Branch Points, Cuts & Riemann Surfaces

Derivation

Branch Points, Cuts & Riemann Surfaces

D-186 Home PU-205 Threads symmetry Depends on Cauchy-Riemann Equations & Harmonic Conjugates

Statement

For a multivalued function built from \( \log z \) or \( z^{\alpha} \), we derive that its multivaluedness is generated entirely by analytic continuation around isolated branch points (here \( z=0 \) and \( z=\infty \)): a loop of winding number one about \( z=0 \) sends \( \log z \mapsto \log z + 2\pi i \). Removing a branch cut joining the branch points yields a simply connected domain on which a single-valued analytic branch exists, and gluing the sheets across the cut produces the Riemann surface on which the function is globally single-valued and holomorphic.

Why it matters

Almost every closed-form special function — logarithms, fractional and complex powers, inverse trigonometric and elliptic functions — is multivalued. Without a systematic account of branch points and cuts you cannot integrate them, cannot state where they are analytic, and cannot even assign them a definite value. The Riemann-surface picture converts a "many-valued" object on \( \mathbb{C} \) into an honest single-valued holomorphic function on a curved domain, restoring all the machinery (Cauchy's theorem, residues, conformality) built for single-valued maps.

Physically the same structure governs the analytic continuation of scattering amplitudes and Green's functions, the \( \sqrt{k^2-k_0^2} \) cuts of dispersion relations, adiabatic (Berry) phases that accumulate on encircling a degeneracy, and the two-sheeted energy surfaces of level crossings — all instances of monodromy about a branch point. That places the topic squarely on the symmetry thread: the deck transformations of the covering surface form a group.

Assumptions

The functions \( \exp \) and \( \arg \) supply the model.We define \( \log \) as a right inverse of the entire function \( \exp \); if instead one only posited a formal power series, the global multivalued structure and the \( 2\pi i \) periodicity of \( \exp \) that generates it would not be visible. Branch points are isolated.We assume the singular set where continuation is obstructed consists of isolated points (plus \( \infty \)); if branch points accumulated or formed a natural boundary, no finite system of cuts would single-value the function. Analytic continuation is path-dependent only through homotopy class.By the monodromy theorem, continuation of a locally analytic germ along a path depends only on the path's homotopy class in the punctured domain; drop simple connectedness of the cut plane and the "branch" would fail to be well defined even locally. The Riemann surface is taken as the maximal analytic continuation.We identify the surface with the connected covering built from all germs; if one instead glued sheets by an arbitrary prescription the result need not be a holomorphic covering and \( f \) need not lift to a single-valued map.

Derivation

\[ z = r\,e^{i\theta}, \qquad r=|z|>0,\quad \theta=\arg z. \]

Polar form of a nonzero complex number; \( \theta \) is defined only modulo \( 2\pi \), which is the seed of everything that follows. A

\[ w=\log z \iff e^{w}=z,\qquad w=u+iv. \]

Define the logarithm as the inverse relation of the exponential, the only definition that respects \( e^{w_1+w_2}=e^{w_1}e^{w_2}\). A

\[ e^{u}e^{iv}=r\,e^{i\theta}\;\Longrightarrow\; e^{u}=r,\quad v=\theta+2\pi k,\ k\in\mathbb{Z}. \]

Match modulus and argument. The modulus fixes \( u \) uniquely but the phase is fixed only up to full turns because \( e^{i\cdot 2\pi k}=1 \). A

\[ \boxed{\;\log z = \ln r + i(\theta + 2\pi k)\;},\qquad k\in\mathbb{Z}. \]

Solve for \( w=u+iv \). The countable family indexed by \( k \) exhibits the multivaluedness explicitly; \( k=0 \) with \( \theta\in(-\pi,\pi] \) is the principal branch \( \operatorname{Log} z \). A

\[ \gamma(t)=r\,e^{i(\theta_0+2\pi t)},\quad t:0\to1 \;\Longrightarrow\; \theta:\theta_0\to\theta_0+2\pi. \]

Take a closed loop of winding number \( +1 \) about the origin and continue \( \log \) analytically along it. Continuity forces \( \theta \) to increase continuously, not to jump. B

\[ \Delta(\log z)=\oint_{\gamma}\frac{dz}{z}=\oint_{\gamma} d(\ln r + i\theta)=2\pi i. \]

Since \( (\log z)'=1/z \) (from step 2, or verify below), the change on returning equals the contour integral of \( 1/z \); \( \ln r \) is single-valued so only the phase contributes \( i\cdot 2\pi \). This is the monodromy. B

\[ \text{loop not enclosing }0:\ \oint\frac{dz}{z}=0 \;\Longrightarrow\; \text{value unchanged}. \]

A contractible loop (winding number \( 0 \)) returns \( \log \) to itself. Hence the obstruction is concentrated at \( z=0 \): it is a branch point. Inversion \( z\mapsto1/z \) shows \( z=\infty \) is the second branch point. B

\[ \operatorname{Log} z=\underbrace{\tfrac12\ln(x^2+y^2)}_{u}+\,i\,\underbrace{\arg(x+iy)}_{v},\quad \frac{\partial u}{\partial x}=\frac{x}{x^2+y^2}=\frac{\partial v}{\partial y},\ \ \frac{\partial u}{\partial y}=\frac{y}{x^2+y^2}=-\frac{\partial v}{\partial x}. \]

On the cut plane \( \mathbb{C}\setminus(-\infty,0] \) the principal branch satisfies the Cauchy–Riemann equations (prior result), so it is holomorphic there with \( \operatorname{Log}'z=u_x+iv_x=\dfrac{x-iy}{x^2+y^2}=\dfrac1z \). The cut is the locus where \( \arg \) is discontinuous. C

\[ \text{cut plane }\mathbb{C}\setminus\{\text{ray }0\to\infty\}\ \text{is simply connected}\;\Longrightarrow\; \text{branch single-valued \& analytic.} \]

Deleting a curve joining the two branch points forbids loops of nonzero winding, killing the monodromy; the monodromy theorem then yields a genuine single-valued analytic branch on the remainder. B

\[ \mathcal{R}=\bigcup_{k\in\mathbb{Z}} S_k,\qquad \text{upper edge of }S_k \ \sim\ \text{lower edge of }S_{k+1}, \]

\[ \Phi:\mathcal{R}\to\mathbb{C},\quad \Phi(z,k)=\ln r+i(\theta+2\pi k)\ \text{single-valued, holomorphic, bijective onto }\mathbb{C}. \]

Glue a countable stack of cut planes \( S_k \), joining each to the next across the cut so that crossing the cut advances \( k\to k+1 \). On this Riemann surface (a helicoid over the punctured plane) \( \log \) lifts to a single-valued holomorphic function; the deck group \( k\mapsto k+n \) is \( \mathbb{Z} \). C

\[ z^{\alpha}:=e^{\alpha\log z}=e^{\alpha(\ln r+i\theta)}\,e^{2\pi i\alpha k}. \]

Powers inherit their monodromy from \( \log \). The distinct values are labelled by \( e^{2\pi i\alpha k} \): for \( \alpha=p/q \) (lowest terms) exactly \( q \) values, so the branch point has finite order \( q \) and the surface has \( q \) sheets; for irrational or complex \( \alpha \), infinitely many. C

Result

\[ \log z = \ln|z| + i\big(\arg z + 2\pi k\big),\quad z\!\underset{n}{\circlearrowleft}\!0 \Rightarrow \Delta\log z = 2\pi i\,n,\qquad z^{p/q}\ \text{has}\ q\ \text{sheets}. \]

Reading. The logarithm's value is fixed up to integer multiples of \( 2\pi i \); which multiple you hold depends on how many times your path has wound around the branch point \( z=0 \). Cutting the plane from \( 0 \) to \( \infty \) makes a single branch well defined; stacking the branches across the cut builds the Riemann surface on which the function is a single-valued holomorphic map. A power \( z^{p/q} \) closes up after \( q \) turns, giving a \( q \)-sheeted surface.

Units check. The arguments of \( \log \) and of \( z^{\alpha} \) must be dimensionless: \( \ln|z| \) needs \( |z| \) pure, and the additive constant \( 2\pi i \) is dimensionless (radians of phase), so both sides of each equation carry no physical units. In applications one always logs a ratio, e.g. \( \ln(r/r_0) \), restoring dimensionlessness.

Limiting cases

\( \alpha=n\in\mathbb{Z} \): \( z^{n}=e^{n\log z} \) is single-valued (the \( 2\pi i n k \) phase is trivial) — integer powers have no branch point, one sheet.
\( \alpha=1/q \): exactly \( q \) sheets; \( q=2 \) is the double cover \( w^2=z \), the model square-root Riemann surface.
\( k=0,\ \theta\in(-\pi,\pi] \): recovers the principal branch \( \operatorname{Log} z \), agreeing with \( \ln x \) on the positive real axis.
\( r\to\infty \) or \( r\to0 \): \( \operatorname{Re}\log z=\ln r\to\pm\infty \) — the branch points at \( 0 \) and \( \infty \) are logarithmic singularities, not poles.
\( \alpha\to0 \): \( z^{\alpha}=1+\alpha\log z+O(\alpha^2)\to1 \), and the multivaluedness weakens as the residual \( \alpha\log z \) term.

Breaks when

Path crosses a branch point. Analytic continuation is defined only along paths avoiding the branch points; a contour dragged through \( z=0 \) has no well-defined continuation and the \( 2\pi i \) bookkeeping is meaningless there.
Branch points coalesce or accumulate. If two branch points merge (e.g. the discriminant of \( \sqrt{(z-a)(z-b)} \) as \( a\to b \)) the local order changes discontinuously; and functions with an accumulating set of branch points (a natural boundary) admit no finite cut system, so no global single-valued branch exists.
Cut endpoints don't pair the branch points correctly. A cut that fails to connect branch points whose combined monodromy is nontrivial leaves a residual loop with nonzero winding, so the "branch" is still multivalued (see the \( \log \) sum in the problems).
Non-dimensionless argument. Writing \( \log(x) \) for \( x \) carrying units is ill-posed; the expansion \( \ln(1+\epsilon) \) presumes a dimensionless \( \epsilon \).

Failure modes

Log-of-a-product slip: asserting \( \log(z_1 z_2)=\log z_1+\log z_2 \) as an equality of principal values. It holds only modulo \( 2\pi i \); e.g. \( \operatorname{Log}((-1)(-1))=0 \neq \operatorname{Log}(-1)+\operatorname{Log}(-1)=2\pi i \).
Power laws abused: writing \( (z_1 z_2)^{\alpha}=z_1^{\alpha}z_2^{\alpha} \) or \( (z^{a})^{b}=z^{ab} \) for the principal branch. Both fail across cuts because each side carries its own monodromy factor \( e^{2\pi i\alpha k} \).
Fixed cut assumed canonical: treating the negative real axis as "the" cut of \( \log \). The cut is a modelling choice; any curve from \( 0 \) to \( \infty \) works, and physics problems often demand a different one.
Miscounting sheets: claiming \( z^{2/6} \) has six sheets. Reduce to lowest terms \( 1/3 \) first — three sheets.
Forgetting \( \infty \): analysing \( \sqrt{z} \) with a branch point only at \( 0 \). The second branch point at \( \infty \) is what the cut's far end attaches to.
Sign of the jump: taking \( \Delta\log=+2\pi i \) regardless of orientation. Clockwise encirclement gives \( -2\pi i \); the sign is the winding number.

Discussion

The deepest message is that multivaluedness is a statement about topology, not about the formula. The germs of \( \log \) at a base point form the fibre of a covering space of the punctured plane \( \mathbb{C}\setminus\{0\} \); analytic continuation is the lift of paths, and monodromy is the induced action of the fundamental group \( \pi_1(\mathbb{C}\setminus\{0\})\cong\mathbb{Z} \) on that fibre. For \( \log \) the action is a free \( \mathbb{Z} \)-action (the universal cover, a helicoid); for \( z^{1/q} \) it factors through \( \mathbb{Z}/q\mathbb{Z} \), a \( q \)-fold cyclic cover. This is why the "symmetry" thread owns the topic: the deck-transformation group is a genuine symmetry group of the Riemann surface, and the number of sheets equals its order.

The Riemann surface reconciles two viewpoints. Locally, each branch is an ordinary holomorphic function obeying the Cauchy–Riemann equations, so all local calculus survives. Globally, the surface remembers the winding that a single copy of \( \mathbb{C} \) forgets, so contour integrals encircling a branch point return the honest \( 2\pi i \) increment rather than an ambiguity. Cauchy's residue theorem, applied on the surface with the cut respected, then computes real integrals such as \( \int_0^\infty \frac{x^{s-1}}{1+x}\,dx \) by exploiting exactly the \( e^{2\pi i s} \) jump across the cut.

At the sharp end, the surface is the natural home of the function, and its genus encodes the global analytic data. For \( w^2=P(z) \) with \( P \) a degree-\( (2g+1) \) or \( (2g+2) \) polynomial with distinct roots, the branch points are the roots (paired by cuts), and the compactified two-sheeted cover is a hyperelliptic Riemann surface of genus \( g \); the holomorphic differentials \( z^{j}\,dz/w \), \( j=0,\dots,g-1 \), span a \( g \)-dimensional space by Riemann–Roch. Thus branch-point combinatorics on the sphere translate directly into the topology (handles) of the surface — the same mechanism that turns \( \sqrt{(z-e_1)(z-e_2)(z-e_3)} \) into the torus underlying the Weierstrass \( \wp \)-function and elliptic integrals.

Common misconceptions. A branch cut is not a physical discontinuity of the function — it is a bookkeeping curve we choose so that one sheet can be displayed on the plane; on the Riemann surface the function is perfectly smooth as you slide from one sheet to the next. Likewise "the value of \( \log(-1) \)" is not \( i\pi \) intrinsically; \( i\pi \) is merely the principal representative of the whole set \( i(2k+1)\pi \).

Worked examples

\[ \text{Evaluate all values of }\ \log(-1)\ \text{and identify the principal one.} \]

Task: apply the master formula with \( z=-1 \). A

\[ |{-1}|=1,\quad \arg(-1)=\pi \;\Rightarrow\; r=1,\ \theta=\pi. \]

Read off modulus and a representative argument. A

\[ \log(-1)=\ln 1 + i(\pi+2\pi k)=i(2k+1)\pi,\quad k\in\mathbb{Z}. \]

Substitute; \( \ln 1=0 \) removes the real part. A

\[ \log(-1)=i(2k+1)\pi=\{\dots,-i\pi,\ \boxed{i\pi},\ 3i\pi,\dots\}. \]

Reading. Purely imaginary values spaced by \( 2\pi i \); principal value \( \operatorname{Log}(-1)=i\pi \) (\( k=0 \)). Consistent with Euler: \( e^{i\pi}=-1 \).

Units check. Dimensionless argument \( -1 \); result is a phase in radians times \( i \), dimensionless.

\[ \text{Evaluate all values of }\ i^{\,i}. \]

Task: use \( z^{\alpha}=e^{\alpha\log z} \) with \( z=\alpha=i \). B

\[ \log i=\ln 1 + i\!\left(\tfrac{\pi}{2}+2\pi k\right)=i\!\left(\tfrac{\pi}{2}+2\pi k\right). \]

Apply the master formula: \( |i|=1,\ \arg i=\pi/2 \). A

\[ i^{\,i}=e^{\,i\log i}=\exp\!\Big[i\cdot i\big(\tfrac{\pi}{2}+2\pi k\big)\Big]=\exp\!\Big[-\big(\tfrac{\pi}{2}+2\pi k\big)\Big]. \]

Multiply the exponent by \( i \); \( i^2=-1 \) makes it real and negative. B

\[ k=0:\ e^{-\pi/2}\approx 0.20788,\qquad k=-1:\ e^{3\pi/2}\approx 111.3,\ \dots \]

Enumerate values; principal value at \( k=0 \). A

\[ i^{\,i}=e^{-(\pi/2+2\pi k)}\in\mathbb{R}_{>0},\qquad \boxed{\ (i^{\,i})_{\text{principal}}=e^{-\pi/2}\approx0.2079\ }. \]

Reading. A famously real, positive, and infinitely-multivalued result: each extra winding of \( \log i \) rescales the answer by \( e^{-2\pi}\approx1.87\times10^{-3} \).

Units check. Base and exponent dimensionless; the exponent \( -(\pi/2+2\pi k) \) is a pure number, so \( i^{i} \) is dimensionless.

Problems

Find all values of \( \log(1+i) \) and its principal value.
Solution
\( |1+i|=\sqrt2 \), \( \arg(1+i)=\pi/4 \). Hence \( \log(1+i)=\tfrac12\ln 2 + i\big(\tfrac{\pi}{4}+2\pi k\big) \). Principal value \( \operatorname{Log}(1+i)=\tfrac12\ln 2 + i\tfrac{\pi}{4}\approx 0.3466 + 0.7854\,i \).
How many sheets does the Riemann surface of \( z^{4/6} \) have, and what is the order of its branch point at \( z=0 \)?
Solution
Reduce \( 4/6=2/3 \) to lowest terms. The distinct values are \( e^{2\pi i(2/3)k}=e^{4\pi i k/3} \), which cycle with period \( 3 \) (\( k=0,1,2 \)). So the branch point at \( 0 \) has order \( 3 \) and the surface has \( 3 \) sheets. (Also a branch point of order \( 3 \) at \( \infty \).)
Continue \( \sqrt{z} \) once counter-clockwise around the unit circle starting from the branch with \( \sqrt{1}=+1 \). What value returns, and how many turns restore \( +1 \)?
Solution
On the circle \( z=e^{i\theta} \), \( \sqrt z=e^{i\theta/2} \). Start \( \theta=0\to+1 \). After one turn \( \theta=2\pi \): \( \sqrt z=e^{i\pi}=-1 \). The value has flipped sign — you are on the second sheet. A second turn (\( \theta=4\pi \)) gives \( e^{2\pi i}=+1 \), restoring the start. Thus \( 2 \) turns close the double cover, consistent with order \( 2 \).
Find all cube roots of \( -8 \) via \( (-8)^{1/3}=e^{\frac13\log(-8)} \).
Solution
\( |-8|=8,\ \arg(-8)=\pi \), so \( \log(-8)=\ln 8 + i(\pi+2\pi k) \). Then \( (-8)^{1/3}=e^{\frac13\ln 8}\,e^{\frac{i}{3}(\pi+2\pi k)}=2\,e^{i(\pi+2\pi k)/3} \). For \( k=0,1,2 \): \( 2e^{i\pi/3}=1+i\sqrt3 \); \( 2e^{i\pi}=-2 \); \( 2e^{i5\pi/3}=1-i\sqrt3 \). Three values, the real one \( -2 \) being the principal-real cube root.
For \( f(z)=\log(z-a)+\log(z-b) \) with \( a\neq b \), locate the branch points and determine the monodromy about a large loop enclosing both. Can a single finite cut joining \( a \) to \( b \) single-value \( f \)? Contrast with \( g(z)=\log\frac{z-a}{z-b} \).
Solution
Branch points of \( f \) at \( z=a \) and \( z=b \) (each contributing a \( \log \)), plus one at \( \infty \). Encircle both once: each logarithm gains \( 2\pi i \), so \( \Delta f = 2\pi i + 2\pi i = 4\pi i \neq 0 \). A large loop is nontrivial, so a finite cut joining \( a \) to \( b \) does not single-value \( f \); one needs two cuts running from \( a \) and \( b \) out to \( \infty \) (equivalently, cut to infinity). For \( g \), the loop gives \( \Delta g = 2\pi i - 2\pi i = 0 \): the large loop is trivial, so a single finite cut from \( a \) to \( b \) suffices, and \( g \) is single-valued outside it. This is the standard two-branch-point double-cut versus finite-cut distinction.