Holomorphic Functions

What does it mean for a function of a complex variable to be differentiable? The question sounds like a formality. You might think we simply replace real numbers by complex ones in the definition of a derivative and move on. But the consequences are extraordinary. A complex-differentiable function turns out to be infinitely differentiable. It equals its own Taylor series near every point. It is also rigid in a way that has no real analogue: two holomorphic functions agreeing on a tiny arc must agree everywhere they are both defined. This single hypothesis of complex differentiability at every point of an open set generates the entire architecture of complex analysis. The reason for this rigidity is geometric. Differentiability in one real variable only asks that a function look linear along a line. Differentiability in the complex plane asks for more. It requires a function to look linear as you approach a point from every direction at once. Multiplication by a complex number is a rotation-and-scaling, not merely a stretch. This extra structure is encoded algebraically by the The Cauchy-Riemann Equations. It forces the real and imaginary parts of the function to be harmonic and mutually determined. This ties complex analysis to potential theory, conformal geometry, and the topology of the plane. The big picture is this: holomorphy is a local differential condition. But its consequences are deeply global — power series expansions, the identity theorem, the maximum modulus principle, Cauchy's integral formula. This article develops the definition of holomorphy on open subsets of C\mathbb{C} from first principles. It proves the elementary algebraic and structural properties every analytic function enjoys. It also shows exactly how the naive real-variable intuition breaks down. This happens without the open-set hypothesis and the full disk-of-approach requirement. By the end you will have a complete, self-contained account of what "holomorphic" means. You will understand why the definition is stated on open sets. You will see how the arithmetic of derivatives transfers from calculus. You will also get a first glimpse of why holomorphic functions are so much more constrained — and so much more powerful — than their real-variable cousins.

Key takeaways
  • A function f:U→Cf:U\to\mathbb{C} on an open set U⊆CU\subseteq\mathbb{C} is holomorphic if the complex derivative exists at every point of UU.
  • The complex derivative is defined by the same difference-quotient limit as in single-variable calculus, but the limit must exist as h→0h\to 0 through complex hh, i.e. from every direction.
  • Complex differentiability at a point is equivalent to real differentiability plus the The Cauchy-Riemann Equations holding there.
  • Sums, products, quotients (where defined), and compositions of holomorphic functions are holomorphic — the usual calculus rules hold verbatim.
  • Holomorphy must be checked on an open set; differentiability "at a single point" or on a set with empty interior does not generate the rich theory.
  • Key idea: requiring one complex limit to exist is secretly requiring infinitely many real directional derivatives to agree, which is why holomorphic functions are so much more rigid than real-differentiable ones.

Where this lives today

The notion of holomorphy is the entry point to all of complex analysis and radiates outward into many fields. In pure mathematics it is the foundation for Cauchy's Theorem and the residue calculus used to evaluate real integrals, for the theory of harmonic functions (since real and imaginary parts of holomorphic functions are automatically harmonic), and for Riemann surface theory, where holomorphy is defined chart-by-chart on more general one-complex-dimensional manifolds. It underlies the construction of the complex exponential and logarithm, power series, and analytic continuation, the mechanism by which a function determined only on a small disk can be extended, essentially uniquely, over vastly larger domains. In applied contexts, conformal maps built from holomorphic functions solve two-dimensional Laplace and Poisson equations in electrostatics, fluid dynamics (potential flow around airfoils), and heat conduction, because harmonic functions are exactly the real (or imaginary) parts of holomorphic functions. Signal processing and control theory use holomorphic transfer functions and the location of their poles and zeros to characterize system stability. In physics, holomorphic (and anti-holomorphic) structure organizes two-dimensional conformal field theory and string theory worldsheets. In number theory, holomorphic modular forms and LL-functions are central objects whose analytic continuation properties (holomorphy except at controlled poles) encode deep arithmetic information. Even numerical analysis benefits: fast, stable algorithms for computing special functions often exploit holomorphy to justify contour-shifting and series-acceleration techniques.

Ask the tutor: Can you show me step by step why f(z)=∣z∣2f(z)=|z|^2 is complex differentiable only at z=0z=0, using the same Cauchy–Riemann method as in this article?

DefinitionHolomorphic function

Let U⊆CU \subseteq \mathbb{C} be an open set and let f:U→Cf : U \to \mathbb{C} be a function. Fix z0∈Uz_0 \in U. Since UU is open there is r>0r>0 with the disk D(z0,r)⊂UD(z_0,r) \subset U, so f(z0+h)f(z_0+h) is defined for all sufficiently small h∈C∖{0}h \in \mathbb{C}\setminus\{0\}. We say ff is complex differentiable at z0z_0 if the limit

f′(z0)  =  lim⁡h→0f(z0+h)−f(z0)hf'(z_0) \;=\; \lim_{h \to 0} \frac{f(z_0+h)-f(z_0)}{h}

exists as a limit over h∈C∖{0}h \in \mathbb{C}\setminus\{0\}, h→0h \to 0 (i.e. ∣h∣→0|h|\to 0 with hh approaching 00 along an arbitrary path in C\mathbb{C}). The value f′(z0)∈Cf'(z_0) \in \mathbb{C} is called the derivative of ff at z0z_0. The function ff is called holomorphic on UU (equivalently, analytic on UU, at this stage of the theory used as a synonym) if ff is complex differentiable at every point of UU. A function holomorphic on all of C\mathbb{C} is called entire. If A⊆CA \subseteq \mathbb{C} is an arbitrary set, we say ff is holomorphic on AA if ff extends to a holomorphic function on some open set containing AA.

Intuition

The definition looks identical to the definition of the derivative from first-year calculus, and algebraically it is. The entire content is hidden in one word: hh is now a complex number, so h→0h \to 0 can happen along a line through the origin at any angle, along a spiral, along a parabola — along literally any path. In real analysis, h→0h\to 0 only has two directions (left and right), and requiring a two-sided limit is already a meaningful constraint. In the plane, requiring the same limit along a continuum of directions is a dramatically stronger condition. Concretely, approaching along the real axis (h=t∈Rh = t \in \mathbb{R}, t→0t\to 0) computes the partial derivative in the xx-direction (up to normalization), while approaching along the imaginary axis (h=ith = it, t→0t \to 0) computes the partial derivative in the yy-direction, divided by ii. Demanding these two special limits agree is exactly the Cauchy–Riemann system; demanding all directional limits agree is, remarkably, no stronger than demanding those two agree once real differentiability is assumed — this is precisely the content of the theorem below, and it is why the Cauchy-Riemann equations serve as a checkable algebraic proxy for the geometric limit condition. Geometrically, complex differentiability at z0z_0 with f′(z0)≠0f'(z_0)\neq 0 means that near z0z_0, ff looks like the affine map z↦f(z0)+f′(z0)(z−z0)z \mapsto f(z_0) + f'(z_0)(z-z_0), which is a rotation by arg⁡f′(z0)\arg f'(z_0) composed with a scaling by ∣f′(z0)∣|f'(z_0)|. Such a map preserves angles between curves through z0z_0. This is the seed of conformality, one of the signature geometric features that distinguishes holomorphic maps from generic smooth maps R2→R2\mathbb{R}^2 \to \mathbb{R}^2.

Figure
Approaching z0z_0 along different directions must give the same limiting quotient
Re Im z₀ h = t (real axis) h = it (imag. axis) h along any ray h along a curve All paths h → 0 must yield the SAME limit f′(z₀).
TheoremComplex differentiability and the Cauchy–Riemann equations

Let U⊆CU \subseteq \mathbb{C} be open, write z=x+iyz = x+iy and f(z)=u(x,y)+iv(x,y)f(z) = u(x,y) + iv(x,y) for real-valued functions u,vu,v on UU (identified with an open subset of R2\mathbb{R}^2). Fix z0=x0+iy0∈Uz_0 = x_0+iy_0 \in U. Then ff is complex differentiable at z0z_0 if and only if uu and vv are (real) differentiable at (x0,y0)(x_0,y_0) and satisfy the The Cauchy-Riemann Equations

ux(x0,y0)=vy(x0,y0),uy(x0,y0)=−vx(x0,y0).u_x(x_0,y_0) = v_y(x_0,y_0), \qquad u_y(x_0,y_0) = -v_x(x_0,y_0).

In that case,

f′(z0)=ux(x0,y0)+ivx(x0,y0).f'(z_0) = u_x(x_0,y_0) + i v_x(x_0,y_0).
Proof

We use the direct ε\varepsilon–δ\delta verification of differentiability, translating between the complex linear map "multiplication by f′(z0)f'(z_0)" and its representation as a real 2×22\times 2 matrix.

(⇒\Rightarrow) Suppose ff is complex differentiable at z0z_0 with derivative f′(z0)=a+ibf'(z_0) = a+ib. By definition of the limit, for every ε>0\varepsilon>0 there is δ>0\delta>0 such that for all h∈Ch\in\mathbb{C} with 0<∣h∣<δ0<|h|<\delta,

∣f(z0+h)−f(z0)h−(a+ib)∣<ε,\left| \frac{f(z_0+h)-f(z_0)}{h} - (a+ib)\right| < \varepsilon,

which we rewrite as

f(z0+h)−f(z0)=(a+ib)h+η(h)h,η(h)→0 as h→0.(1)f(z_0+h) - f(z_0) = (a+ib)h + \eta(h)h, \qquad \eta(h) \to 0 \text{ as } h \to 0. \tag{1}

Write h=s+ith = s+it with s,t∈Rs,t\in\mathbb{R}. Multiplication by a+iba+ib acting on hh, viewed as an R\mathbb{R}-linear map R2→R2\mathbb{R}^2 \to \mathbb{R}^2, is represented by the matrix

(a−bba),\begin{pmatrix} a & -b \\ b & a \end{pmatrix},

since (a+ib)(s+it)=(as−bt)+i(bs+at)(a+ib)(s+it) = (as - bt) + i(bs+at). Equation (1), separated into real and imaginary parts, states exactly that

(u(x0+s,y0+t)−u(x0,y0)v(x0+s,y0+t)−v(x0,y0))=(a−bba)(st)+o(s2+t2)\begin{pmatrix} u(x_0+s,y_0+t) - u(x_0,y_0) \\ v(x_0+s,y_0+t)-v(x_0,y_0)\end{pmatrix} = \begin{pmatrix} a & -b \\ b& a\end{pmatrix} \begin{pmatrix} s \\ t\end{pmatrix} + o(\sqrt{s^2+t^2})

as (s,t)→(0,0)(s,t) \to (0,0), where the o(⋅)o(\cdot) term is Re(η(h)h)\mathrm{Re}(\eta(h)h) and Im(η(h)h)\mathrm{Im}(\eta(h)h) respectively, both of which are o(∣h∣)=o(s2+t2)o(|h|) = o(\sqrt{s^2+t^2}) because η(h)→0\eta(h)\to 0. This is precisely the statement that the map (x,y)↦(u(x,y),v(x,y))(x,y)\mapsto(u(x,y),v(x,y)) is (Fréchet) differentiable at (x0,y0)(x_0,y_0) with total derivative matrix (a−bba)\begin{pmatrix} a & -b\\ b& a\end{pmatrix}. Reading off entries of the Jacobian matrix

Df(x0,y0)=(uxuyvxvy)Df(x_0,y_0) = \begin{pmatrix} u_x & u_y \\ v_x & v_y \end{pmatrix}

against (a−bba)\begin{pmatrix} a & -b \\ b & a\end{pmatrix} gives

ux=a=vy,vx=b=−uy,u_x = a = v_y, \qquad v_x = b = -u_y,

which are exactly the Cauchy–Riemann equations, and f′(z0)=a+ib=ux+ivxf'(z_0)=a+ib = u_x + iv_x, as claimed.

(⇐\Leftarrow) Conversely, suppose u,vu,v are differentiable at (x0,y0)(x_0,y_0) and the Cauchy–Riemann equations hold there. Set a:=ux(x0,y0)=vy(x0,y0)a := u_x(x_0,y_0) = v_y(x_0,y_0) and b:=vx(x0,y0)=−uy(x0,y0)b:= v_x(x_0,y_0) = -u_y(x_0,y_0). Differentiability of uu at (x0,y0)(x_0,y_0) means

u(x0+s,y0+t)−u(x0,y0)=uxs+uyt+ε1(s,t),u(x_0+s,y_0+t) - u(x_0,y_0) = u_x s + u_y t + \varepsilon_1(s,t),

with ε1(s,t)/s2+t2→0\varepsilon_1(s,t)/\sqrt{s^2+t^2} \to 0 as (s,t)→(0,0)(s,t)\to(0,0), and similarly for vv with error ε2\varepsilon_2. Substituting ux=au_x=a, uy=−bu_y = -b, vx=bv_x = b, vy=av_y = a:

u(x0+s,y0+t)−u(x0,y0)=as−bt+ε1(s,t),u(x_0+s,y_0+t)-u(x_0,y_0) = as - bt + \varepsilon_1(s,t),
v(x0+s,y0+t)−v(x0,y0)=bs+at+ε2(s,t).v(x_0+s,y_0+t) - v(x_0,y_0) = bs + at + \varepsilon_2(s,t).

Forming f(z0+h)−f(z0)f(z_0+h) - f(z_0) with h=s+ith=s+it, and using the matrix identity above in reverse,

f(z0+h)−f(z0)=(a+ib)(s+it)+(ε1(s,t)+iε2(s,t))=(a+ib)h+ε(h),f(z_0+h)-f(z_0) = (a+ib)(s+it) + \big(\varepsilon_1(s,t)+i\varepsilon_2(s,t)\big) = (a+ib)h + \varepsilon(h),

where ε(h)=ε1+iε2\varepsilon(h) = \varepsilon_1+i\varepsilon_2 satisfies ∣ε(h)∣/∣h∣→0|\varepsilon(h)|/|h| \to 0 as h→0h\to 0 (because ∣h∣=s2+t2|h| = \sqrt{s^2+t^2} and each εj(s,t)/s2+t2→0\varepsilon_j(s,t)/\sqrt{s^2+t^2}\to 0). Hence

f(z0+h)−f(z0)h−(a+ib)=ε(h)h→0as h→0,\frac{f(z_0+h)-f(z_0)}{h} - (a+ib) = \frac{\varepsilon(h)}{h} \to 0 \quad \text{as } h\to 0,

so ff is complex differentiable at z0z_0 with f′(z0)=a+ib=ux+ivxf'(z_0)=a+ib = u_x+iv_x evaluated at (x0,y0)(x_0,y_0). ■\blacksquare

∎
CorollaryAlgebra of holomorphic functions

Let U⊆CU\subseteq \mathbb{C} be open and let f,g:U→Cf,g: U \to \mathbb{C} be holomorphic on UU, with c∈Cc\in\mathbb{C} a constant. Then: 1. f+gf+g and cfcf are holomorphic on UU, with (f+g)′=f′+g′(f+g)' = f'+g' and (cf)′=cf′(cf)'=cf'. 2. fgfg is holomorphic on UU, with (fg)′=f′g+fg′(fg)' = f'g+fg'. 3. If g(z)≠0g(z)\neq 0 for all z∈Uz\in U, then f/gf/g is holomorphic on UU, with (f/g)′=f′g−fg′g2(f/g)' = \dfrac{f'g - fg'}{g^2}. 4. If V⊆CV\subseteq\mathbb{C} is open, g:U→Vg:U\to V is holomorphic on UU, and f:V→Cf:V\to\mathbb{C} is holomorphic on VV, then f∘gf\circ g is holomorphic on UU, with (f∘g)′(z)=f′(g(z)) g′(z)(f\circ g)'(z) = f'(g(z))\,g'(z). Every polynomial p(z)=anzn+⋯+a0p(z) = a_nz^n + \cdots + a_0 is entire, and every rational function is holomorphic on the complement of its (finite) set of poles.

Proof

The proof is the verbatim transcription of the real-variable difference-quotient proofs, valid because C\mathbb{C} is a field and the algebraic manipulations use only field axioms and the triangle inequality — none of the real-variable-specific structure (order, one-dimensionality) is used. (1) Immediate from linearity of limits: for fixed z0∈Uz_0\in U and h≠0h\neq 0 small,

(f+g)(z0+h)−(f+g)(z0)h=f(z0+h)−f(z0)h+g(z0+h)−g(z0)h→f′(z0)+g′(z0).\frac{(f+g)(z_0+h) - (f+g)(z_0)}{h} = \frac{f(z_0+h)-f(z_0)}{h} + \frac{g(z_0+h)-g(z_0)}{h} \to f'(z_0)+g'(z_0).

The scalar rule is similar and easier. (2) Write, for h≠0h\neq 0,

f(z0+h)g(z0+h)−f(z0)g(z0)h=f(z0+h)−f(z0)h g(z0+h)+f(z0) g(z0+h)−g(z0)h.\frac{f(z_0+h)g(z_0+h) - f(z_0)g(z_0)}{h} = \frac{f(z_0+h)-f(z_0)}{h}\,g(z_0+h) + f(z_0)\,\frac{g(z_0+h)-g(z_0)}{h}.

As h→0h\to 0: the first factor of the first term tends to f′(z0)f'(z_0); the factor g(z0+h)g(z_0+h) tends to g(z0)g(z_0) because complex differentiability at z0z_0 implies continuity of gg there (indeed g(z0+h)−g(z0)=h⋅g(z0+h)−g(z0)h→0⋅g′(z0)=0g(z_0+h)-g(z_0) = h\cdot\frac{g(z_0+h)-g(z_0)}{h} \to 0\cdot g'(z_0) = 0 using that a convergent difference quotient is bounded near 00); the second term's quotient tends to g′(z0)g'(z_0). Products and sums of convergent limits converge to the product/sum of the limits (standard limit laws, valid in C\mathbb{C} exactly as in R\mathbb{R} since C\mathbb{C} is a normed field). This gives (fg)′(z0)=f′(z0)g(z0)+f(z0)g′(z0)(fg)'(z_0) = f'(z_0)g(z_0) + f(z_0)g'(z_0). (3) First treat 1/g1/g where g(z0)≠0g(z_0)\neq 0. Since gg is continuous at z0z_0 (by the differentiability-implies-continuity remark above) and g(z0)≠0g(z_0)\neq 0, there is a punctured neighborhood where g≠0g\neq 0 (openness of C∖{0}\mathbb{C}\setminus\{0\} pulled back by continuity), so 1/g1/g is defined near z0z_0. Compute

1h(1g(z0+h)−1g(z0))=−1g(z0+h)g(z0)⋅g(z0+h)−g(z0)h  ⟶  −g′(z0)g(z0)2\frac{1}{h}\left(\frac{1}{g(z_0+h)} - \frac{1}{g(z_0)}\right) = \frac{-1}{g(z_0+h)g(z_0)}\cdot\frac{g(z_0+h)-g(z_0)}{h} \;\longrightarrow\; \frac{-g'(z_0)}{g(z_0)^2}

as h→0h\to 0, using continuity of gg at z0z_0 and the limit laws again. Then apply the product rule (2) to f⋅(1/g)f\cdot(1/g) to get the quotient rule as stated. (4) Fix z0∈Uz_0\in U, let w0=g(z0)∈Vw_0 = g(z_0)\in V. Define

Φ(w):={f(w)−f(w0)w−w0,w≠w0f′(w0),w=w0.\Phi(w) := \begin{cases} \dfrac{f(w)-f(w_0)}{w-w_0}, & w\neq w_0 \\ f'(w_0), & w = w_0.\end{cases}

By definition of f′(w0)f'(w_0), Φ\Phi is continuous at w0w_0, and by construction f(w)−f(w0)=Φ(w)(w−w0)f(w)-f(w_0) = \Phi(w)(w-w_0) for all w∈Vw\in V (trivially true at w=w0w=w_0, and true by definition of Φ\Phi elsewhere). Substituting w=g(z)w = g(z):

f(g(z))−f(g(z0))=Φ(g(z)) (g(z)−g(z0))for all z∈U.f(g(z)) - f(g(z_0)) = \Phi(g(z))\,\big(g(z)-g(z_0)\big) \qquad \text{for all } z \in U.

Divide by h=z−z0≠0h = z-z_0 \neq 0 and let h→0h\to 0: g(z)→g(z0)=w0g(z)\to g(z_0)=w_0 by continuity of gg at z0z_0 (again, differentiability ⇒\Rightarrow continuity), so Φ(g(z))→Φ(w0)=f′(w0)\Phi(g(z)) \to \Phi(w_0) = f'(w_0) by continuity of Φ\Phi at w0w_0; and g(z)−g(z0)h→g′(z0)\frac{g(z)-g(z_0)}{h}\to g'(z_0) by hypothesis. Hence

f(g(z))−f(g(z0))z−z0→f′(w0) g′(z0)=f′(g(z0))g′(z0),\frac{f(g(z))-f(g(z_0))}{z-z_0} \to f'(w_0)\,g'(z_0) = f'(g(z_0))g'(z_0),

proving (f∘g)(f\circ g) is complex differentiable at z0z_0 with the stated derivative. Since z0∈Uz_0\in U was arbitrary, f∘gf\circ g is holomorphic on UU. Finally, the constant function f(z)=cf(z)=c is holomorphic with derivative 00 (the difference quotient is identically 00), and f(z)=zf(z) = z is holomorphic with f′(z)=1f'(z)=1 (the difference quotient is identically 11). Repeated application of (1) and (2) shows every monomial znz^n (n≥0n\ge 0) is entire with derivative nzn−1nz^{n-1}, and every polynomial, being a finite linear combination of monomials, is entire. By (3), a rational function p(z)/q(z)p(z)/q(z) (with p,qp,q polynomials, q≢0q\not\equiv 0) is holomorphic on C\mathbb{C} minus the finite zero set of qq, since a nonzero polynomial has finitely many roots. ■\blacksquare

∎

Worked examples

Example 1: An asymmetric polynomial-times-conjugate expression. Determine all points z∈Cz\in\mathbb{C} where

f(z)=z2zˉ+3zf(z) = z^2\bar z + 3z

is complex differentiable, and find f′f' there. Write z=x+iyz=x+iy. Then z2zˉ=z⋅zzˉ=z∣z∣2=(x+iy)(x2+y2)z^2 \bar z = z\cdot z\bar z = z|z|^2 = (x+iy)(x^2+y^2), so

f(z)=(x+iy)(x2+y2)+3(x+iy)=[x(x2+y2)+3x]+i[y(x2+y2)+3y].f(z) = (x+iy)(x^2+y^2) + 3(x+iy) = \big[x(x^2+y^2) + 3x\big] + i\big[y(x^2+y^2)+3y\big].

Thus u(x,y)=x3+xy2+3xu(x,y) = x^3+xy^2+3x and v(x,y)=x2y+y3+3yv(x,y) = x^2y+y^3+3y. Both are polynomials in (x,y)(x,y), hence differentiable everywhere on R2\mathbb{R}^2; we only need to locate where the Cauchy–Riemann equations hold, by the Cauchy–Riemann theorem above. Compute the partials:

ux=3x2+y2+3,uy=2xy,u_x = 3x^2+y^2+3, \qquad u_y = 2xy,
vx=2xy,vy=x2+3y2+3.v_x = 2xy, \qquad v_y = x^2+3y^2+3.

The equation uy=−vxu_y = -v_x reads 2xy=−2xy2xy = -2xy, i.e. 4xy=04xy = 0, i.e. x=0x=0 or y=0y=0. The equation ux=vyu_x = v_y reads

3x2+y2+3=x2+3y2+3  ⟺  2x2=2y2  ⟺  y=±x.3x^2+y^2+3 = x^2+3y^2+3 \iff 2x^2 = 2y^2 \iff y = \pm x.

We need both equations simultaneously. Combining x=0x=0 or y=0y=0 with y=±xy=\pm x: if x=0x=0, then y=±x=0y=\pm x=0; if y=0y=0, then x=±y=0x=\pm y = 0. In either branch we force x=y=0x=y=0. So the only point where both Cauchy–Riemann equations hold is z=0z=0. Since u,vu,v are (everywhere) real-differentiable, complex differentiability of ff holds exactly at z=0z=0, by the theorem. At that single point,

f′(0)=ux(0,0)+ivx(0,0)=3+i⋅0=3.f'(0) = u_x(0,0) + iv_x(0,0) = 3 + i\cdot 0 = 3.

Note carefully: ff is complex differentiable at the single point z=0z=0 but not on any open disk around 00 (since away from 00 the Cauchy–Riemann equations fail), so ff is not holomorphic at 00 or anywhere — differentiability at an isolated point is not enough for holomorphy, which requires an open set of differentiability. This distinguishes "complex differentiable at a point" from "holomorphic," and is exactly why the definition of holomorphy is stated on open sets.

Example 2: Quotient and chain rules on a genuinely non-symmetric rational/compositional function. Let

g(z)=z3−2iz2+1,h(z)=(g(z))2+5z.g(z) = \frac{z^3 - 2i}{z^2+1}, \qquad h(z) = \big(g(z)\big)^2 + 5z.

Find where hh is holomorphic and compute h′(2−i)h'(2-i). The denominator z2+1=(z−i)(z+i)z^2+1 = (z-i)(z+i) vanishes exactly at z=±iz=\pm i. By the corollary (part 3, applied on U=C∖{i,−i}U = \mathbb{C}\setminus\{i,-i\}, which is open), gg is holomorphic on UU, and by the quotient rule,

g′(z)=3z2(z2+1)−(z3−2i)(2z)(z2+1)2=3z4+3z2−2z4+4iz(z2+1)2=z4+3z2+4iz(z2+1)2.g'(z) = \frac{3z^2(z^2+1) - (z^3-2i)(2z)}{(z^2+1)^2} = \frac{3z^4+3z^2 - 2z^4 + 4iz}{(z^2+1)^2} = \frac{z^4+3z^2+4iz}{(z^2+1)^2}.

Since w↦w2w\mapsto w^2 is entire (a polynomial) and z↦5zz\mapsto 5z is entire, and compositions and sums of holomorphic functions are holomorphic (corollary, parts 1 and 4), h=g2+5zh = g^2 + 5z is holomorphic exactly where gg is, namely on U=C∖{i,−i}U=\mathbb{C}\setminus\{i,-i\}. By the chain and sum rules,

h′(z)=2g(z)g′(z)+5.h'(z) = 2g(z)g'(z) + 5.

Now evaluate at z0=2−iz_0 = 2-i. First,

z02=(2−i)2=4−4i+i2=3−4i,z02+1=4−4i.z_0^2 = (2-i)^2 = 4 - 4i + i^2 = 3-4i, \qquad z_0^2+1 = 4-4i.

Next,

z03=z02⋅z0=(3−4i)(2−i)=6−3i−8i+4i2=6−11i−4=2−11i,z_0^3 = z_0^2\cdot z_0 = (3-4i)(2-i) = 6 - 3i - 8i + 4i^2 = 6 - 11i - 4 = 2 - 11i,

so

z03−2i=2−11i−2i=2−13i.z_0^3 - 2i = 2 - 11i - 2i = 2 - 13i.

Hence

g(z0)=2−13i4−4i.g(z_0) = \frac{2-13i}{4-4i}.

Rationalize: multiply numerator and denominator by 4−4i‾=4+4i\overline{4-4i} = 4+4i, and note (4−4i)(4+4i)=16+16=32(4-4i)(4+4i) = 16+16=32:

(2−13i)(4+4i)=8+8i−52i−52i2=8−44i+52=60−44i,(2-13i)(4+4i) = 8 + 8i - 52i - 52i^2 = 8 - 44i + 52 = 60 - 44i,
g(z0)=60−44i32=15−11i8.g(z_0) = \frac{60-44i}{32} = \frac{15 - 11i}{8}.

Next we need g′(z0)=z04+3z02+4iz0(z02+1)2g'(z_0) = \dfrac{z_0^4+3z_0^2+4iz_0}{(z_0^2+1)^2}. Compute z04=(z02)2=(3−4i)2=9−24i+16i2=9−24i−16=−7−24iz_0^4 = (z_0^2)^2 = (3-4i)^2 = 9 - 24i + 16i^2 = 9-24i-16 = -7-24i. Then

3z02=3(3−4i)=9−12i,4iz0=4i(2−i)=8i−4i2=4+8i.3z_0^2 = 3(3-4i) = 9-12i, \qquad 4iz_0 = 4i(2-i) = 8i - 4i^2 = 4+8i.

Sum:

z04+3z02+4iz0=(−7−24i)+(9−12i)+(4+8i)=(−7+9+4)+(−24−12+8)i=6−28i.z_0^4 + 3z_0^2 + 4iz_0 = (-7-24i) + (9-12i) + (4+8i) = (-7+9+4) + (-24-12+8)i = 6 - 28i.

Denominator: (z02+1)2=(4−4i)2=16−32i+16i2=16−32i−16=−32i(z_0^2+1)^2 = (4-4i)^2 = 16 - 32i + 16i^2 = 16-32i-16 = -32i. So

g′(z0)=6−28i−32i.g'(z_0) = \frac{6-28i}{-32i}.

Multiply numerator and denominator by ii: (6−28i)i−32i⋅i=6i−28i2−32i2=28+6i32=14+3i16\dfrac{(6-28i)i}{-32i\cdot i} = \dfrac{6i - 28i^2}{-32i^2} = \dfrac{28+6i}{32} = \dfrac{14+3i}{16}. Finally,

h′(z0)=2g(z0)g′(z0)+5=2⋅15−11i8⋅14+3i16+5=(15−11i)(14+3i)64+5.h'(z_0) = 2g(z_0)g'(z_0) + 5 = 2\cdot\frac{15-11i}{8}\cdot\frac{14+3i}{16} + 5 = \frac{(15-11i)(14+3i)}{64} + 5.

Expand (15−11i)(14+3i)=210+45i−154i−33i2=210−109i+33=243−109i(15-11i)(14+3i) = 210 + 45i - 154i - 33i^2 = 210 - 109i + 33 = 243 - 109i. So

h′(2−i)=243−109i64+5=243−109i+32064=563−109i64.h'(2-i) = \frac{243-109i}{64} + 5 = \frac{243 - 109i + 320}{64} = \frac{563 - 109i}{64}.

This example shows the corollary's rules composing correctly through a genuinely asymmetric rational-then-polynomial construction, with every intermediate arithmetic step tracked exactly.

CounterexampleWhy the "for every path h→0h\to 0" clause cannot be replaced by "along the real and imaginary axes only"

The theorem crucially requires the full two-real-variable limit defining f′(z0)f'(z_0) — equivalently, real differentiability of (u,v)(u,v), not merely existence of the partial derivatives ux,uy,vx,vyu_x,u_y,v_x,v_y satisfying Cauchy–Riemann. If we weaken "real differentiable at (x0,y0)(x_0,y_0)" to merely "partial derivatives exist at (x0,y0)(x_0,y_0) and satisfy Cauchy–Riemann there," the conclusion can fail: the complex derivative need not exist. Define

f(x+iy)=u(x,y)+iv(x,y),u(x,y)=v(x,y)={xy(x+y)x2+y2,(x,y)≠(0,0)0,(x,y)=(0,0).f(x+iy) = u(x,y)+iv(x,y), \qquad u(x,y) = v(x,y) = \begin{cases} \dfrac{xy(x+y)}{x^2+y^2}, & (x,y)\neq(0,0) \\ 0, & (x,y)=(0,0).\end{cases}

(This is a standard non-differentiable-but-partials-exist function, used for both real and imaginary parts.) Partial derivatives at the origin exist and satisfy Cauchy–Riemann. Along the xx-axis (y=0y=0), u(x,0)=0u(x,0)=0 for all xx, so ux(0,0)=lim⁡x→0u(x,0)−u(0,0)x=0u_x(0,0) = \lim_{x\to0} \frac{u(x,0)-u(0,0)}{x} = 0. Along the yy-axis, u(0,y)=0u(0,y)=0, so uy(0,0)=0u_y(0,0)=0. Identically vx(0,0)=vy(0,0)=0v_x(0,0)=v_y(0,0)=0 since v=uv=u. Then trivially

ux(0,0)=0=vy(0,0),uy(0,0)=0=−vx(0,0),u_x(0,0) = 0 = v_y(0,0), \qquad u_y(0,0) = 0 = -v_x(0,0),

so the Cauchy–Riemann equations hold at the origin, and both partials of u,vu,v vanish there. Yet ff is not complex differentiable at 00. If f′(0)f'(0) existed it would have to equal ux(0,0)+ivx(0,0)=0u_x(0,0)+iv_x(0,0) = 0 by the same computation as in the theorem's proof (this identity does not require differentiability, only existence of partials along the axes, so it is legitimate to use as the candidate value). Test the limit along the ray h=t(1+i)h = t(1+i), t∈R∖{0}→0t\in\mathbb{R}\setminus\{0\}\to 0: here x=y=tx=y=t, so

u(t,t)=t⋅t⋅(2t)2t2=t,v(t,t)=t,u(t,t) = \frac{t\cdot t\cdot(2t)}{2t^2} = t, \qquad v(t,t) = t,

so f(t+it)=t+it=hf(t+it) = t+it = h, giving

f(0+h)−f(0)h=hh=1for all such h.\frac{f(0+h)-f(0)}{h} = \frac{h}{h} = 1 \quad \text{for all such } h.

This tends to 11 as t→0t\to 0 along this ray — but the candidate value forced by the axis computation was 00. Since 1≠01 \neq 0, the limit defining f′(0)f'(0) does not exist (different paths give different values): ff is not complex differentiable at 00, even though the pointwise Cauchy–Riemann equations hold there. The failure is entirely due to dropping full (Fréchet) real differentiability of u,vu,v at (0,0)(0,0): this u=vu=v is continuous and has directional derivatives along the axes, but is not differentiable at the origin as a map R2→R\mathbb{R}^2\to\mathbb{R} (it fails to be well-approximated by any single linear map, which is exactly why different directions hh give different quotient limits). This shows the "differentiable, not just partial derivatives exist" hypothesis in the theorem cannot be dropped.

Scope, edge cases & common traps

SituationWhat holdsWhat fails / must be checked
ff complex differentiable at one isolated point z0z_0f′(z0)f'(z_0) can exist and equal a specific number (Example 1)ff need not be holomorphic anywhere; holomorphy needs an open set of differentiability
Cauchy–Riemann equations hold at (x0,y0)(x_0,y_0), partials existNecessary conditionNot sufficient without full real differentiability of u,vu,v (see counterexample)
UU not open (e.g. a line segment, a single point, a closed disk)"Differentiable on UU" is not standardly defined via the two-sided complex limitThe definition of holomorphy requires interior points on all sides to make sense of h→0h\to 0 from every direction
ff real-differentiable (as a map R2→R2\mathbb{R}^2\to\mathbb{R}^2) everywhereu,vu,v smooth, Jacobian existsDoes not imply holomorphic; e.g. f(z)=zˉf(z)=\bar z is real-differentiable everywhere but nowhere complex differentiable, since ux=1≠−1=vyu_x=1\neq -1 = v_y... (checking: u=x,v=−yu=x,v=-y, so ux=1,vy=−1u_x=1, v_y=-1, CR fails everywhere)
Quotient rule f/gf/gHolomorphic wherever g≠0g\neq0Domain shrinks to UU minus zero set of gg; must re-verify openness of the new domain

The mathematician behind it

Origins · 1 minAugustin-Louis Cauchy and how this was discovered
AC
Augustin-Louis Cauchy
1789–1857 · Early nineteenth-century France, École Polytechnique and the Paris Academy of Sciences

Cauchy was the principal architect of rigorous complex analysis, introducing the modern epsilon-based definition of limits and differentiability and, crucially, the complex-variable derivative and the associated integral calculus that bears his name. Working through decades of memoirs on "functions of an imaginary variable," he formulated the differentiability condition central to this article, connected it to path integrals, and — together with the later systematic work of Bernhard Riemann on the geometric and equation-based (Cauchy–Riemann) formulation, and Karl Weierstrass on power-series foundations — built complex analysis into the coherent, rigorous theory used today.

History
- 1814 — Cauchy presents early work on complex integrals to the Paris Academy, seeding what becomes the theory of the complex integral.
- 1821 — Cauchy's Cours d'Analyse formalizes rigorous limit-based definitions of continuity and convergence, the foundation later applied to complex differentiability.
- 1825 — Cauchy's memoir on integrals between imaginary limits lays groundwork connecting differentiability and integration in the complex plane.
- 1851 — Riemann's dissertation formulates the geometric, equation-based approach to complex differentiability, crystallizing the Cauchy–Riemann conditions in the form used today.
- 1876 — Weierstrass's lectures popularize the power-series (locally convergent Taylor series) definition of analyticity, later proved equivalent to Cauchy's differentiability-based definition.
Is "holomorphic" the same as "analytic"?

At the introductory level treated here, yes: both terms describe a function that is complex differentiable at every point of an open set. Later in the theory one proves the remarkable fact that holomorphic functions are automatically representable by a convergent power series near every point of their domain — that is where the term "analytic" (meaning "locally equal to a convergent power series") gets its independent life. Once that equivalence is established (via Cauchy's integral formula), the two words become interchangeable for functions on open subsets of C\mathbb{C}, and most authors use them synonymously without further comment.

Why isn't f(z)=zˉf(z)=\bar z holomorphic anywhere, even though it looks so simple?

Write f(x+iy)=x−iyf(x+iy) = x - iy, so u=xu=x, v=−yv=-y. Then ux=1u_x = 1 and vy=−1v_y = -1, so ux≠vyu_x \neq v_y at every point: the Cauchy–Riemann equations fail everywhere. By the theorem, ff is nowhere complex differentiable. Geometrically, conjugation is a reflection, which reverses orientation, while a nonzero complex-linear map (multiplication by f′(z0)f'(z_0)) is always orientation-preserving (a rotation and scaling); reflections cannot be approximated by any complex-linear map, which is the deep reason conjugation can never be holomorphic.

Does complex differentiability at a point imply continuity there, and is that fact used anywhere above?

Yes: if f′(z0)f'(z_0) exists, then f(z0+h)−f(z0)=h⋅f(z0+h)−f(z0)h→0⋅f′(z0)=0f(z_0+h)-f(z_0) = h\cdot\frac{f(z_0+h)-f(z_0)}{h} \to 0 \cdot f'(z_0) = 0 as h→0h \to 0, since a convergent difference quotient is bounded near h=0h=0 and is multiplied by h→0h\to 0. This shows ff is continuous at z0z_0. This fact is used explicitly in the proof of the product, quotient, and chain rules above, where continuity of one factor is needed to pass limits through products.

Can a function be holomorphic on a set that is not open, like a closed disk or the real line?

Strictly, the definition given here only applies to open sets, because the difference quotient's limit needs hh to range over a full punctured neighborhood of 00. When people say "ff is holomorphic on a closed set AA," they mean, by convention, that ff extends to a holomorphic function on some open set containing AA — exactly the convention stated in the definition block above. Without this convention, "differentiable" along, say, the real line only tests one direction of approach and misses the essential two-real-dimensional character of the definition.

Why do complex-differentiable functions behave so much more rigidly than real-differentiable ones?

A real-differentiable function of one variable can be as wild as it likes beyond first order — e.g. C1C^1 but not C2C^2. A complex-differentiable function, by contrast, is forced (via later theorems built on holomorphy, such as Cauchy's integral formula) to be infinitely differentiable and locally equal to its Taylor series. The root cause is exactly what this article proves: complex differentiability packages two real partial-derivative conditions into a single complex limit condition holding in every direction simultaneously, which is a far stronger constraint than any finite-dimensional real differentiability requirement.

Is every holomorphic function automatically harmonic?

The real and imaginary parts u,vu,v of a holomorphic function are each harmonic (satisfy Laplace's equation uxx+uyy=0u_{xx}+u_{yy}=0), but this requires knowing that u,vu,v are twice continuously differentiable, which is not part of the elementary definition given here — it is a deeper consequence of holomorphy proved later using Cauchy's integral formula. Once that regularity is established, differentiating the Cauchy–Riemann equations ux=vyu_x=v_y, uy=−vxu_y=-v_x and using equality of mixed partials gives uxx+uyy=0u_{xx}+u_{yy}=0 directly; the full argument belongs to Harmonic Functions and Holomorphy.

What goes wrong if we only require the Cauchy-Riemann equations to hold, without assuming u,vu, v are differentiable?

This is exactly the content of the counterexample above. If u,vu,v merely have partial derivatives satisfying Cauchy–Riemann at a point, without being differentiable there in the full real-variable (Fréchet) sense, the complex limit defining f′(z0)f'(z_0) can fail to exist — different paths of approach can give different values, as shown explicitly. The theorem's hypothesis of real differentiability of u,vu,v (not just existence of partials) is therefore essential and cannot be weakened.

Before this

Where this leads