorphic functions are those real two-variable functions whose first-order behaviour never contains a conjugation component. ## Components of a Complex Function To turn this warning into a criterion, we first need to describe the real and imaginary parts of a complex-valued function on a plane domain. The notation is simple, but it matters because the equations compare partial derivatives of these two real components. [definition: Real and Imaginary Parts of a Complex Function] Let $\Omega \subset \mathbb C$ be open, and let $f: \Omega \to \mathbb C$ be a function. The real and imaginary parts of $f$ are the functions $u, v: \Omega \to \mathbb R$ defined by \begin{align*} f(x+iy) = u(x,y) + i v(x,y) \end{align*} for all $x+iy \in \Omega$. [/definition] Once $f$ is written as $u+iv$, complex differentiability becomes a statement about the real derivative of the map $(x,y) \mapsto (u(x,y),v(x,y))$. The next definition isolates the first-order equations that make this real derivative compatible with multiplication by a complex number. ## Definition The point of the definition is not to invent two mysterious equations, but to name the exact failure detected by the opening example. If the derivative in the real direction and the derivative in the imaginary direction are to come from multiplication by a single complex number, then the four real partial derivatives of $u$ and $v$ cannot vary independently; they must fit together with one equality for the scaling part and one equality for the rotation part. [definition: Cauchy-Riemann Equations] Let $\Omega \subset \mathbb C$ be open, let $f: \Omega \to \mathbb C$, and write $f = u + iv$ with $u, v: \Omega \to \mathbb R$. The function $f$ satisfies the Cauchy-Riemann equations at $z_0 = x_0 + iy_0 \in \Omega$ if the partial derivatives $\partial_x u(z_0)$, $\partial_y u(z_0)$, $\partial_x v(z_0)$, and $\partial_y v(z_0)$ exist and \begin{align*} \partial_x u(z_0) &= \partial_y v(z_0), \\ \partial_y u(z_0) &= -\partial_x v(z_0). \end{align*} The function $f$ satisfies the Cauchy-Riemann equations on $\Omega$ if it satisfies them at every point of $\Omega$. [/definition] These equations are often remembered as two formulas, but their meaning is structural. They say that the Jacobian matrix of $(u,v)$ has the form \begin{align*} Jf_{z_0} = \begin{pmatrix} \partial_x u(z_0) & \partial_y u(z_0) \\ \partial_x v(z_0) & \partial_y v(z_0) \end{pmatrix} = \begin{pmatrix} a & -b \\ b & a \end{pmatrix}, \end{align*} which is the real matrix of multiplication by $a+ib$. ## Complex Differentiability ### The Difference Quotient The real derivative of a map $\mathbb R^2\to\mathbb R^2$ can depend on direction in many ways, but a complex difference quotient divides by the same complex increment used to approach the point. For the limit to exist, all approaches to the point must agree on one complex number. This is the obstruction that the definition isolates. [definition: Complex Derivative] Let $\Omega \subset \mathbb C$ be open, let $f: \Omega \to \mathbb C$, and let $z_0 \in \Omega$. The complex derivative of $f$ at $z_0$ is the complex number $f'(z_0)$, when it exists, defined by \begin{align*} f'(z_0) = \lim_{h \to 0} \frac{f(z_0+h)-f(z_0)}{h}, \end{align*} where $h \in \mathbb C \setminus \{0\}$ and $z_0+h \in \Omega$. [/definition] ### Holomorphicity on a Domain Which functions should complex analysis study on an open domain, rather than at a single isolated point? We need a name for functions whose complex derivative exists everywhere in the region, because contour integration, [power series](/page/Power%20Series) expansions, and conformal geometry all begin with this local-everywhere condition. [definition: Holomorphic Function] Let $\Omega \subset \mathbb C$ be open. A function $f: \Omega \to \mathbb C$ is holomorphic on $\Omega$ if the complex derivative $f'(z)$ exists for every $z \in \Omega$. [/definition] The definition says nothing about power series or contour integrals, yet those are consequences of holomorphy. The Cauchy-Riemann equations are the first bridge from real multivariable calculus to this much stronger complex theory. ## The Linear Algebra Behind the Equations ### Complex-Linear Maps The quickest way to understand the Cauchy-Riemann equations is to forget limits for a moment and ask which real linear maps $\mathbb R^2 \to \mathbb R^2$ can be multiplication by a complex number. This linear algebra is the local model for every holomorphic map. A real linear map can stretch the $x$-axis and $y$-axis independently, shear, rotate, or reflect. Multiplication by $a+ib$ can only scale and rotate at the same time. It sends $1$ to $a+ib$ and sends $i$ to $i(a+ib)=-b+ia$, so the image of the second basis vector is forced by the image of the first. [definition: Complex-Linear Real Map] A real linear map $A: \mathbb R^2 \to \mathbb R^2$ is complex-linear if there exists $\lambda \in \mathbb C$ such that, under the identification $\mathbb R^2 \cong \mathbb C$, one has \begin{align*} A(z) = \lambda z \end{align*} for every $z \in \mathbb C$. [/definition] To use this definition as a computational test, we need to recognize complex-linearity from the entries of a real matrix. The key observation is that if $A(1)=a+ib$, then complex-linearity forces $A(i)=i(a+ib)=-b+ia$. Thus the first column of the real matrix determines the second column. The next result gives that recognition rule and accounts for the signs in the Cauchy-Riemann equations. [quotetheorem:4987] This theorem explains why the equations appear with exactly those signs. A complex-linear first-order map has only two real degrees of freedom, not four: once the image of the real direction is known, the image of the imaginary direction is forced by multiplication by $i$. The Cauchy-Riemann equations are the same constraint applied to the Jacobian of a function. ### The Model Example [example: Multiplication by a Complex Number] Let $\lambda = a+ib \in \mathbb C$, and define \begin{align*} f: \mathbb C &\to \mathbb C \\ z &\mapsto \lambda z. \end{align*} For $z=x+iy$, expansion of the product gives \begin{align*} f(x+iy) &= (a+ib)(x+iy) \\ &= ax + aiy + ibx + i^2by \\ &= ax + iay + ibx - by \\ &= (ax-by) + i(ay+bx) \\ &= (ax-by) + i(bx+ay). \end{align*} Hence the real and imaginary parts are \begin{align*} u(x,y) &= ax-by, \\ v(x,y) &= bx+ay. \end{align*} Their first partial derivatives are \begin{align*} \partial_x u(x,y) &= \partial_x(ax-by)=a, \\ \partial_y v(x,y) &= \partial_y(bx+ay)=a, \\ \partial_y u(x,y) &= \partial_y(ax-by)=-b, \\ \partial_x v(x,y) &= \partial_x(bx+ay)=b, \end{align*} so \begin{align*} \partial_x u(x,y) &= \partial_y v(x,y), \\ \partial_y u(x,y) &= -\partial_x v(x,y) \end{align*} for every $(x,y) \in \mathbb R^2$. Thus the Cauchy-Riemann equations hold everywhere. The complex derivative can also be read directly from the difference quotient. For every $z \in \mathbb C$ and every $h \ne 0$, \begin{align*} \frac{f(z+h)-f(z)}{h} &= \frac{\lambda(z+h)-\lambda z}{h} \\ &= \frac{\lambda z+\lambda h-\lambda z}{h} \\ &= \frac{\lambda h}{h} \\ &= \lambda. \end{align*} Therefore the quotient has limit $\lambda$ as $h \to 0$, so $f'(z)=\lambda$ at every point. This is the basic model for a complex-linear first-order map: it scales and rotates by the fixed multiplier $\lambda$. [/example] The example is the model case. Every [holomorphic function](/page/Holomorphic%20Function) looks like this to first order at each point, although the multiplier may vary from point to point. ## Necessity: Holomorphic Functions Satisfy the Equations ### Directional Limits Force the Equations The first serious result is that complex differentiability forces the Cauchy-Riemann equations. The reason is that the same difference quotient must have the same limit along the real and imaginary axes. [quotetheorem:333] The theorem should be read as a diagnostic. Complex differentiability is stronger than the existence of partial derivatives: it asks for one complex-linear first-order approximation, independent of the direction of approach. If a proposed holomorphic function fails even one of the two equations, then that complex-linear approximation cannot exist. This is often the fastest way to detect hidden dependence on $\bar z$. ### A Diagnostic Failure [example: The Conjugation Map Fails the Test] For \begin{align*} f: \mathbb C &\to \mathbb C \\ z &\mapsto \bar z, \end{align*} take $z=x+iy$. Since $\overline{x+iy}=x-iy$, we have \begin{align*} f(x+iy)=x-iy=x+i(-y), \end{align*} so the real and imaginary parts are \begin{align*} u(x,y)&=x,\\ v(x,y)&=-y. \end{align*} At an arbitrary point $(x_0,y_0)$, the four first partial derivatives are \begin{align*} \partial_x u(x_0,y_0) &=\lim_{t\to 0}\frac{u(x_0+t,y_0)-u(x_0,y_0)}{t} \\ &=\lim_{t\to 0}\frac{(x_0+t)-x_0}{t} \\ &=1,\\ \partial_y v(x_0,y_0) &=\lim_{t\to 0}\frac{v(x_0,y_0+t)-v(x_0,y_0)}{t} \\ &=\lim_{t\to 0}\frac{-(y_0+t)-(-y_0)}{t} \\ &=-1, \end{align*} and \begin{align*} \partial_y u(x_0,y_0) &=\lim_{t\to 0}\frac{u(x_0,y_0+t)-u(x_0,y_0)}{t} \\ &=\lim_{t\to 0}\frac{x_0-x_0}{t} \\ &=0,\\ \partial_x v(x_0,y_0) &=\lim_{t\to 0}\frac{v(x_0+t,y_0)-v(x_0,y_0)}{t} \\ &=\lim_{t\to 0}\frac{(-y_0)-(-y_0)}{t} \\ &=0. \end{align*} Thus \begin{align*} \partial_y u(x_0,y_0)&=-\partial_x v(x_0,y_0), \end{align*} because both sides are $0$, but \begin{align*} \partial_x u(x_0,y_0)&=1 \ne -1=\partial_y v(x_0,y_0). \end{align*} The first Cauchy-Riemann equation fails at every point, so the conjugation map is not complex differentiable anywhere, even though as a real map it is the linear map $(x,y)\mapsto (x,-y)$. [/example] The failure of $z \mapsto \bar z$ is not a marginal pathology. It is the prototype for the anti-holomorphic direction. The equations distinguish the $z$ direction from the $\bar z$ direction. ## Sufficiency and the Regularity Hypothesis The converse direction is more delicate. The Cauchy-Riemann equations involve only partial derivatives, while complex differentiability requires a full two-dimensional limit. Partial derivatives alone do not control all directions of approach. A regularity hypothesis is needed to convert the equations into a genuine linear approximation. [quotetheorem:333] This is the practical theorem used in calculations. The key hypothesis is real differentiability, not just the pointwise existence of the four partial derivatives. Under that hypothesis, the Cauchy-Riemann equations say that the real first-order approximation is complex-linear, so the function has a complex derivative. In routine examples, continuity of the first partial derivatives is a convenient way to guarantee the needed real differentiability. [example: A Polynomial Checked by the Equations] Let \begin{align*} f: \mathbb C &\to \mathbb C \\ z &\mapsto z^2. \end{align*} We compute its real and imaginary parts and then check the Cauchy-Riemann equations. For $z=x+iy$, \begin{align*} f(x+iy) &= (x+iy)^2 \\ &= (x+iy)(x+iy) \\ &= x^2 + xiy + iyx + (iy)(iy) \\ &= x^2 + ixy + ixy + i^2y^2 \\ &= x^2 + 2ixy - y^2 \\ &= (x^2-y^2) + i(2xy). \end{align*} Thus \begin{align*} u(x,y)&=x^2-y^2,\\ v(x,y)&=2xy. \end{align*} The four first partial derivatives are \begin{align*} \partial_x u(x,y) &=\partial_x(x^2-y^2) \\ &=2x,\\ \partial_y v(x,y) &=\partial_y(2xy) \\ &=2x, \end{align*} and \begin{align*} \partial_y u(x,y) &=\partial_y(x^2-y^2) \\ &=-2y,\\ \partial_x v(x,y) &=\partial_x(2xy) \\ &=2y. \end{align*} Therefore \begin{align*} \partial_x u(x,y)&=2x=\partial_y v(x,y),\\ \partial_y u(x,y)&=-2y=-\partial_x v(x,y) \end{align*} for every $(x,y)\in \mathbb R^2$. Since $u$ and $v$ are polynomials in $x$ and $y$, their first partial derivatives are continuous on $\mathbb R^2$, so the real differentiability hypothesis in the Cauchy-Riemann characterization is satisfied. Hence $f$ is holomorphic on $\mathbb C$, and \begin{align*} f'(z) &=\partial_x u(x,y)+i\partial_x v(x,y) \\ &=2x+i(2y) \\ &=2(x+iy) \\ &=2z. \end{align*} So the Cauchy-Riemann equations recover the expected derivative of the squaring map at every point. [/example] The continuity condition in the criterion is not cosmetic. It supplies the real differentiability estimate needed to assemble the two coordinate derivatives into a complex derivative. [example: Pointwise Equations Without Holomorphy] Define \begin{align*} f: \mathbb C &\to \mathbb C \\ z &\mapsto \begin{cases} \dfrac{x^3}{x^2+y^2} + i\dfrac{y^3}{x^2+y^2}, & z=x+iy \ne 0, \\ 0, & z=0. \end{cases} \end{align*} Thus $f=u+iv$, where $u(0,0)=v(0,0)=0$ and, for $(x,y)\ne(0,0)$, \begin{align*} u(x,y)&=\frac{x^3}{x^2+y^2},\\ v(x,y)&=\frac{y^3}{x^2+y^2}. \end{align*} At the origin, the $x$-partials are \begin{align*} \partial_x u(0,0) &=\lim_{t\to 0}\frac{u(t,0)-u(0,0)}{t} \\ &=\lim_{t\to 0}\frac{\dfrac{t^3}{t^2+0^2}-0}{t} \\ &=\lim_{t\to 0}\frac{t-0}{t} \\ &=1, \end{align*} and \begin{align*} \partial_x v(0,0) &=\lim_{t\to 0}\frac{v(t,0)-v(0,0)}{t} \\ &=\lim_{t\to 0}\frac{\dfrac{0^3}{t^2+0^2}-0}{t} \\ &=\lim_{t\to 0}\frac{0}{t} \\ &=0. \end{align*} The $y$-partials are \begin{align*} \partial_y u(0,0) &=\lim_{t\to 0}\frac{u(0,t)-u(0,0)}{t} \\ &=\lim_{t\to 0}\frac{\dfrac{0^3}{0^2+t^2}-0}{t} \\ &=\lim_{t\to 0}\frac{0}{t} \\ &=0, \end{align*} and \begin{align*} \partial_y v(0,0) &=\lim_{t\to 0}\frac{v(0,t)-v(0,0)}{t} \\ &=\lim_{t\to 0}\frac{\dfrac{t^3}{0^2+t^2}-0}{t} \\ &=\lim_{t\to 0}\frac{t}{t} \\ &=1. \end{align*} Therefore \begin{align*} \partial_x u(0,0)&=1=\partial_y v(0,0),\\ \partial_y u(0,0)&=0=-\partial_x v(0,0), \end{align*} so the Cauchy-Riemann equations hold at $0$. We now test the complex derivative at $0$. Since $f(0)=0$, the difference quotient is $f(h)/h$ for $h\ne 0$. Along the real axis, write $h=t$ with $t\in\mathbb R$ and $t\ne 0$. Then \begin{align*} f(t) &=\frac{t^3}{t^2+0^2}+i\frac{0^3}{t^2+0^2} \\ &=t, \end{align*} so \begin{align*} \frac{f(t)}{t} &=\frac{t}{t} \\ &=1. \end{align*} Along the diagonal, write $h=t+it$ with $t\in\mathbb R$ and $t\ne 0$. Then $x=t$ and $y=t$, so \begin{align*} f(t+it) &=\frac{t^3}{t^2+t^2}+i\frac{t^3}{t^2+t^2} \\ &=\frac{t^3}{2t^2}+i\frac{t^3}{2t^2} \\ &=\frac{t}{2}+i\frac{t}{2}. \end{align*} Hence \begin{align*} \frac{f(t+it)}{t+it} &=\frac{\frac{t}{2}+i\frac{t}{2}}{t+it} \\ &=\frac{\frac{t}{2}(1+i)}{t(1+i)} \\ &=\frac{1}{2}. \end{align*} The same difference quotient has path value $1$ along the real axis and path value $1/2$ along the diagonal, so its limit as $h\to 0$ does not exist. Thus $f$ satisfies the Cauchy-Riemann equations at the origin but is not complex differentiable there. [/example] This example shows why the theorem asks for more than pointwise equations. The Cauchy-Riemann equations are a first-order compatibility condition, but they become a differentiability criterion only when the real first-order approximation is under control. ## Wirtinger Derivatives and Dependence on $z$ versus $\bar z$ The equations become more transparent if we introduce coordinates adapted to $z$ and $\bar z$. Instead of treating $x$ and $y$ as the only basic variables, Wirtinger calculus separates the holomorphic and anti-holomorphic directions. [definition: Wirtinger Derivatives] Let $\Omega \subset \mathbb C$ be open. The Wirtinger derivative operators are the maps \begin{align*} \partial_z &: C^1(\Omega;\mathbb C) \to C^0(\Omega;\mathbb C), \\ \partial_{\bar z} &: C^1(\Omega;\mathbb C) \to C^0(\Omega;\mathbb C) \end{align*} defined for $f \in C^1(\Omega;\mathbb C)$ by \begin{align*} \partial_z f &= \frac{1}{2}\left(\partial_x f - i\partial_y f\right), \\ \partial_{\bar z} f &= \frac{1}{2}\left(\partial_x f + i\partial_y f\right). \end{align*} [/definition] The notation treats $z$ and $\bar z$ as independent first-order variables. To make this more than notation, we need a criterion that identifies which of these two derivatives obstructs complex differentiability. Although the operators above are defined globally on $C^1(\Omega;\mathbb C)$, the same formulas can also be read pointwise whenever the four first partial derivatives at the point exist. The next result rewrites the Cauchy-Riemann equations as the vanishing of the anti-holomorphic derivative in that pointwise sense. [quotetheorem:4990] This form is indispensable in several complex variables and partial differential equations. The operator $\partial_{\bar z}$ measures exactly the obstruction to holomorphy. [example: Separating $z$ and $\bar z$] Let \begin{align*} f: \mathbb C &\to \mathbb C \\ z &\mapsto z^2\bar z. \end{align*} We compute $\partial_{\bar z}f$ from the Wirtinger definition. Since $z=x+iy$ and $\bar z=x-iy$, \begin{align*} \partial_{\bar z}z &=\frac{1}{2}\left(\partial_x z+i\partial_y z\right) \\ &=\frac{1}{2}\left(\partial_x(x+iy)+i\partial_y(x+iy)\right) \\ &=\frac{1}{2}\left(1+i(i)\right) \\ &=\frac{1}{2}(1-1) \\ &=0, \end{align*} while \begin{align*} \partial_{\bar z}\bar z &=\frac{1}{2}\left(\partial_x \bar z+i\partial_y \bar z\right) \\ &=\frac{1}{2}\left(\partial_x(x-iy)+i\partial_y(x-iy)\right) \\ &=\frac{1}{2}\left(1+i(-i)\right) \\ &=\frac{1}{2}(1+1) \\ &=1. \end{align*} Using the product rule, \begin{align*} \partial_{\bar z}(z^2\bar z) &=\partial_{\bar z}(z^2)\bar z+z^2\partial_{\bar z}(\bar z) \\ &=2z(\partial_{\bar z}z)\bar z+z^2\cdot 1 \\ &=2z\cdot 0\cdot \bar z+z^2 \\ &=z^2. \end{align*} Thus \begin{align*} \partial_{\bar z}f(z)=0 \end{align*} if and only if $z^2=0$, which happens if and only if $z=0$. By the *[Wirtinger Form of the Cauchy-Riemann Equations](/theorems/4990)*, the Cauchy-Riemann equations hold only at $0$ and fail at every nonzero point. This computation separates the holomorphic factor $z^2$ from the anti-holomorphic factor $\bar z$: the dependence on $\bar z$ is exactly what survives in $\partial_{\bar z}f$. [/example] In one complex variable this notation may look optional. In higher dimensions it becomes the language of the $\bar\partial$ problem, where solving $\bar\partial u=f$ is a central analytic task. ## Harmonicity and Conjugate Functions The Cauchy-Riemann equations do more than characterize holomorphicity. They connect complex analysis to potential theory by forcing the real and imaginary parts of holomorphic functions to be harmonic. [definition: Harmonic Function] Let $\Omega \subset \mathbb R^2$ be open. A function $u \in C^2(\Omega;\mathbb R)$ is harmonic on $\Omega$ if \begin{align*} \Delta u = \partial_x^2 u + \partial_y^2 u = 0 \end{align*} on $\Omega$. [/definition] Once harmonicity has been named, the natural question is why holomorphic functions produce such functions. The Cauchy-Riemann equations give the answer: they pair the $x$ and $y$ derivatives so that the two second-derivative contributions in the Laplacian cancel. In the full theory, holomorphic functions are automatically smooth, so the theorem can be stated directly for holomorphic functions. [quotetheorem:336] This theorem points in one direction: holomorphic functions produce harmonic functions. The reverse direction asks for a harmonic function to be the real part of a holomorphic function. The missing data is the imaginary part, and the Cauchy-Riemann equations prescribe its gradient. [definition: Harmonic Conjugate] Let $\Omega \subset \mathbb C$ be open, and let $u: \Omega \to \mathbb R$ be harmonic. A harmonic conjugate of $u$ on $\Omega$ is a function $v: \Omega \to \mathbb R$ such that $u+iv: \Omega \to \mathbb C$ is holomorphic. [/definition] The definition raises an existence problem: if the equations prescribe $\partial_x v=-\partial_y u$ and $\partial_y v=\partial_x u$, when can those prescribed derivatives be integrated to a single-valued function $v$? The local answer is positive, because harmonicity is exactly the compatibility condition for this first-order system. [quotetheorem:4992] The local theorem is enough for many computations, but global conjugates require a condition on periods. The annulus gives the standard obstruction. [example: A Harmonic Function With No Global Conjugate] Let $\Omega=\mathbb C\setminus\{0\}$ and define $u(z)=\log |z|$. For $z=x+iy\ne 0$, \begin{align*} u(x,y) &=\log\sqrt{x^2+y^2} \\ &=\frac{1}{2}\log(x^2+y^2). \end{align*} Its first partial derivatives are \begin{align*} \partial_x u(x,y) &=\frac{1}{2}\cdot \frac{2x}{x^2+y^2} =\frac{x}{x^2+y^2},\\ \partial_y u(x,y) &=\frac{1}{2}\cdot \frac{2y}{x^2+y^2} =\frac{y}{x^2+y^2}. \end{align*} Differentiating once more gives \begin{align*} \partial_x^2 u(x,y) &=\partial_x\left(\frac{x}{x^2+y^2}\right) \\ &=\frac{(x^2+y^2)-x(2x)}{(x^2+y^2)^2} \\ &=\frac{y^2-x^2}{(x^2+y^2)^2}, \end{align*} and \begin{align*} \partial_y^2 u(x,y) &=\partial_y\left(\frac{y}{x^2+y^2}\right) \\ &=\frac{(x^2+y^2)-y(2y)}{(x^2+y^2)^2} \\ &=\frac{x^2-y^2}{(x^2+y^2)^2}. \end{align*} Therefore \begin{align*} \Delta u(x,y) &=\partial_x^2u(x,y)+\partial_y^2u(x,y) \\ &=\frac{y^2-x^2}{(x^2+y^2)^2} +\frac{x^2-y^2}{(x^2+y^2)^2} \\ &=0, \end{align*} so $u$ is harmonic on $\Omega$. Locally, choose a continuous branch of the argument and write $v(z)=\arg z$ on that neighbourhood. Then \begin{align*} \log z=\log |z|+i\arg z \end{align*} on the chosen branch, so $v$ is a local harmonic conjugate of $u$. There cannot be a single-valued harmonic conjugate on all of $\Omega$. Suppose such a function $v:\Omega\to\mathbb R$ existed. Since $u+iv$ would be holomorphic, the Cauchy-Riemann equations would give \begin{align*} \partial_x v(x,y)&=-\partial_y u(x,y)=-\frac{y}{x^2+y^2},\\ \partial_y v(x,y)&=\partial_x u(x,y)=\frac{x}{x^2+y^2}. \end{align*} Along the unit circle $\gamma(t)=e^{it}=\cos t+i\sin t$, with $0\le t\le 2\pi$, the chain rule gives \begin{align*} \frac{d}{dt}v(\gamma(t)) &=\partial_xv(\cos t,\sin t)(-\sin t) +\partial_yv(\cos t,\sin t)(\cos t) \\ &=(-\sin t)(-\sin t)+(\cos t)(\cos t) \\ &=\sin^2 t+\cos^2 t \\ &=1. \end{align*} Hence \begin{align*} v(\gamma(2\pi))-v(\gamma(0)) &=\int_0^{2\pi}\frac{d}{dt}v(\gamma(t))\,dt \\ &=\int_0^{2\pi}1\,dt \\ &=2\pi. \end{align*} But $\gamma(2\pi)=\gamma(0)=1$, so the left side is $v(1)-v(1)=0$, a contradiction. Thus $\log|z|$ has local harmonic conjugates, but no single-valued harmonic conjugate on all of $\mathbb C\setminus\{0\}$. [/example] This example shows that the Cauchy-Riemann equations are local differential equations, while holomorphic primitives and conjugates may depend on global topology. ## Geometry of Holomorphic Maps The matrix form of the equations has a geometric consequence: wherever the derivative is nonzero, a holomorphic function preserves angles and orientation. This is the beginning of conformal geometry. [definition: Conformal Map] Let $\Omega_1, \Omega_2 \subset \mathbb C$ be open. A function $f: \Omega_1 \to \Omega_2$ is conformal if $f$ is holomorphic, bijective, and $f'(z) \ne 0$ for every $z \in \Omega_1$. [/definition] The nonvanishing derivative condition prevents local collapse. Without it, distinct tangent directions can be crushed together, as happens at a critical point such as the origin for $z\mapsto z^2$. When the derivative is nonzero, the first-order model is multiplication by a nonzero complex number, so every tangent vector is rotated and scaled by the same amount. The remaining question is whether this infinitesimal model really controls the angle between curves through the point. [quotetheorem:334] This theorem explains why the Cauchy-Riemann equations are geometric, not only computational. They eliminate shears and reflections from the first-order picture. [example: The Squaring Map Near and At the Origin] Let \begin{align*} f: \mathbb C &\to \mathbb C \\ z &\mapsto z^2. \end{align*} At an arbitrary point $z_0\in \mathbb C$, the difference quotient is \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*} for $h\ne 0$. Therefore \begin{align*} f'(z_0) &=\lim_{h\to 0}(2z_0+h) \\ &=2z_0. \end{align*} If $z_0\ne 0$, then $2z_0\ne 0$, so by *Holomorphic Maps Preserve Angles Where the Derivative Is Nonzero*, the squaring map preserves oriented angles between $C^1$ curves passing through $z_0$ with nonzero tangent vectors. At the origin, the same formula gives \begin{align*} f'(0)=2\cdot 0=0, \end{align*} so the nonzero-derivative hypothesis is absent. To see the geometric effect, fix an angle $\theta$ and consider the ray \begin{align*} \gamma_\theta(t)=te^{i\theta} \end{align*} for $t\ge 0$. Its image under $f$ is \begin{align*} (f\circ\gamma_\theta)(t) &=f(te^{i\theta}) \\ &=(te^{i\theta})^2 \\ &=t^2 e^{i\theta}e^{i\theta} \\ &=t^2e^{2i\theta}. \end{align*} Thus a ray with direction angle $\theta$ is sent to a ray with direction angle $2\theta$. For example, the two rays with angles $0$ and $\pi/4$ meet at angle $\pi/4$, while their images have angles $0$ and $\pi/2$, so the image angle is $\pi/2=2(\pi/4)$. The squaring map is conformal away from $0$, but at the critical point $0$ its first-order part vanishes and angles at the origin are doubled rather than preserved. [/example] The example marks the boundary of conformality. Holomorphy controls the form of the derivative, but a zero derivative removes the invertible first-order approximation. ## Beyond and Connected Topics The Cauchy-Riemann equations are the entry point to the larger structure of one-variable complex analysis. In [Cambridge IB Complex Analysis](/page/Cambridge%20IB%20Complex%20Analysis), they sit beside contour integration, [Cauchy's theorem](/page/Cauchy's%20Theorem), power series, residues, and [conformal maps](/page/Conformal%20Maps). The equations provide the local differential test; Cauchy's integral theory explains why that local condition has global consequences. In several complex variables, the single equation $\partial_{\bar z}f=0$ becomes a system $\partial_{\bar z_j}f=0$ for $j=1,\ldots,n$. The geometry changes sharply: domains of holomorphy, pseudoconvexity, and extension phenomena have no direct analogue in one variable. The course [Androma Several Complex Variables I: Domains and Holomorphy](/page/Androma%20Several%20Complex%20Variables%20I%3A%20Domains%20and%20Holomorphy) is the natural continuation from the one-variable equations to higher-dimensional holomorphic geometry. The $\bar\partial$ operator also leads to sheaf-theoretic methods. In [Several Complex Variables II: Sheaves and Stein Theory](/page/Several%20Complex%20Variables%20II%3A%20Sheaves%20and%20Stein%20Theory), the local solvability and gluing of holomorphic functions are organized through sheaves and cohomology. The Cauchy-Riemann equations remain present, but the emphasis shifts from local derivatives to global analytic structure. A further direction is the analytic theory of the $\bar\partial$ equation with estimates. The course [Several Complex Variables III: L² Methods and Applications](/page/Several%20Complex%20Variables%20III%3A%20L%C2%B2%20Methods%20and%20Applications) develops [Hilbert space](/page/Hilbert%20Space) methods for solving $\bar\partial u=f$. There the Cauchy-Riemann equations become part of an operator-theoretic framework involving adjoints, closed range estimates, and applications to complex geometry. ## 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 II: Sheaves and Stein Theory](/page/Several%20Complex%20Variables%20II%3A%20Sheaves%20and%20Stein%20Theory). Androma, [Several Complex Variables III: L² Methods and Applications](/page/Several%20Complex%20Variables%20III%3A%20L%C2%B2%20Methods%20and%20Applications). Ahlfors, *Complex Analysis* (1979). Conway, *Functions of One Complex Variable I* (1978). Stein and Shakarchi, *Complex Analysis* (2003).