Complex logarithm

Poll without page-refresh

Article on other languages:

	del.icio.us
	Digg
	Furl
	Reddit
	Rojo
	Add to OnlyWire

complex logarithm drawn with a color map

In complex analysis, a complex logarithm function is an "inverse" of the complex exponential function, just as the natural logarithm ln x is the inverse of the real exponential function e^x. So a logarithm of z is a complex number w such that e^w = z.^[1] The notation for such a w is log z. But because every nonzero complex number z has infinitely many logarithms,^[1] care is required to give this notation an unambiguous meaning.

If z = re^iθ with r > 0 (polar form), then w = ln r + iθ is one logarithm of z; adding integer multiples of 2πi gives all the others.^[1]

1 An inverse of the exponential function?
2 Definition of principal value
3 Branches of the complex logarithm
4 The complex logarithm as a conformal map
5 The associated Riemann surface
6 Applications
7 Generalizations
- 7.1 Logarithms to other bases
- 7.2 Logarithms of holomorphic functions
8 Derivation from the Taylor series
9 Log(z) as the inverse of the exponential function
10 Log(z) as a multi-valued function
11 The complex logarithm as a conformal mapping
- 11.1 Logarithms to other bases
12 Plots of the complex logarithm function (principal branch)
13 See also
14 Notes
15 References

An inverse of the exponential function?

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

$\ldots,\;w-4\pi i, \;w-2\pi i, \;w, \;w + 2\pi i, \;w+4\pi i, \;\ldots,$

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.^[2]^[3]

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 θ 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, but a Riemann surface 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 e^w = 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 = re^iθ. 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. Then the principal value of the logarithm^[1] can be defined by

Log z : = ln r + i θ = ln | z | + i Arg z .

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 to the punctured complex plane $\mathbb{C}^* = \mathbb{C} - \{0\}$ , and the inverse of this restriction is $\text{Log}\colon \mathbb{C}^* \to S$ . 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^[1] 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^Log z = z for all z ≠ 0 (this is what it means for Log z to be a logarithm of z), but the identity Log e^z = z fails for z outside the strip S. For this reason, one cannot always apply Log to both sides of an identity e^z = e^w to deduce z = w. Also, the identity Log(z₁z₂) = Log z₁ + Log z₂ can fail: the two sides can differ by by an integer multiple of 2πi : for instance,

$\text{Log}(-i) = -\frac{\pi i}{2} \ne \frac{\pi i}{2} + \pi i = \text{Log}(i) + \text{Log}(-1).$

The function Log z is discontinuous at each negative real number, but continuous everywhere else in $\mathbb{C}^*$ . 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. Crossing the negative real axis is like crossing the International Date Line!

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 $\mathbb{C}^*$ ? Unfortunately, the answer is no. To see why, imagine tracking such a logarithm function along the unit circle, by evaluating L at e^iθ 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(e^iθ) must agree with iθ as θ increases (the difference is a continuous function of θ taking values in the discrete set $2\pi i \mathbb{Z}$ ). In particular, L(e^2πi) = 2πi, but e^2πi = 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 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. Also, it is reasonable to assume that U is connected, 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 L(z) defined on a connected open subset U of the complex plane such that L(z) is a logarithm of z for each z in U.^[1]

For example, the principal value defines a branch on the open set where it is continuous, which is the set $\mathbb{C}-\mathbb{R}_{\le 0}$ obtained by removing 0 and all negative real numbers from the complex plane.

Another example: The Mercator series

$\log(1+u)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} u^n = u - \frac{u^2}{2} + \frac{u^3}{3} - \cdots \,$

converges for |u| < 1, so setting z = 1+u defines a branch of log z on the open disk of radius 1 centered 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 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 applies. It shows that L(z) is holomorphic at each z in U, and L′(z) = 1/z.^[1] Another way to prove this is to check the Cauchy-Riemann equations in polar coordinates.^[1]

Constructing branches via integration

The function ln x for x > 0 can be constructed by the formula

$\ln x = \int_1^x \frac{1}{u}\,du.$

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

$\ln x = \ln a + \int_a^x \frac{1}{u}\,du$

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

If U is a simply connected open subset of $\mathbb{C}$ 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

$L(z) := b + \int_a^z \frac{1}{w}\,dw.$

for each z in U.^[4]

The complex logarithm as a conformal map

The circles Re(Log z) = constant and the rays Im(Log z) = constant in the complex z-plane.

Any holomorphic map $f\colon U \to \C$ satisfying $f'(z) \ne 0$ for all $z \in U$ is a conformal 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.

For example, the principal branch w = Log z, viewed as a mapping from $\mathbb{C}-\mathbb{R}_{\le 0}$ 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^[5] 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

A visualization of the Riemann surface of log z. The surface appears to spiral around a vertical line corresponding to the origin of the complex plane. The actual surface extends arbitrarily far both horizontally and vertically, but is cut off in this image.

Construction

The various branches of log z cannot be glued to give a single function $\log \colon \mathbb{C}^* \to \mathbb{C}$ because two branches may give different values at a point where both are defined. Compare, for example, the principal branch Log(z) on $\mathbb{C}-\mathbb{R}_{\le 0}$ with imaginary part θ in (−π,π) and the branch L(z) on $\mathbb{C}-\mathbb{R}_{\ge 0}$ 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 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 $\mathbb{C} \times \mathbb{R} \approx \mathbb{R}^3$ .

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 $\log_R \colon R \to \mathbb{C}$ .^[6] 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.

There is a "projection map" from R down to $\mathbb{C}^*$ that "flattens" the spiral, sending (z,θ) to z. For any $z \in \mathbb{C}^*$ , if one takes all the points (z,θ) of R lying "directly above" z and evaluates log_R 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 $L_1\colon U_1 \to \mathbb{C}$ and $L_2\colon U_2 \to \mathbb{C}$ along the largest open subset of $U_1 \cap U_2$ on which L₁ and L₂ agree. This yields the same Riemann surface R and function log_R 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 $\mathbb{C}^*$ , then the restriction of log_R 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 $R \to \mathbb{C}^*$ realizes R as a covering space of $\mathbb{C}^*$ . In fact, it is a Galois covering with deck transformation group isomorphic to $\mathbb{Z}$ , generated by the homeomorphism sending (z,θ) to (z,θ+2π).

As a complex manifold, R is biholomorphic with $\mathbb{C}$ via log_R. (The inverse map sends z to (e^z,Im z).) This shows that R is simply connected, so R is the universal cover of $\mathbb{C}^*$ .

Applications

The complex logarithm is needed to define exponentiation 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 a^b = e^b Log a. One can also replace Log a by other logarithms of a to obtain other values of a^b.
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.^{[citation needed]}

Generalizations

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

$\log_i e = \frac{\log e}{\log i} = \frac{1}{\pi i/2} = -\frac{2i}{\pi}.$

Logarithms of holomorphic functions

If f is a holomorphic function on a connected open subset U of $\mathbb{C}$ , then a branch of log f on U is a holomorphic function g on U such that e^g(z) = f(z) for all z in U.

If U is a simply connected open subset of $\mathbb{C}$ , 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

$g(z) := b + \int_a^z \frac{f'(w)}{f(w)}\,dw.$

for each z in U.^[1]

Derivation from the Taylor series

From the viewpoint of analysis, the most natural way^{[citation needed]} to define log(z) is by the method of analytic continuation, from the positive real number line (where log(x) = ln(x)) into the complex plane. Here's how that works.

The Taylor series (also known as the Mercator series)

$\log(1+z)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} z^n = z - \frac{z^2}{2} + \frac{z^3}{3} - \cdots \,$

locally converges uniformly for all complex numbers z with absolute value less than 1. And by applying Abel's test to the Taylor series for log(1 − z), it can be shown that the series for log(1 + z) converges when |z| = 1, except when z = −1 (in other words, log(0) does not exist).

A careful analysis of the power series for

$\log \frac{1+z}{1-z} = 2\left(z + \frac{z^3}{3} + \frac{z^5}{5} + \cdots\right) = 2\sum_{n=0}^\infty \frac{z^{2n+1}}{2n+1}.\,$

shows that this series converges for |z| ≤ 1, except when z = ±1.^[7] By plugging

$\frac{1+i}{1-i} = i \qquad \frac{1-i}{1+i} = -i$

into the power series direct calculation reveals that

$\log(i) = -\log(-i) = 2i \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} +- \cdots \right) = 2i \arctan(1) = i\frac{\pi}{2} \,$

where the last equality follows from the well-known series for the (real) inverse tangent function.

By supposing that the laws of (real) logarithms apply to complex logarithms we obtain two results:^[8]

$i^2 = -1 \Rightarrow \log(-1) = 2\log(i) = i\pi \qquad (-i)^2 = -1 \Rightarrow \log(-1) = 2\log(-i) = -i\pi .\,$

The fact that log(−1) has two values can be explained this way. The complex logarithm is only continuous in a finite neighborhood of a point z, and log(0) does not exist. The "path" from z = 1 through z = i to z = −1 leads, by continuity, to one value of log(−1), while the "path" that passes through z = −i leads, by continuity, to the other value. The two paths lead to two different values because they encircle a branch point at z = 0. This multi-valued aspect of the complex logarithm is traditionally handled by making a "cut" in the complex plane all along the negative real axis. And the preceding explanation is, historically, why the argument of a complex number is often specified in the range from −π to π.^[9]

Log(z) as the inverse of the exponential function

If the non-zero complex number z is expressed in polar coordinates as z = re^iθ with r > 0 and −π < θ ≤ π, then

$\log(z) = \ln(r) + i\theta \,$

where ln(r) is the usual natural logarithm of a real number.

So defined, log (z) is holomorphic for all complex numbers which are not real numbers ≤ 0, and it has the property

$e^{\log(z)} = z \,$

for all nonzero z. Some familiar properties of the real-valued natural logarithm are no longer valid for this complex extension. For example, log(e^z) does not always equal z, and log(zw) does not always equal log(z) + log(w) – in either case, the result might have to be adjusted modulo 2πi to stay within the range of this principal branch of the complex log function.

Log(z) as a multi-valued function

A somewhat more natural definition of log(z) interprets it as a multi-valued function, or, more precisely, as a multi-valued relationship.^[10] For z = re^iθ, it would be possible to choose

$\log(z) = \ln(r) + i(\theta + 2 \pi k) \,$

for any integer k. There is no other complex number u for which e^u = z, because the period of e^z is 2πi (see Euler's identity).

The standard way to deal with multi-valued relationships such as this in complex analysis is via Riemann surfaces: the function log(z) is then defined not on the complex plane, but on a suitable Riemann surface having countably many "leaves" or "sheets" instead. The function is then single-valued on each sheet, and the values of the function differ by 2πi from one sheet to the next.

More precisely, the logarithm has a branch point at z = 0 and another at the point at infinity, where the point at infinity is understood to be the point that compactifies the complex plane into the Riemann sphere. There is a branch cut between these two points; by convention, it is placed so that the cut runs off to the left, along the negative real axis, between z = 0 and z = −∞. The location of this cut is by convention only: it could be placed running from the origin in any direction, and some textbooks show the cut running downwards.

At the cut, different sheets join together, and a continuous path moving through the cut simply goes from one sheet to the next. The gluing of the sheets is such that the phase differs by 2π between two sheets at any point that is not on the cut, and by 0 at the cut itself. On the Riemann surface, then, log(−x) for positive, real x, can be defined uniquely, as long as it is made clear which sheet one is on; otherwise, by convention, log(−x) lies in the branch cut and remains undefined.

The complex logarithm as a conformal mapping

Since the derivative of log(z), z⁻¹, is a meromorphic function with one simple pole at z = 0, the principal branch of log(z) is a conformal map from ℂ/{0} into an infinitely long strip of height 2π, −π < Im(w) ≤ π. We can gain insight into the way this mapping works by writing it as

$w = \log(z) = \ln(|z|) + i\arg(z)$

and then analyzing the images in the w-plane of certain circles and straight lines in the z-plane.^[11]

The unit circle in the z-plane is mapped into a segment of the imaginary axis in the w-plane, running from −πi to πi. Notice that the length of the resulting straight line segment equals the circumference of the original circle.

The interior of the unit circle is mapped into the left-hand portion of the strip; that is, into the region where Re(w) < 0. More precisely, each circle centered on the origin and of radius r < 1, |z| = r, is mapped into the straight line segment Re(w) = ln(|z|). Each of these line segments is longer than the circle whose image it is, and the smaller the circle, the more it gets stretched out.

The exterior of the unit circle is mapped into the right-hand portion of the strip; each circle |z| = r > 1 is straightened out and compressed to fit on the line segment Re(w) = ln(|z|).

The positive real axis in the z-plane is mapped into the entire real axis in the w-plane. More precisely, if z₀ is real and z₀ > 0, its image w₀ is given by w₀ = ln(z₀). The negative real axis is mapped into the straight line Im(w) = π, with one notable difference – its direction is reversed. That is to say, the negative real axis, which runs from right to left in the z-plane, is mapped into a straight line running from left to right in the w-plane.

All the rays originating at z = 0 are mapped into parallel horizontal lines in the w-plane. Rays that are "close" to the positive real axis lie close together in the w-plane, and rays that are '"close" to the negative real axis are farther apart, because the image of each ray arg(z) = θ is the straight line Im(w) = θ.

Since the mapping w = log(z) transforms circles into vertical straight line segments, it is useful in engineering applications involving an annulus, or doughnut-shaped region; the image of the annulus is a rectangle of height 2π, whose width depends on the thickness of the annulus. By the same token, the inverse mapping z = e^w transforms a rectangle into an annulus.

Log(z) also serves as a particularly simple illustration of the principle that conformal maps preserve angles. All the circles and radial lines discussed above are mutually perpendicular, as are the vertical and horizontal lines into which they are transformed.

Finally, the complete map of the corkscrew Riemann surface on which log(z) is holomorphic into the complex plane is easily visualized – each "sheet" of that surface is mapped into an infinitely wide strip of height 2π, and those strips cover the entire complex plane like wallpaper.

Logarithms to other bases

$\log_i e = \frac{\log e}{\log i} = \frac{1}{\pi i/2} = -\frac{2i}{\pi}.$

Plots of the complex logarithm function (principal branch)

z = Re(Log(x + iy))

z = |Im(Log(x + iy))|

z = |Log(x + iy)|

Superposition of the previous three graphs

Notes

^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ Donald Sarason, Complex function theory, 2nd ed., Amer. Math. Society, 2007, Section IV.9.
^ John B. Conway, Functions of one complex variable, second edition, Springer, 1978, p. 39.
^ Another interpretation of this is that the "inverse" of the complex exponential function is a multivalued function taking each nonzero complex number z to the set of all logarithms of z.
^ Serge Lang, Complex analysis, 3rd edition, Springer-Verlag, 1993, p. 121.
^ Strictly speaking, the point on each circle on the negative real axis should be discarded, or the principal value should be used there.
^ The notations R and log_R are not standard outside this article.
^ The point z = 0 and the point at infinity are branch points for the complex logarithm function precisely because this series does not converge when z = ±1.
^ These two results can also be obtained by expanding log(z) as a Taylor series about the points z = ±i, and then evaluating log(−1) with those two infinite series.
^ (Whittaker and Watson, 1927, Appendix).
^ A mathematical function is defined to be a single-valued relationship, so that a multi-valued function is a contradiction in terms, strictly speaking.
^ This mapping can also be studied in conjunction with the inverse mapping w = e^z. See (Moretti, 1964, pp. 351-355), for example.

References

Gino Moretti, Functions of a Complex Variable, Prentice-Hall, Inc., 1964.
E. T. Whittaker and G. N. Watson, A Course in Modern Analysis, fourth edition, Cambridge University Press, 1927.

Retrieved from "http://en.wikipedia.org/wiki/Complex_logarithm"

Categories: Complex analysis | Logarithms

This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.

Giant Panda

Mercedes Car

This site monitored by SitePinger.net

Complex logarithm

Contents

An inverse of the 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

Generalizations

Logarithms to other bases

Logarithms of holomorphic functions

Derivation from the Taylor series

Log(z) as the inverse of the exponential function

Log(z) as a multi-valued function

The complex logarithm as a conformal mapping

Logarithms to other bases

Plots of the complex logarithm function (principal branch)

See also

Notes

References