Complex logarithm

Logarithm of a complex number

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 Such a number is denoted by 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.

For each nonzero complex number , the principal value is the logarithm whose imaginary part lies in the interval .

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
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 .

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 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 .

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 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 .

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 The expression has a single value if and only if is an integer. 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. In electrical engineering, the propagation constant involves a complex logarithm.