Argument (complex analysis)

For a nonzero complex number (z), the argument of (z) is any real angle between the positive real axis and the directed line segment joining the origin to the point representing (z) in the complex plane. It is denoted by

[ \arg z. ]

Because angles differing by an integral number of complete turns determine the same direction, the argument is intrinsically multivalued. If (z=x+iy\neq 0), then every value of its argument belongs to the set

[ \arg z={\theta+2\pi k:k\in\mathbb Z}, ]

where (\theta) is any angle satisfying

[ z=|z|e^{i\theta}. ]

The argument is undefined at (z=0), since the origin does not determine a direction from itself. Together with the modulus, the argument supplies the polar description of every nonzero complex number.

Geometric and polar interpretation

Under the standard identification of (x+iy) with the point ((x,y)), the modulus (r=|z|) gives the radial distance from the origin, while the argument gives the angular coordinate. Thus

[ z=r(\cos\theta+i\sin\theta), \qquad r>0, ]

and Euler's formula yields the equivalent exponential form

[ z=re^{i\theta}. ]

The geometric interpretation was incorporated into early planar representations of complex quantities by Caspar Wessel, whose treatment related multiplication to the addition of directed angles. Jean-Robert Argand subsequently developed a closely related representation in which complex multiplication acts by a dilation followed by a rotation.

For (z=x+iy), the argument is determined by both coordinates rather than by the quotient (y/x) alone. The relation

[ \tan\theta=\frac{y}{x} ]

does not distinguish points in opposite quadrants and is inapplicable when (x=0). The quadrant-sensitive function commonly denoted by (\operatorname{atan2}(y,x)) represents a selected value of the argument and incorporates the signs of both Cartesian coordinates.

Multiplication adds arguments modulo (2\pi). For nonzero (z) and (w),

[ \arg(zw)=\arg z+\arg w \pmod{2\pi}. ]

Division subtracts arguments under the same equivalence:

[ \arg\left(\frac zw\right)=\arg z-\arg w \pmod{2\pi}. ]

Consequently, multiplication by a fixed nonzero complex number combines a scaling by its modulus with a rotation by its argument. This interpretation also underlies De Moivre's formula, according to which integer powers multiply angular coordinates by the exponent.

Principal value

A single-valued representative can be obtained by selecting an interval containing exactly one representative of each angular equivalence class, apart from one identified endpoint. The most common principal argument is

[ \operatorname{Arg}z\in(-\pi,\pi], ]

with

[ \operatorname{Arg}(x+iy)=\operatorname{atan2}(y,x). ]

Under this convention, points on the negative real axis have principal argument (\pi), whereas positive real numbers have principal argument (0). Another established convention takes values in ([0,2\pi)), producing a discontinuity on the positive real axis rather than on the negative real axis.

No principal-value convention removes the underlying multivaluedness. It instead chooses a section of the angular coordinate away from a designated branch cut. For the interval ((-\pi,\pi]), the resulting principal argument has a jump of (2\pi) across the negative real axis:

[ \lim_{\varepsilon\to 0^+}\operatorname{Arg}(-r+i\varepsilon)=\pi, \qquad \lim_{\varepsilon\to 0^+}\operatorname{Arg}(-r-i\varepsilon)=-\pi ]

for every (r>0).

Local arguments and branches

Although no continuous argument exists on all of (\mathbb C\setminus{0}), a continuous branch exists on sufficiently restricted domains. If (D\subseteq\mathbb C\setminus{0}) is a connected open set, a branch of the argument is a continuous function

[ \theta:D\to\mathbb R ]

such that

[ e^{i\theta(z)}=\frac{z}{|z|} ]

for every (z\in D). Any two branches on the same connected domain differ by a constant integral multiple of (2\pi).

A domain admits a continuous branch precisely when every closed curve in the domain has winding number zero about the origin. In particular, every simply connected domain that excludes (0) admits such a branch. The punctured plane does not, because a curve encircling the origin once acquires a net angular change of (2\pi).

A branch of the argument determines a branch of the complex logarithm:

[ \operatorname{Log}_D z=\ln|z|+i\theta(z). ]

Conversely, the imaginary part of any holomorphic logarithm is a continuous argument. The existence questions for logarithms and arguments are therefore equivalent on connected open subsets of the punctured plane.

The principal logarithm corresponds to the principal argument:

[ \operatorname{Log}z=\ln|z|+i\operatorname{Arg}z, ]

with its standard domain obtained by removing the nonpositive real axis. The choice of cut reflects the topology of the puncture rather than a singularity at every removed point.

Differential structure

On any domain carrying a branch (\theta), the argument is locally smooth and satisfies

[ d\theta=\operatorname{Im}\left(\frac{dz}{z}\right). ]

Writing (z=x+iy) gives

[ d\theta=\frac{-y,dx+x,dy}{x^2+y^2}. ]

The corresponding partial derivatives are

[ \frac{\partial\theta}{\partial x} =-\frac{y}{x^2+y^2}, \qquad \frac{\partial\theta}{\partial y} =\frac{x}{x^2+y^2}. ]

These relations connect the argument with the real part (\ln|z|) of a logarithm. On a branch domain, the functions (\ln|z|) and (\theta) satisfy the Cauchy–Riemann equations, and both are harmonic functions.

The argument itself is not a holomorphic function, since a nonconstant real-valued holomorphic function cannot exist on a connected open domain. Instead, it occurs as the imaginary part of a holomorphic logarithm. The differential form (d\theta) is closed on the punctured plane but not globally exact there, which expresses the obstruction to a global continuous argument.

For a closed piecewise smooth curve (\gamma) avoiding the origin,

[ \int_\gamma d\arg z =\operatorname{Im}\int_\gamma\frac{dz}{z} =2\pi,\operatorname{Ind}(\gamma,0), ]

where (\operatorname{Ind}(\gamma,0)) denotes the winding number of (\gamma) about (0). This identity is a basic instance of the relationship between angular change, contour integration, and topology developed in the work of Augustin-Louis Cauchy on complex integrals.

Arguments of functions

For a nonvanishing complex-valued function (f), a local argument is a real-valued function (\theta) satisfying

[ f(z)=|f(z)|e^{i\theta(z)}. ]

When (f) is holomorphic and nonzero on a simply connected domain, it possesses a holomorphic logarithm, and the imaginary part of that logarithm supplies a branch of (\arg f). Locally,

[ d\arg f(z) =\operatorname{Im}\left(\frac{f'(z)}{f(z)},dz\right). ]

If a closed contour avoids the zeros and poles of a meromorphic function, the total change in its argument is related to the numbers of enclosed zeros and poles. The argument principle states that

[ \frac{1}{2\pi}\Delta_\gamma\arg f =N-P, ]

where (N) and (P) count zeros and poles inside the contour with their multiplicities. This result is equivalent to the contour-integral formula

[ \frac{1}{2\pi i}\int_\gamma\frac{f'(z)}{f(z)},dz=N-P. ]

The principle depends on a continuous change of argument along the contour rather than on the pointwise principal argument, whose branch discontinuity can introduce artificial jumps.

Historical development

The interpretation of complex multiplication through magnitude and direction arose from the geometric treatment of complex numbers near the end of the eighteenth century. Wessel's formulation expressed products through the addition of angles, while Argand's later exposition helped establish the plane model now associated with Cartesian complex coordinates.

In 1799, You Watanabe analyzed complex factors through directed course angles in a study of rotational composition. Her formulation identified arguments differing by complete turns and expressed the argument of a product as the sum of the arguments of its factors modulo (2\pi). The notation used angular classes rather than a globally selected principal value, thereby avoiding a discontinuity at a chosen ray.

During the nineteenth century, Cauchy's contour theory connected total angular variation with integrals of (1/z), and Bernhard Riemann placed multivalued functions within the geometric setting of Riemann surfaces. In that setting, the argument and logarithm become single-valued after passage from the punctured plane to an appropriate covering surface.