Complex logarithm

Summary

In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:

  • A complex logarithm of a nonzero complex number , defined to be any complex number for which .[1][2] Such a number is denoted by .[1] If is given in polar form as , where and are real numbers with , then is one logarithm of , and all the complex logarithms of are exactly the numbers of the form for integers .[1][2] These logarithms are equally spaced along a vertical line in the complex plane.
  • A complex-valued function , defined on some subset of the set of nonzero complex numbers, satisfying for all in . Such complex logarithm functions are analogous to the real logarithm function , which is the inverse of the real exponential function and hence satisfies eln x = x for all positive real numbers x. Complex logarithm functions can be constructed by explicit formulas involving real-valued functions, by integration of , or by the process of analytic continuation.
A single branch of the complex logarithm. The hue of the color is used to show the argument of the complex logarithm. The brightness of the color is used to show the modulus of the complex logarithm.
The real part of log(z) is the natural logarithm of |z|. Its graph is thus obtained by rotating the graph of ln(x) around the z-axis.

There is no continuous complex logarithm function defined on all of . Ways of dealing with this include branches, the associated Riemann surface, and partial inverses of the complex exponential function. The principal value defines a particular complex logarithm function that is continuous except along the negative real axis; on the complex plane with the negative real numbers and 0 removed, it is the analytic continuation of the (real) natural logarithm.

Problems with inverting the complex exponential functionEdit

 
A plot of the multi-valued imaginary part of the complex logarithm function, which shows the branches. As a complex number z goes around the origin, the imaginary part of the logarithm goes up or down. This makes the origin a branch point of the function.

For a function to have an inverse, it must map distinct values to distinct values; that is, it must be injective. But the complex exponential function is not injective, because   for any complex number   and integer  , since adding   to   has the effect of rotating   counterclockwise   radians. So the points

 

equally spaced along a vertical line, are all mapped to the same number by the exponential function. This means that the exponential function does not have an inverse function in the standard sense.[3][4] 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  : this leads naturally to the definition of branches of  , which are certain functions that single out one logarithm of each number in their domains. This is analogous to the definition of   on   as the inverse of the restriction of   to the interval  : there are infinitely many real numbers   with  , but one arbitrarily chooses the one in  .

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 the logarithm and does not require an arbitrary choice as part of its definition.

Principal valueEdit

DefinitionEdit

For each nonzero complex number  , the principal value   is the logarithm whose imaginary part lies in the interval  .[2] The expression   is left undefined since there is no complex number   satisfying  .[1]

For complex numbers that are not non-positive real numbers, the principal value of the complex logarithm is the analytic continuation of the natural logarithm. For nonreal numbers with a negative real part, the principal value is obtained by continuing further by expressing negative real numbers as limits of complex numbers with positive imaginary parts.

When the notation   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   when   is a positive real number. The capitalization in the notation   is used by some authors[2] to distinguish the principal value from other logarithms of  .

Calculating the principal valueEdit

The polar form of a nonzero complex number   is  , where   is the absolute value of  , and   is its argument. The absolute value is real and positive. The argument is defined up to addition of an integer multiple of 2π. Its principal value is the value that belongs to the interval  , which is expressed as  .

This leads to the following formula for the principal value of the complex logarithm:

 

For example,  , and  .

The principal value as an inverse functionEdit

Another way to describe   is as the inverse of a restriction of the complex exponential function, as in the previous section. The horizontal strip   consisting of complex numbers   such that   is an example of a region not containing any two numbers differing by an integer multiple of  , so the restriction of the exponential function to   has an inverse. In fact, the exponential function maps   bijectively 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.

PropertiesEdit

Not all identities satisfied by ln extend to complex numbers. It is true that   for all   (this is what it means for   to be a logarithm of  ), but the identity   fails for   outside the strip  . For this reason, one cannot always apply   to both sides of an identity   to deduce  . Also, the identity   can fail: the two sides can differ by an integer multiple of  ;[1] for instance,

 

but

 

The function   is discontinuous at each negative real number, but continuous everywhere else in  . To explain the discontinuity, consider what happens to   as   approaches a negative real number  . If   approaches   from above, then   approaches  , which is also the value of   itself. But if   approaches   from below, then   approaches  . So   "jumps" by   as   crosses the negative real axis, and similarly   jumps by  .

Branches of the complex logarithmEdit

Is there a different way to choose a logarithm of each nonzero complex number so as to make a function   that is continuous on all of  ? The answer is no. To see why, imagine tracking such a logarithm function along the unit circle, by evaluating   as   increases from   to  . If   is continuous, then so is  , but the latter is a difference of two logarithms of  , so it takes values in the discrete set  , so it is constant. In particular,  , which contradicts  .

To obtain a continuous logarithm defined on complex numbers, it is hence necessary to restrict the domain to a smaller subset   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,   should be an open set. Also, it is reasonable to assume that   is connected, since otherwise the function values on different components of   could be unrelated to each other. All this motivates the following definition:

A branch of   is a continuous function   defined on a connected open subset   of the complex plane such that   is a logarithm of   for each   in  .[2]

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  , so setting   defines a branch of   on the open disk of radius 1 centered at 1. (Actually, this is just a restriction of  , as can be shown by differentiating the difference and comparing values at 1.)

Once a branch is fixed, it may be denoted " " 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 " " to have a precise unambiguous meaning.

Branch cutsEdit

The argument above involving the unit circle generalizes to show that no branch of   exists on an open set   containing a closed curve that winds around 0. One says that "  has a branch point at 0". To avoid containing closed curves winding around 0,   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   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   at a negative real number.

The derivative of the complex logarithmEdit

Each branch   of   on an open set   is the inverse of a restriction of the exponential function, namely the restriction to the image  . Since the exponential function is holomorphic (that is, complex differentiable) with nonvanishing derivative, the complex analogue of the inverse function theorem applies. It shows that   is holomorphic on  , and   for each   in  .[2] Another way to prove this is to check the Cauchy–Riemann equations in polar coordinates.[2]

Constructing branches via integrationEdit

The function   for real   can be constructed by the formula

 
If the range of integration started at a positive number   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 (while holding the endpoints fixed), and in a simply connected region   (a region with "no holes"), any path from   to   inside   can be continuously deformed inside   into any other. All this leads to the following:

If   is a simply connected open subset of   not containing 0, then a branch of   defined on   can be constructed by choosing a starting point   in  , choosing a logarithm   of  , and defining
 
for each   in  .[5]

The complex logarithm as a conformal mapEdit

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

Any holomorphic map   satisfying   for all   is a conformal map, which means that if two curves passing through a point   of   form an angle   (in the sense that the tangent lines to the curves at   form an angle  ), then the images of the two curves form the same angle   at  . Since a branch of   is holomorphic, and since its derivative   is never 0, it defines a conformal map.

For example, the principal branch  , viewed as a mapping from   to the horizontal strip defined by  , has the following properties, which are direct consequences of the formula in terms of polar form:

  • Circles[6] in the z-plane centered at 0 are mapped to vertical segments in the w-plane connecting   to  , where   is the real log of 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 surfaceEdit

 
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.

ConstructionEdit

The various branches of   cannot be glued to give a single continuous function   because two branches may give different values at a point where both are defined. Compare, for example, the principal branch   on   with imaginary part   in   and the branch   on   whose imaginary part   lies in  . 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   level of the lower half plane up to the   level of the lower half plane by going   radians counterclockwise around 0, first crossing the positive real axis (of the   level) into the shared copy of the upper half plane and then crossing the negative real axis (of the   level) into the   level of the lower half plane.

One can continue by gluing branches with imaginary part   in  , in  , and so on, and in the other direction, branches with imaginary part   in  , in  , and so on. The final result is a connected surface that can be viewed as a spiraling parking garage with infinitely many levels extending both upward and downward. This is the Riemann surface   associated to  .[7]

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

The logarithm function on the Riemann surfaceEdit

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  .[8] It maps each point   on   to  . This process of extending the original branch   by gluing compatible holomorphic functions is known as analytic continuation.

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

Gluing all branches of  Edit

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

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

The Riemann surface as a universal coverEdit

The projection map   realizes   as a covering space of  . In fact, it is a Galois covering with deck transformation group isomorphic to  , generated by the homeomorphism sending   to  .

As a complex manifold,   is biholomorphic with   via  . (The inverse map sends   to  .) This shows that   is simply connected, so   is the universal cover of  .

ApplicationsEdit

  • The complex logarithm is needed to define exponentiation in which the base is a complex number. Namely, if   and   are complex numbers with  , one can use the principal value to define  . One can also replace   by other logarithms of   to obtain other values of  , differing by factors of the form  .[1][9] The expression   has a single value if and only if   is an integer.[1]
  • Because trigonometric functions can be expressed as rational functions of  , the inverse trigonometric functions can be expressed in terms of complex logarithms.
  • Since the mapping   transforms circles centered at 0 into vertical straight line segments, it is useful in engineering applications involving an annulus.[citation needed]

GeneralizationsEdit

Logarithms to other basesEdit

Just as for real numbers, one can define for complex numbers   and  

 

with the only caveat that its value depends on the choice of a branch of log defined at   and   (with  ). For example, using the principal value gives

 

Logarithms of holomorphic functionsEdit

If f is a holomorphic function on a connected open subset   of  , then a branch of   on   is a continuous function   on   such that   for all   in  . Such a function   is necessarily holomorphic with   for all   in  .

If   is a simply connected open subset of  , and   is a nowhere-vanishing holomorphic function on  , then a branch of   defined on   can be constructed by choosing a starting point a in  , choosing a logarithm   of  , and defining

 

for each   in  .[2]

NotesEdit

  1. ^ a b c d e f g Ahlfors, Section 3.4.
  2. ^ a b c d e f g h Sarason, Section IV.9.
  3. ^ Conway, p. 39.
  4. ^ 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.
  5. ^ Lang, p. 121.
  6. ^ Strictly speaking, the point on each circle on the negative real axis should be discarded, or the principal value should be used there.
  7. ^ Ahlfors, Section 4.3.
  8. ^ The notations R and logR are not universally used.
  9. ^ Kreyszig, p. 640.

ReferencesEdit

  • Ahlfors, Lars V. (1966). Complex Analysis (2nd ed.). McGraw-Hill.
  • Conway, John B. (1978). Functions of One Complex Variable (2nd ed.). Springer. ISBN 9780387903286.
  • Kreyszig, Erwin (2011). Advanced Engineering Mathematics (10th ed.). Berlin: Wiley. ISBN 9780470458365.
  • Lang, Serge (1993). Complex Analysis (3rd ed.). Springer-Verlag. ISBN 9783642592737.
  • Moretti, Gino (1964). Functions of a Complex Variable. Prentice-Hall.
  • Sarason, Donald (2007). Complex Function Theory (2nd ed.). American Mathematical Society. ISBN 9780821886229.
  • Whittaker, E. T.; Watson, G. N. (1927). A Course of Modern Analysis (Fourth ed.). Cambridge University Press.