Complex logarithm - AbsoluteAstronomy.com

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

, a complex logarithm function is an "inverse

Inverse function

In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...

" of the complex exponential function, just as the natural logarithm

Natural logarithm

The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

ln x is the inverse of the real exponential function

Exponential function

In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

e^x. Thus, a logarithm of z is a complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

w such that e^w = z. The notation for such a w is log z. But because every nonzero complex number z has infinitely many logarithms, care is required to give this notation an unambiguous meaning.

If z = reiθ with r > 0 (polar form), then w = ln r + iθ is one logarithm of z; adding integer multiples of 2πi gives all the others.

Problems with inverting the complex exponential function
For a function to have an inverse
Inverse function
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...

, it must map distinct values to distinct values. But the complex exponential function does not have this property: ew+2πi = ew for any w, since adding iθ to w has the effect of rotating ew counterclockwise θ radians. Even worse, the infinitely many numbers

forming a sequence of equally spaced points along a vertical line, are all mapped to the same number by the exponential function. So the exponential function does not have an inverse function in the standard sense.

There are two solutions to this problem.

One is to restrict the domain of the exponential function to a region that does not contain any two numbers differing by an integer multiple of 2πi: this leads naturally to the definition of branches of log z, which are certain functions that single out one logarithm of each number in their domains. This is analogous to the definition of sin⁻¹x on [−1,1] as the inverse of the restriction of sin θSine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....

to the interval [−π/2,π/2]: there are many real numbers θ with sin θ = x, but one (somewhat arbitrarily) chooses the one in [−π/2,π/2].

Another way to resolve the indeterminacy is to view the logarithm as a function whose domain is not a region in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

, but a Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

that covers the punctured complex plane in an infinite-to-1 way.

Branches have the advantage that they can be evaluated at complex numbers. On the other hand, the function on the Riemann surface is elegant in that it packages together all branches of log z and does not require any choice for its definition.

Definition of principal value
For each nonzero complex number z, the principal value Log z is the logarithm whose imaginary part lies in the interval (−π,π]. The expression Log 0 is left undefined since there is no complex number w satisfying ew = 0.

The principal value can be described also in a few other ways.

To give a formula for Log z, begin by expressing z in polar form, z = reiθ. Given z, the polar form is not quite unique, because of the possibility of adding an integer multiple of 2π to θ, but it can be made unique by requiring θ to lie in the interval (−π,π]; this θ is called the principal value of the argument, and is sometimes written Arg z
Arg (mathematics)
In mathematics, arg is a function operating on complex numbers . It gives the angle between the line joining the point to the origin and the positive real axis, shown as in figure 1 opposite, known as an argument of the point In mathematics, arg is a function operating on complex numbers...

. Then the principal value of the logarithm can be defined by

For example, Log(-3i) = ln 3 − πi/2.

Another way to describe Log z is as the inverse of a restriction of the complex exponential function, as in the previous section.
The horizontal strip S consisting of complex numbers w = x+yi such that −π < y ≤ π is an example of a region not containing any two numbers differing by an integer multiple of 2πi, so the restriction of the exponential function to S has an inverse. In fact, the exponential function maps S bijectively
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

to the punctured complex plane , and the inverse of this restriction is . The conformal mapping section below explains the geometric properties of this map in more detail.

When the notation log z appears without any particular logarithm having been specified, it is generally best to assume that the principal value is intended. In particular, this gives a value consistent with the real value of ln z when z is a positive real number.
The capitalization in the notation Log is used by some authors to distinguish the principal value from other logarithms of z.

A common source of errors in dealing with complex logarithms is to assume that identities satisfied by ln extend to complex numbers. It is true that e^Logz = z for all z ≠ 0 (this is what it means for Log z to be a logarithm of z), but the identity Log ez = z fails for z outside the strip S. For this reason, one cannot always apply Log to both sides of an identity ez = ew to deduce z = w. Also, the identity Log(z₁z₂) = Log z₁ + Log z₂ can fail: the two sides can differ by an integer multiple of 2πi : for instance,

The function Log z is discontinuous at each negative real number, but continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

everywhere else in . To explain the discontinuity, consider what happens to Arg z as z approaches a negative real number a. If z approaches a from above, then Arg z approaches π, which is also the value of Arg a itself. But if z approaches a from below, then Arg z approaches −π. So Arg z "jumps" by 2π as z crosses the negative real axis, and similarly Log z jumps by 2πi.

Branches of the complex logarithm
Is there a different way to choose a logarithm of each nonzero complex number so as to make a function L(z) that is continuous on all of ? Unfortunately, the answer is no. To see why, imagine tracking such a logarithm function along the unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

, by evaluating L at eiθ as θ increases from 0 to 2π. For simplicity, suppose that the starting value L(1) is 0. Then for L(z) to be continuous, L(eiθ) must agree with iθ as θ increases (the difference is a continuous function of θ taking values in the discrete set ). In particular, L(e²πi) = 2πi, but e²πi = 1, so this contradicts L(1) = 0.

To obtain a continuous logarithm defined on complex numbers, it is hence necessary to restrict the domain to a smaller subset U of the complex plane. Because one of the goals is to be able to differentiate
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

the function, it is reasonable to assume that the function is defined on a neighborhood of each point of its domain; in other words, U should be an open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

. Also, it is reasonable to assume that U is connected
Connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected...

, since otherwise the function on different components of U would be unrelated to each other. All this motivates the following definition:

A branch of log z is a continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

L(z) defined on a connected open subset
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

U of the complex plane such that L(z) is a logarithm of z for each z in U.

For example, the principal value defines a branch on the open set where it is continuous, which is the set

obtained by removing 0 and all negative real numbers from the complex plane.

Another example: The Mercator series

converges locally uniformly for |u| < 1, so setting z = 1+u defines a branch of log z on the open disk of radius 1 centered at 1. (Actually, this is just a restriction of Log z, as can be shown by differentiating the difference and comparing values at 1.)

Once a branch is fixed, it may be denoted "log z" if no confusion can result. Different branches can give different values for the logarithm of a particular complex number, however, so a branch must be fixed in advance (or else the principal branch must be understood) in order for "log z" to have a precise unambiguous meaning.

Branch cuts
The argument above involving the unit circle generalizes to show that no branch of log z exists on an open set U containing a closed curve that winds
Winding number
In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...

around 0. To foil this argument, U is typically chosen as the complement of a ray or curve in the complex plane going from 0 (inclusive) to infinity in some direction. In this case, the curve is known as a branch cut. For example, the principal branch has a branch cut along the negative real axis.

If the function L(z) is extended to be defined at a point of the branch cut, it will necessarily be discontinuous there; at best it will be continuous "on one side", like Log z at a negative real number.

The derivative of the complex logarithm
Each branch L(z) of log z on an open set U is an inverse of a restriction of the exponential function, namely the restriction to the image of U under L. Since the exponential function is holomorphic (i.e., complex differentiable) with nonvanishing derivative, the complex analogue of the inverse function theorem
Inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain...

applies. It shows that L(z) is holomorphic at each z in U, and L′(z) = 1/z. Another way to prove this is to check the Cauchy-Riemann equations in polar coordinates.

Constructing branches via integration
The function ln x for x > 0 can be constructed by the formula

If the range of integration started at a positive number a other than 1, the formula would have to be

instead.

In developing the analogue for the complex logarithm, there is an additional complication: the definition of the complex integral requires a choice of path. Fortunately, if the integrand is holomorphic, then the value of the integral is unchanged by deforming the path
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...

(while holding the endpoints fixed), and in a simply connected region U (a region with "no holes") any path from a to z inside U can be continuously deformed
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...

inside U into any other. All this leads to the following:

If U is a simply connected open subset of not containing 0, then a branch of log z defined on U can be constructed by choosing a starting point a in U, choosing a logarithm b of a, and defining

for each z in U.

The complex logarithm as a conformal map

Any holomorphic map satisfying for all is a conformal map
Conformal map
In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...

, which means that if two curves passing through a point a of U form an angle α (in the sense that the tangent lines to the curves at a form an angle α), then the images of the two curves form the same angle α at f(a).
Since a branch of log z is holomorphic, and since its derivative 1/z is never 0, it defines a conformal map
Conformal map
In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...

.

For example, the principal branch w = Log z, viewed as a mapping from to the horizontal strip defined by |Im z| < π, has the following properties, which are direct consequences of the formula in terms of polar form:

Circles in the z-plane centered at 0 are mapped to vertical segments in the w-plane connecting a − πi to a + πi, where a is a real number depending on the radius of the circle.

Rays emanating from 0 in the z-plane are mapped to horizontal lines in the w-plane.

Each circle and ray in the z-plane as above meet at a right angle. Their images under Log are a vertical segment and a horizontal line (respectively) in the w-plane, and these too meet at a right angle. This is an illustration of the conformal property of Log.

The associated Riemann surface

Construction
The various branches of log z cannot be glued to give a single function because two branches may give different values at a point where both are defined. Compare, for example, the principal branch Log(z) on with imaginary part θ in (−π,π) and the branch L(z) on whose imaginary part θ lies in (0,2π). These agree on the upper half plane, but not on the lower half plane. So it makes sense to glue the domains of these branches only along the copies of the upper half plane. The resulting glued domain is connected, but it has two copies of the lower half plane. Those two copies can be visualized as two levels of a parking garage, and one can get from the Log level of the lower half plane up to the L level of the lower half plane by going 360° counterclockwise around 0, first crossing the positive real axis (of the Log level) into the shared copy of the upper half plane and then crossing the negative real axis (of the L level) into the L level of the lower half plane.

One can continue by gluing branches with imaginary part θ in (π,3π), in (2π,4π), and so on, and in the other direction, branches with imaginary part θ in (−2π,0), in (−3π,−π), and so on. The final result is a connected surface that can be viewed as a spiralling parking garage with infinitely many levels extending both upward and downward. This is the Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

R associated to log z.

A point on R can be thought of as a pair (z,θ) where θ is a possible value of the argument of z. In this way, R can be embedded in .

The logarithm function on the Riemann surface
Because the domains of the branches were glued only along open sets where their values agreed, the branches glue to give a single well-defined function . It maps each point (z,θ) on R to ln |z| + iθ. This process of extending the original branch Log by gluing compatible holomorphic functions is known as analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...

.

There is a "projection map" from R down to that "flattens" the spiral, sending (z,θ) to z. For any , if one takes all the points (z,θ) of R lying "directly above" z and evaluates logR at all these points, one gets all the logarithms of z.

Gluing all branches of log z

Instead of gluing only the branches chosen above, one can start with all branches of log z, and simultaneously glue every pair of branches and along the largest open subset of on which L₁ and L₂ agree. This yields the same Riemann surface R and function logR as before. This approach, although slightly harder to visualize, is more natural in that it does not require selecting any particular branches.

If U′ is an open subset of R projecting bijectively to its image U in , then the restriction of logR to U′ corresponds to a branch of log z defined on U. Every branch of log z arises in this way.

The Riemann surface as a universal cover
The projection map realizes R as a covering space of . In fact, it is a Galois covering with deck transformation group isomorphic to , generated by the homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

sending (z,θ) to (z,θ+2π).

As a complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

, R is biholomorphic with via logR. (The inverse map sends z to (ez,Im z).) This shows that R is simply connected, so R is the universal cover of .

Applications

The complex logarithm is needed to define exponentation in which the base is a complex number. Namely, if a and b are complex numbers with a ≠ 0, one can use the principal value to define ab = eb Log a. One can also replace Log a by other logarithms of a to obtain other values of ab.

Since the mapping w = Log z transforms circles centered at 0 into vertical straight line segments, it is useful in engineering applications involving an annulus
Annulus (mathematics)
In mathematics, an annulus is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. Or, it is the area between two concentric circles...

.

Logarithms to other bases
Just as for real numbers, one can define log_{ab = (log}b)/(log a) for complex numbers a and b, the only caveat being that its value depends on the choice of a branch of log defined at a and b (with log a ≠ 0). For example, using the principal value gives

Logarithms of holomorphic functions
If f is a holomorphic function on a connected open subset U of , then a branch of log f on U is a continuous function g on U such that e^g(z) = f(z) for all z in U. Such a function g is necessarily holomorphic with g′(z) = f′(z)/f(z) for all z in U.

If U is a simply connected open subset of

, and f is a nowhere-vanishing holomorphic function on U, then a branch of log f defined on U can be constructed by choosing a starting point a in U, choosing a logarithm b of f(a), and defining

for each z in U.

Problems with inverting the complex exponential function

Definition of principal value

Branches of the complex logarithm

Branch cuts

The derivative of the complex logarithm

Constructing branches via integration

The complex logarithm as a conformal map

The associated Riemann surface

Construction

The logarithm function on the Riemann surface

Gluing all branches of log z

The Riemann surface as a universal cover

Applications

Logarithms to other bases

Logarithms of holomorphic functions

See also