Analytic Functions and the Cauchy-Riemann Equations

The equations are at work wherever a 2D Laplace problem meets an awkward boundary. Because a holomorphic map with $f'\neq0$ preserves both angles and harmonicity, conformal mapping transplants a hard Dirichlet problem on a complicated domain onto a disc or half-plane where the answer is known — the route for the capacitance of odd-shaped conductors, ideal flow past a Joukowski aerofoil, and the field around microstrip and coplanar-waveguide cross-sections, where Schwarz-Christoffel maps turn polygonal conductors into tractable geometry. In mathematical physics the $\bar\partial$ ("dee-bar") operator of equation (5) is a central object: the inhomogeneous equation $\partial_{\bar z}f=g$ underlies pseudo-analytic functions, integrable systems, and the several-complex-variables machinery of quantum field theory. Two-dimensional conformal field theory — the engine of critical phenomena and of the string worldsheet — rests directly on the decoupling of holomorphic from anti-holomorphic parts, which is literally $\partial_{\bar z}f=0$. Numerically, conformal-mapping algorithms flatten cortical surfaces for brain imaging, while Laplacian-growth and Hele-Shaw studies of viscous fingering remain a source of open problems about finite-time singularities of smooth conformal maps.

Zoom in on a holomorphic map until only the neighbourhood of $z_0$ is in view. To first order $f$ becomes a linear map of the tangent plane, and the question is which linear maps are allowed. A general real-linear map has four free entries — it can stretch differently along two axes and shear the angle between them, turning a small circle into a tilted ellipse. Complex differentiability rules almost all of these out. Because the derivative must be one complex number $f'(z_0)$, and multiplication by a complex number is exactly a rotation-plus-scaling, the only survivors spin the neighbourhood by $\arg f'(z_0)$ and magnify it by $|f'(z_0)|$ — no shear, no differential stretch. A circle stays a circle, angles are preserved. This is the whole content of the Cauchy-Riemann equations, read as a constraint on the Jacobian.

Why does the direction of approach carry the whole story? Slide into $z_0$ horizontally, $\Delta z=\Delta x$, and the difference quotient reads $u_x+iv_x$; slide in vertically, $\Delta z=i\,\Delta y$, and dividing by $i$ gives $v_y-iu_y$. For one derivative to exist these must agree, and matching real to real and imaginary to imaginary is the Cauchy-Riemann system. The second figure makes the rotation-and-scaling literal for the simplest nontrivial map, $w=z^2$, at a concrete point where the numbers can be checked by hand.

Because the amplitwist scales all directions equally, the level curves of $u$ and $v$ stay perpendicular after mapping — the geometric face of harmonic conjugacy, and precisely the orthogonality of equipotentials and field lines in 2D electrostatics. One equation, three readings: an analytic condition on partials, a geometric amplitwist, and a physical statement about potentials.

The complex derivative is defined by the same limit as in real calculus, with the decisive difference that $\Delta z$ is a complex increment and the limit is required to exist independently of how $\Delta z$ approaches zero.

where: $f$ — complex function $f:\Omega\subset\mathbb{C}\to\mathbb{C}$; $z_0$ — interior point of the domain $\Omega$; $\Delta z$ — complex increment, $\Delta z\to0$ from any direction; $f'(z_0)$ — the single complex number the quotient must approach for every $\arg\Delta z$. Reading: if two approach directions give different answers, $f'(z_0)$ does not exist — this is the sieve everything else passes through. Evaluating that one limit along the real and imaginary axes and demanding they agree gives the Cauchy-Riemann equations, the local coordinate form of holomorphy.

where: $u=\operatorname{Re}f$ and $v=\operatorname{Im}f$ as functions of $(x,y)$; $u_x,u_y,v_x,v_y$ — first partial derivatives. Reading: the two equations couple the parts so tightly that fixing $u$ almost fixes $v$ — the source of harmonic conjugacy. Once the equations hold, the derivative can be written four equivalent ways in terms of partials, which is often the fastest route to $f'$ in practice.

where: $f'(z)$ — the complex derivative; all four expressions coincide precisely when Cauchy-Riemann holds. Reading: you only ever need the partials of one of $u$ or $v$ to get $f'$. In polar coordinates $z=re^{i\theta}$ the same conditions take a form better suited to powers, roots, and logarithms.

where: $r,\theta$ — modulus and argument of $z$; $u,v$ — real and imaginary parts as functions of $(r,\theta)$. Reading: for $z^n$ or $\log z$ the polar form is where the algebra is cleanest. Regarding $z$ and $\bar z$ as independent variables through the Wirtinger operators compresses both equations of (2) into a single elegant statement.

where: $\partial/\partial\bar z$ — the Wirtinger derivative, treating $z,\bar z$ as independent; $\partial/\partial z=\tfrac12(\partial_x-i\partial_y)$ equals $f'$ for holomorphic $f$. Reading: holomorphy is "no $\bar z$ dependence", which instantly flags $\bar z$, $\operatorname{Re}z$, and $|z|^2$ as non-analytic. Differentiating the Cauchy-Riemann pair and using equality of mixed partials makes both parts harmonic — the bridge to potential theory.

where: $\nabla^2$ — the two-dimensional Laplacian. Reading: every analytic function manufactures a pair of harmonic functions, and every harmonic $u$ is locally the real part of some analytic $f$. The real Jacobian of the map $(x,y)\mapsto(u,v)$ inherits the amplitwist pattern, and its determinant is the squared derivative modulus — always non-negative, so orientation is preserved.

where: $J$ — Jacobian of the underlying real map; $\det J$ — local area magnification. Reading: $\det J=|f'|^2\ge0$ is why conformal maps never flip handedness, and $f'\neq0$ is exactly the condition for local invertibility. Finally, the deepest fact: on an open set, holomorphy is equivalent to analyticity — local representability by a convergent power series — a rigidity with no real-variable counterpart.

where: $a_n$ — Taylor coefficients; $\Omega$ — open connected domain; the series converges on the largest disc in $\Omega$ about $z_0$. Reading: one complex derivative on an open set silently grants infinitely many, plus a Taylor series — the whole of the following chapter rests here.

Special structural cases sharpen what analyticity permits and forbids — several surprising precisely because no real-variable analogue exists.

1 · The amplitwist of z² at a point (confidence builder)

Take $f(z)=z^2$ at $z_0=1+i$. Verify the Cauchy-Riemann equations, compute $f'(z_0)$, and state the local geometric action — the rotation angle, the scale factor, and the area magnification.

Step 1. Split into real and imaginary parts: $z^2=(x+iy)^2=(x^2-y^2)+i(2xy)$, so $u=x^2-y^2$ and $v=2xy$. The partials are $u_x=2x,\ v_y=2x,\ u_y=-2y,\ v_x=2y$, giving $u_x=v_y$ and $u_y=-v_x$ for all $(x,y)$ — with continuous partials, so $z^2$ is entire.

Step 2. The derivative from equation (3): $f'(z)=u_x+iv_x=2x+i\,2y=2(x+iy)=\mathbf{2z}$. At $z_0=1+i$ this is $f'(z_0)=2(1+i)=\mathbf{2+2i}$, and the image point is $f(z_0)=(1+i)^2=\mathbf{2i}$.

Step 3. Read off the amplitwist. The modulus $|f'(z_0)|=|2+2i|=2\sqrt2\approx\mathbf{2.83}$ is the local scale factor and the argument $\arg(2+2i)=\mathbf{45°}$ is the local rotation, so a tiny figure planted at $1+i$ is turned $45°$ counter-clockwise and magnified $2.83\times$ on its way to $2i$.

Step 4. Cross-check the area: the Jacobian determinant is $\det J=|f'(z_0)|^2=(2\sqrt2)^2=\mathbf{8}$, consistent with a linear scale of $2.83$ since $2.83^2\approx8$. f′(z₀) = 2 + 2i, scale 2.83, rotation 45°, area ×8.

2 · Is the exponential analytic, and what is its derivative?

Test $f(z)=e^z$ for holomorphy using the Cauchy-Riemann equations, and identify $f'$.

Step 1. With $e^z=e^{x+iy}=e^x(\cos y+i\sin y)$, read off $u=e^x\cos y$ and $v=e^x\sin y$. The partials are $u_x=e^x\cos y,\ u_y=-e^x\sin y,\ v_x=e^x\sin y,\ v_y=e^x\cos y$.

Step 2. Check the equations: $u_x=e^x\cos y=v_y$ and $u_y=-e^x\sin y=-v_x$, both holding for every $(x,y)$ with continuous partials. Hence $e^z$ is entire.

Step 3. Compute the derivative: $f'(z)=u_x+iv_x=e^x\cos y+i\,e^x\sin y=e^x(\cos y+i\sin y)=e^x e^{iy}=\mathbf{e^z}$. The exponential is its own derivative in the complex plane, exactly as on the real line — a first hint that the real and complex exponentials are one analytic object continued off the axis.

3 · Building a harmonic conjugate

Given $u(x,y)=x^3-3xy^2$, confirm it is harmonic, construct a conjugate $v$ so that $f=u+iv$ is analytic, and identify $f$.

Step 1. Check harmonicity: $u_{xx}=6x$ and $u_{yy}=-6x$, so $\nabla^2u=6x-6x=0$. A harmonic $u$ is a legitimate real part, so a conjugate exists.

Step 2. Integrate the first Cauchy-Riemann equation. From $v_y=u_x=3x^2-3y^2$, integrate in $y$: $v=3x^2y-y^3+g(x)$, where $g$ is a function of $x$ alone (the "constant" of $y$-integration).

Step 3. Fix $g$ with the second equation. Now $v_x=6xy+g'(x)$ must equal $-u_y=6xy$, forcing $g'(x)=0$, so $g$ is a constant. Hence $\mathbf{v=3x^2y-y^3+C}$.

Step 4. Recognise the function. Assembling $f=u+iv=(x^3-3xy^2)+i(3x^2y-y^3)=(x+iy)^3=\mathbf{z^3}$ (up to the additive constant $iC$). The conjugate was fixed uniquely up to that constant — the rigidity of harmonic conjugacy. v = 3x²y − y³ + C, and f = z³.

The Cauchy-Riemann equations are older than the theory named after them. They surfaced first in hydrodynamics, were used as a computational trick for real integrals, and only later became the defining axiom of a whole branch of mathematics — a story that runs from d'Alembert's fluids through Cauchy's rigour to Riemann's geometry.

Texts. Stone and Goldbart, Mathematics for Physics, Ch. 17 (Complex Analysis I) · Arfken, Weber and Harris, Mathematical Methods for Physicists, 7th ed., Ch. 11 · Riley, Hobson and Bence, Mathematical Methods for Physics and Engineering, 3rd ed., Ch. 24 · Ahlfors, Complex Analysis, 3rd ed., Ch. 2 · Needham, Visual Complex Analysis, Ch. 4 (the amplitwist picture) · Byron and Fuller, Mathematics of Classical and Quantum Physics, Ch. 6.

Provenance. Definitions and theorems follow Stone and Goldbart Ch. 17, Arfken-Weber-Harris Ch. 11, and Riley-Hobson-Bence Ch. 24; the sufficiency theorem is cross-checked against Ahlfors Ch. 2, and the amplitwist reading against Needham Ch. 4. A verifier should: (1) re-derive the equations by equating the horizontal and vertical difference quotients, recovering $f'=u_x+iv_x$; (2) confirm the sufficiency proof's fusion step $u_y+iv_y=i(u_x+iv_x)$ under Cauchy-Riemann; (3) verify the polar form via the chain rule; (4) check the worked numbers — $z^2$ at $1+i$ giving $f'=2+2i$, $|f'|=2\sqrt2\approx2.83$, $\det J=8$; $e^z$ entire with $f'=e^z$; and $u=x^3-3xy^2$ yielding conjugate $v=3x^2y-y^3$ and $f=z^3$; (5) confirm $\det J=|f'|^2$ and $\nabla^2u=\nabla^2v=0$ by Clairaut. No empirical constants appear; every result is standard and textbook-verifiable.

Builds on. Complex Numbers and the Complex Plane · Functions of a Complex Variable and Limits. Leads to. Harmonic Functions and Conjugates · Elementary Complex Functions · Complex Integration and Contour Integrals · Cauchy's Integral Theorem · Taylor Series of Analytic Functions.

Ask the tutor: Want to see why "one complex derivative" secretly forces a whole Taylor series — or work through building the harmonic conjugate for a potential of your choosing?