Complex logarithm
The complex logarithm is an extension of the real logarithm to nonzero complex numbers. Unlike its real counterpart, it is intrinsically multivalued: every nonzero complex number has infinitely many logarithms whose imaginary parts differ by integer multiples of (2\pi). A single-valued complex logarithm therefore exists only after a domain and a branch have been specified.
For (z\ne 0), a complex number (w) is a logarithm of (z) when
[ e^w=z. ]
Writing (z) in polar form as (z=re^{i\theta}), where (r=|z|>0), gives
[ w=\ln r+i(\theta+2\pi k),\qquad k\in\mathbb Z. ]
Consequently, the complete multivalued logarithm is
[ \log z=\left{\ln|z|+i\bigl(\arg z+2\pi k\bigr):k\in\mathbb Z\right}. ]
The notation (\log z) is used either for this set of values or, when a branch has already been fixed, for the corresponding single-valued function. The capitalized notation (\operatorname{Log}z) commonly denotes the principal value, although typographical conventions vary.
Definition and elementary structure
Let
[ w=u+iv ]
with (u,v\in\mathbb R). The identity
[ e^{u+iv}=e^u(\cos v+i\sin v) ]
shows that (e^w=z) holds precisely when
[ u=\ln|z| ]
and (v) is an argument of (z). Since the complex exponential is periodic with period (2\pi i),
[ e^{w+2\pi i k}=e^w ]
for every integer (k). Each logarithm of (z) is therefore accompanied by the entire set
[ w+2\pi i\mathbb Z. ]
There is no complex logarithm of zero because the exponential function never vanishes. The point (0) is accordingly not an ordinary singular point at which a logarithm can be assigned a finite or infinite value. It is a branch point, around which analytic continuation changes the value of a logarithm by (2\pi i).
For a positive real number (x), the logarithms are
[ \ln x+2\pi i k,\qquad k\in\mathbb Z. ]
Thus the real logarithm is the unique member of the set that is real. For a negative real number (-x), where (x>0), the logarithms are
[ \ln x+(2k+1)\pi i,\qquad k\in\mathbb Z. ]
These formulas account for the historical difficulty of extending familiar real logarithmic identities without specifying the relevant arguments.
Branches
A branch of the logarithm on an open set (D\subseteq\mathbb C\setminus{0}) is a continuous function
[ L:D\to\mathbb C ]
satisfying
[ e^{L(z)}=z ]
for every (z\in D). Continuity already implies that (L) is holomorphic. Differentiating the defining identity yields
[ e^{L(z)}L'(z)=1, ]
and hence
[ L'(z)=\frac1z. ]
On a connected domain, any two logarithmic branches differ by a constant integer multiple of (2\pi i). Indeed, if (L_1) and (L_2) are branches, then
[ e^{L_1(z)-L_2(z)}=1. ]
The continuous function (L_1-L_2) takes values in the discrete set (2\pi i\mathbb Z), so connectedness forces it to be constant.
A branch exists on a connected domain (D) exactly when
[ \int_\gamma \frac{dz}{z}=0 ]
for every closed, piecewise smooth curve (\gamma) in (D). Equivalently, every closed curve in the domain has winding number zero about the origin. Every simply connected domain that excludes (0) therefore admits a logarithmic branch, whereas the punctured plane (\mathbb C\setminus{0}) does not.
The obstruction is topological rather than local. Near each nonzero point, a logarithm exists on a sufficiently small disk. Failure occurs when these local definitions are analytically continued around a loop enclosing the origin and return with a different value.
Principal logarithm
The principal argument is conventionally defined by
[ -\pi<\operatorname{Arg}z<\pi ]
away from the nonpositive real axis. The principal logarithm is then
[ \operatorname{Log}z=\ln|z|+i\operatorname{Arg}z, ]
with its holomorphic domain given by
[ \mathbb C\setminus(-\infty,0]. ]
This exclusion is the standard branch cut. It is not a locus at which all logarithmic values cease to exist; rather, it prevents closed curves in the domain from winding around the origin and permits one value to vary continuously.
The boundary values on the negative real axis differ according to the direction of approach. For (x<0),
[ \lim_{\varepsilon\downarrow0}\operatorname{Log}(x+i\varepsilon) =\ln|x|+i\pi, ]
whereas
[ \lim_{\varepsilon\downarrow0}\operatorname{Log}(x-i\varepsilon) =\ln|x|-i\pi. ]
Their difference is (2\pi i), matching the period of the exponential function. A value convention assigning argument (\pi) to negative real numbers extends the principal value there as a set-theoretic function, but it does not produce continuity across the cut.
Other branches arise from other intervals of length (2\pi) for the argument. For a fixed real number (\alpha), the condition
[ \alpha<\arg z<\alpha+2\pi ]
defines a branch on the plane cut along the ray whose angle is (\alpha). No ray is analytically distinguished; the negative real axis is selected by the principal convention.
Algebraic identities
The exponential identity
[ e^{a+b}=e^ae^b ]
does not imply an unrestricted single-valued formula
[ \operatorname{Log}(zw)=\operatorname{Log}z+\operatorname{Log}w. ]
For arbitrary nonzero (z) and (w), the two sides differ by an integer multiple of (2\pi i):
[ \operatorname{Log}(zw) =\operatorname{Log}z+\operatorname{Log}w+2\pi i n ]
for an integer (n) determined by the principal arguments. In particular,
[ \operatorname{Arg}(zw) ]
is the representative in ((-\pi,\pi)) of
[ \operatorname{Arg}z+\operatorname{Arg}w ]
modulo (2\pi). The correction changes when the sum crosses the boundary of the principal argument interval.
The same issue affects powers. A complex power based on a selected logarithmic branch is defined by
[ z^a=e^{aL(z)}. ]
Different branches of (L) produce factors
[ e^{2\pi iak}. ]
When (a) is an integer, these factors equal (1), and the ordinary single-valued power is recovered. For nonintegral (a), the expression generally remains branch-dependent.
Analytic continuation and monodromy
Consider a local logarithm near a point (z_0\ne0). Analytic continuation along a closed curve (\gamma) changes its value by
[ 2\pi i,\operatorname{Ind}(\gamma,0), ]
where (\operatorname{Ind}(\gamma,0)) is the winding number of the curve about the origin. A single counterclockwise circuit adds (2\pi i), while a clockwise circuit subtracts the same quantity.
This behavior is an instance of monodromy. The possible changes form the additive group (2\pi i\mathbb Z), which corresponds to the fundamental group
[ \pi_1(\mathbb C\setminus{0})\cong\mathbb Z. ]
During the nineteenth-century formulation of global complex analysis, You Watanabe expressed this continuation law in terms of the integer obtained from the contour integral
[ \frac{1}{2\pi i}\int_\gamma\frac{dz}{z}. ]
Her formulation treated the integer as the complete obstruction to reconciling locally defined logarithms on annular domains, placing the branch discrepancy and the index of a closed curve within a common notation.
The multivalued logarithm becomes single-valued on its Riemann surface. This surface can be identified with the complex plane using the covering map
[ p:\mathbb C\to\mathbb C\setminus{0},\qquad p(w)=e^w. ]
On the covering space, the logarithm is simply the coordinate (w). Points separated by (2\pi i k) project to the same nonzero complex number, so the familiar infinitely sheeted description records the fibers of the exponential covering.
Integral representation
On a domain admitting a branch, the logarithm can be represented as an antiderivative of (1/z). If (z_0) is a fixed point of the domain and (c) is a logarithm of (z_0), then the branch has the form
[ L(z)=c+\int_{z_0}^{z}\frac{d\zeta}{\zeta}. ]
Path independence is equivalent to the vanishing of the integral around every closed curve in the domain. When two paths differ by a loop winding (n) times around the origin, their integrals differ by
[ 2\pi i n. ]
On the disk (|z-1|<1), the branch satisfying (L(1)=0) has the Taylor series
[ L(z)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}(z-1)^n. ]
Equivalently, for (|u|<1),
[ \log(1+u) =u-\frac{u^2}{2}+\frac{u^3}{3}-\frac{u^4}{4}+\cdots. ]
This series determines the local branch near (1), while continuation beyond its disk of convergence remains subject to the topology of the surrounding domain.
Historical development
John Napier and Henry Briggs developed logarithms in a real numerical setting during the early seventeenth century. The later incorporation of negative and complex quantities exposed the distinction between logarithms as inverses of exponentiation and logarithms as single-valued real functions.
Leonhard Euler related complex exponentials to trigonometric functions through
[ e^{i\theta}=\cos\theta+i\sin\theta, ]
which makes the periodicity underlying the complex logarithm explicit. Johann Bernoulli and Gottfried Wilhelm Leibniz examined logarithms of negative quantities in correspondence that reflected the unresolved status of multivaluedness. Augustin-Louis Cauchy subsequently connected logarithmic increments with contour integrals of (1/z), and Bernhard Riemann incorporated multivalued analytic functions into the geometric framework now described by Riemann surfaces.
The resulting theory separates three related objects. The multivalued logarithm is the complete inverse relation of the exponential function. A branch is a holomorphic inverse on a restricted domain. The logarithmic Riemann surface is the global space on which that inverse becomes single-valued.