Complex geometry, differentiation and branches

1. Why one complex derivative is stronger than two real partial derivatives

A complex number is a pair of real numbers with the multiplication rule $i^2=-1$. Write $z=x+iy$, $\bar z=x-iy$, $\operatorname{Re}z=x$, $\operatorname{Im}z=y$, and $|z|=\sqrt{x^2+y^2}$. Then $z\bar z=|z|^2$ and, for $z\ne0$, $z^{-1}=\bar z/|z|^2$. The triangle inequality follows from

$$|z+w|^2=|z|^2+2\operatorname{Re}(z\bar w)+|w|^2\le(|z|+|w|)^2.$$

Consequently $\big||z|-|w|\big|\le|z-w|$. Distance in the complex plane is ordinary Euclidean distance.

The open disk $D(a,r)$ consists of points with $|z-a|<r$. A set is open if each of its points lies in some disk contained in the set. A domain is a nonempty open connected set. Limits and continuity use the distance $|z-w|$; approaching along one or several paths can disprove a limit, but agreement on finitely many paths does not prove one.

Multiplication by $re^{i\theta}=r(\cos\theta+i\sin\theta)$ rotates through $\theta$ and scales by $r\ge0$. Angles are determined modulo $2\pi$, not uniquely as real numbers. De Moivre's identity follows by multiplying the trigonometric addition formulas. Thus the $n$ distinct solutions of $z^n=\rho e^{i\theta}$, for $\rho>0$ and integer $n\ge1$, are

$$\rho^{1/n}e^{i(\theta+2\pi k)/n},\qquad 0\le k<n.$$

There are no others: any root has the stated modulus, and its angle satisfies the indicated congruence. The equation $z^n=0$ has the single root zero, with multiplicity $n$.

Worked example: a geometric inverse image. The condition $|z-1|=|z+1|$ becomes $(x-1)^2+y^2=(x+1)^2+y^2$, hence $x=0$. Points equidistant from two points form their perpendicular bisector. For a mapping problem, compute the inverse-image condition before attempting to sketch it.

A map $z\mapsto az+b$ with $a\ne0$ is a rotation, dilation and translation. The map $z\mapsto\bar z$ is reflection, not a rotation. The distinction becomes decisive under differentiation.

2. Definition and the Cauchy–Riemann equations

Let $f$ be defined in a neighbourhood of $a$. Its complex derivative is

$$f'(a)=\lim_{h\to0,\ h\in\mathbb C\setminus\{0\}}\frac{f(a+h)-f(a)}h,$$

if this limit exists. The same number must work for every complex direction. We call $f$ holomorphic on an open set if the derivative exists at every point of that set. Differentiability at one point is not holomorphy in a neighbourhood. We reserve analytic for locally representable by a convergent power series; a later chapter proves its equivalence with holomorphy.

Write $f=u+iv$. If $f'(a)=\alpha+i\beta$ exists, increments along the real and imaginary axes give

$$u_x=\alpha,\quad v_x=\beta,\quad u_y=-\beta,\quad v_y=\alpha$$

at $a$. Therefore the Cauchy–Riemann equations are necessary:

$$u_x=v_y,\qquad u_y=-v_x.$$

These equations assert that the real derivative, when it exists, is the matrix

$$\begin{pmatrix}\alpha&-\beta\\\beta&\alpha\end{pmatrix},$$

namely multiplication by one complex number. A general real derivative is an arbitrary $2\times2$ real matrix and need not have this shape.

Precise sufficiency theorem. Suppose $f$, regarded as a map $\mathbb R^2\to\mathbb R^2$, is real differentiable at $a$, and its partial derivatives satisfy the Cauchy–Riemann equations there. Then $f'(a)$ exists and equals $u_x(a)+iv_x(a)$. In particular, it is sufficient for $u,v$ to have continuous first partial derivatives in a neighbourhood of $a$ and satisfy the equations at $a$.

Proof. Real differentiability means

$$f(a+h)-f(a)=L(h)+r(h),\qquad |r(h)|/|h|\to0,$$

for a real-linear map $L$. The equations make $L(h)=ch$, where $c=u_x(a)+iv_x(a)$. Division by $h\ne0$ leaves $c+r(h)/h$, whose error has modulus $|r(h)|/|h|\to0$. The usual real differentiability theorem for continuous partial derivatives gives the final assertion. Conversely complex differentiability supplies precisely this real-linear approximation with $L(h)=f'(a)h$. ∎

Why the regularity hypothesis matters. Define $f(0)=0$ and $f(z)=\bar z^2/z$ for $z\ne0$. Because $|f(z)|=|z|$, it is continuous at zero. On the real axis $f(x)=x$, and on the imaginary axis $f(iy)=iy$. All four real partials at zero exist and obey the Cauchy–Riemann equations. Nevertheless

$$\frac{f(z)}z=\frac{\bar z^2}{z^2}$$

is $1$ on the positive real axis and $-1$ on the ray $\arg z=\pi/4$. There is no complex derivative at zero. Existence of the four partials, even together with continuity, is not the missing hypothesis.

3. Calculations that also explain the geometry

Algebra of limits proves the sum, product, quotient and chain rules just as for real derivatives, using complex increments. The quotient rule requires a nonzero denominator at the point. Polynomials are holomorphic on $\mathbb C$; rational functions are holomorphic off their denominator zeros. A cancelled factor can yield an extension, but the original rational expression is still undefined at an excluded point until the extension is explicitly defined.

Worked example: $z^2$. The difference quotient is $2z+h$, so $f'(z)=2z$. In real components, $u=x^2-y^2$ and $v=2xy$; both Cauchy–Riemann equations hold everywhere. At $z\ne0$, the infinitesimal effect is rotation by $\arg(2z)$ and scaling by $2|z|$.

For $f(z)=\bar z$, the quotients along real and imaginary increments are $1$ and $-1$. It is complex differentiable nowhere, although it is real differentiable everywhere. For $f(z)=|z|^2$, the equations require $x=y=0$; at zero the quotient is $\bar h\to0$. Thus it is differentiable only at zero, and holomorphic on no nonempty open set.

A holomorphic map with $f'(a)\ne0$ preserves the oriented angle between two differentiable curves through $a$ whose tangent vectors are nonzero: the chain rule multiplies both tangents by the same nonzero complex number. This is a local statement. It does not say the map is globally one-to-one, nor apply at a critical point where $f'(a)=0$. For example $z^2$ is two-to-one away from zero on the punctured plane, and doubles the angle between rays at zero.

If $u,v$ are $C^2$ and satisfy the Cauchy–Riemann equations in a domain, differentiating gives

$$u_{xx}+u_{yy}=v_{yx}-v_{xy}=0,$$

and similarly $v_{xx}+v_{yy}=0$. A $C^2$ real function satisfying this equation is harmonic. Recovering a harmonic conjugate is a differential-equation problem with possible global topological obstructions; harmonicity alone is not yet a global conjugate theorem.

Worked example: constructing a conjugate. Let $u=x^2-y^2+2x$. The equations require $v_y=2x+2$ and $v_x=2y$. Integration gives $v=2xy+2y+C$. Hence $u+iv=z^2+2z+iC$. The constant in a harmonic conjugate is real, so it adds an imaginary constant to $f$.

4. Exponential and logarithm: local inverses versus global values

Define

$$e^{x+iy}=e^x(\cos y+i\sin y).$$

The addition law follows from the real exponential and trigonometric addition laws. Its components are $C^1$ and satisfy the Cauchy–Riemann equations, giving $(e^z)'=e^z$. Also $e^z\ne0$, $|e^z|=e^{\operatorname{Re}z}$, and $e^z=e^w$ exactly when $z-w=2\pi ik$ for an integer $k$. Thus the exponential has no single inverse on all of $\mathbb C$.

The logarithms of $z\ne0$ are all values

$$\log|z|+i(\theta+2\pi k),\qquad k\in\mathbb Z,$$

where $z=|z|e^{i\theta}$. This is a set of values, not a single-valued function. A holomorphic logarithm branch on an open set $U$ is a holomorphic function $L$ with $e^{L(z)}=z$ for every $z\in U$.

On the slit domain $\Omega=\mathbb C\setminus(-\infty,0]$, choose the unique continuous angle $\operatorname{Arg}z\in(-\pi,\pi)$. The principal holomorphic logarithm is

$$\operatorname{Log}z=\log|z|+i\operatorname{Arg}z\quad(z\in\Omega).$$

The angle has continuous local partials $\theta_x=-y/(x^2+y^2)$ and $\theta_y=x/(x^2+y^2)$. With $u=\tfrac12\log(x^2+y^2)$ these satisfy the Cauchy–Riemann equations, so

$$(\operatorname{Log}z)'=1/z.$$

This choice excludes the nonpositive real axis. A numerical convention assigning angle $\pi$ to a negative real number may define a pointwise principal value there, but does not make that convention holomorphic across the cut.

No global continuous argument on $\mathbb C\setminus\{0\}$. Suppose $\theta$ were continuous and $e^{i\theta(z)}=z/|z|$. On the unit circle set $g(t)=\theta(e^{it})-t$ for $0\le t\le2\pi$. Its values lie in $2\pi\mathbb Z$, so by continuity on an interval it is constant. But $\theta(e^{2\pi i})=\theta(1)$ implies $g(2\pi)=g(0)-2\pi$, contradiction. Therefore there is no global continuous logarithm there either. This proof uses only continuity, not contour integration.

If two logarithm branches $L_1,L_2$ are defined on the same connected domain, their difference lies in $2\pi i\mathbb Z$ and is continuous, hence constant. Without connectedness the integer can differ between components.

A chosen branch defines powers by $z^\alpha=\exp(\alpha L(z))$, with derivative $\alpha z^\alpha/z$ on its domain. This definition is branch dependent. Integer powers are single-valued and agree across branches, but noninteger powers generally do not. For principal values, $\operatorname{Log}(zw)=\operatorname{Log}z+\operatorname{Log}w$ only when the selected arguments add without crossing the cut. For instance, $z=w=e^{3\pi i/4}$ gives principal logarithms whose sum differs from $\operatorname{Log}(zw)$ by $2\pi i$.

5. Exercises: computation, diagnosis and proof

P01 — Foundation. Find every cube root of $-8$ and verify their sum.

P02 — Foundation. Describe $|z-2i|<2$ and its intersection with the real axis. Distinguish the open disk from its boundary.

P03 — Core. Determine exactly where $f(z)=x^2+iy^2$ is complex differentiable and where it is holomorphic.

P04 — Core. For $f(z)=z\bar z+2z$, determine differentiability points and derivatives directly at any exceptional point.

P05 — Core. Find all harmonic conjugates of $u=e^x\cos y$ on $\mathbb C$, and identify the resulting holomorphic functions.

P06 — Core. Evaluate $\operatorname{Log}(i)$ and the principal square root of $-i$ on $\Omega$. Differentiate that square-root branch.

P07 — Diagnosis. A student checks Cauchy–Riemann equations at zero and concludes holomorphy near zero. Give two distinct flaws in the argument and examples illustrating them.

P08 — Proof. Prove that a holomorphic real-valued function on a domain is constant. You may use real differentiability and polygonal path connectedness of open connected subsets of $\mathbb R^2$.

P09 — Proof. If $L$ is a holomorphic logarithm branch on a domain, prove $L'=1/z$ by the chain rule. Explain why zero cannot lie in its domain.

P10 — Extension. Determine when $z\mapsto az+b\bar z$, for complex constants $a,b$, is holomorphic. If it is real differentiable, compute its real Jacobian determinant and interpret the case $a=0$, $b\ne0$.

6. Full solutions

P01. Write $-8=8e^{i\pi}$. The roots are $2e^{i\pi/3}=1+i\sqrt3$, $2e^{i\pi}=-2$, and $2e^{5i\pi/3}=1-i\sqrt3$. Their cubes equal $-8$ and their sum is zero. Angles differing by $2\pi/3$ make them distinct, and the root formula proves exhaustiveness.

P02. It is the open disk of radius two centred at $(0,2)$. For $z=x$ real, $|x-2i|^2=x^2+4\ge4$, so the intersection with the real axis is empty. Zero is on the boundary, not in the disk.

P03. Here $u_x=2x$, $u_y=0$, $v_x=0$, $v_y=2y$. The components are $C^1$, and the equations hold exactly when $x=y$. Thus the derivative exists precisely on that line and equals $2x$. No open disk lies in the line, so there is no nonempty open set of holomorphy. This differentiability set is not itself an open domain.

P04. The $2z$ term is holomorphic everywhere; $|z|^2$ is differentiable only at zero. Thus $f$ is differentiable only at zero and $f'(0)=\lim_{h\to0}(|h|^2+2h)/h=\lim(\bar h+2)=2$. It is holomorphic on no nonempty open set.

P05. The equations give $v_y=e^x\cos y$ and $v_x=e^x\sin y$, hence $v=e^x\sin y+C$. Therefore $f=e^z+iC$ with $C\in\mathbb R$. The difference between any two conjugates has both partials zero and is constant on the connected plane, proving these are all.

P06. $\operatorname{Log}i=i\pi/2$. The selected square root is $\exp(\tfrac12\operatorname{Log}(-i))=e^{-i\pi/4}=(1-i)/\sqrt2$. For $s(z)=\exp(\operatorname{Log}z/2)$, $s'(z)=s(z)/(2z)=1/(2s(z))$ because $s(z)^2=z$ and $s$ is nonzero on $\Omega$.

P07. First, partial derivatives and the equations at a point are not sufficient without real differentiability or another valid regularity condition; $\bar z^2/z$ extended by zero is a counterexample. Second, even a correctly established derivative at one point does not prove derivatives exist on a neighbourhood; $|z|^2$ is differentiable at zero only. These are logically separate flaws.

P08. Write $f=u+i0$. Cauchy–Riemann implies $u_x=u_y=0$. Complex differentiability implies real differentiability, so restricting $u$ to each line segment in the domain and using the chain rule and real mean-value theorem shows it is constant along that segment. Any two points of an open connected subset of the plane can be joined by a finite polygonal path in it; apply this conclusion successively along the segments. Thus the values at any two points agree. Connectivity is essential: different components could carry different constants.

P09. Differentiate $e^{L(z)}=z$: $e^{L(z)}L'(z)=1$, giving $zL'(z)=1$. Since an exponential is never zero, the identity cannot hold at zero. This proof applies to any existing branch; it does not prove that every zero-free domain admits one.

P10. Real linearity makes the real derivative equal to the map itself at every point. Its difference quotient is $a+b\bar h/h$. Along real and purely imaginary increments the values are $a+b$ and $a-b$, so complex differentiability anywhere requires $b=0$, which also suffices everywhere. Write $a=\alpha+i\beta$ and $b=\gamma+i\delta$. The real matrix is $\begin{pmatrix}\alpha+\gamma&-\beta+\delta\\\beta+\delta&\alpha-\gamma\end{pmatrix}$, with determinant $|a|^2-|b|^2$. When $a=0$, $b\ne0$, the map is a reflection composed with a rotation and dilation and reverses orientation, consistently with its negative determinant.

7. Summary and dependency map

Complex differentiation means approximation by complex multiplication, not by an arbitrary real-linear map. Cauchy–Riemann is necessary at a point and sufficient when real differentiability is also known. Holomorphy requires an open neighbourhood. Branches replace a multivalued inverse by a function on a carefully specified domain; a cut is part of the definition, not a removable inconvenience. The next chapter builds contour integration from parameterized curves, without presupposing that holomorphic functions already have power series.