Complex exponential and logarithm Monday, September 30, 2013 1:59 PM Homework 2 posted, due October 11. Introduce complex-valued transcendental functions via the complex exponential defined as: where We'll take this for now as just the definition, and observe that it satisfies desired properties. Another tactic to define transcendental functions like exponential is via Taylor series; we'll emphasize this point of view later. We can see that the definition of exponential has the following properties: Agrees with the real-valued definition of exponential on the real axis (y=0) and the complex exponential is analytic on the whole complex plane. The Principle of Analytic Continuation (which we'll discuss later) shows that the definition we gave is the unique one that has these properties. Remark: exp(z)=e z is a single-valued function, meaning that even if z=m/n (a fraction), then we don't interpret as a multivalued number when viewing as a function. The polar representation of the complex exponential is often convenient: ComplexAnalysis Page 1
We note in particular that e z does not vanish anywhere, but can be negative or positive. The usual algebraic operations on exponentials extend to the complex version: How does the complex exponential behave geometrically? ComplexAnalysis Page 2
This mapping is not one-to-one; it has the following periodicity: That is, the complex exponential is one-to-one when restricted to horizontal strips of the form: Let's now turn to defining a complex version of the logarithm, which is supposed to be an inverse function for the exponential. But this will be tricky because we've just seen that the complex exponential is not 1-1. Let's try to define the logarithm simply by standard inverse function considerations. w=log z (with natural logarithm implied despite the notation which might suggest base 10 logarithm) ComplexAnalysis Page 3
So then, what this implies is that the appropriate definition of logarithnm is: This is a multivalued function because arg z is a multivalued function. Note then how one should compute complex logarithms: As is typical for multivalued functions, one can define single-valued versions of them, at a cost Principal value of the logarithm: Then: (aka the principal branch) ComplexAnalysis Page 4
While this principal value of the logarithm is a canonical single-valued version of the multivalued logarithm, there is nothing particularly special or natural about it, and we will see that there are occasions to define other single-valued versions. With any single-valued version of a multivalued function comes the problem that an artificial discontinuity is introduced; here it's along the negative real axis. The multivalued function itself has no discontinuity there, or anywhere other than the origin. To reconcile these single-valued and multi-valued views of complex functions, it is useful to introduce the concept of Riemann surface, which is a topological representation of the multivalued function. The Riemann surface for the logarithm takes the form of a infinite-sheeted spiral surface glued together at the origin. To every Riemann surface can be associated a group, with the group action corresponding to a path that creates a change in the sheet. For the logarithm, the fundamental group (which is related to topology) is just the group of integers under addition. To work concretely with Riemann surfaces, it is helpful to introduce the following associated concepts: Branch points: Any point in the (extended) complex plane about which arbitrarily close closed paths get mapped to non-closed paths. One can show that branch points are never ComplexAnalysis Page 5
places at which the function is analytic. In other words, branch points can only happen at "singularities" of functions (where they are not analytic). Branch of a multivalued function is a single-valued restriction of the definition of the function. The specification of a branch will inevitably lead to the introduction of discontinuities, and these are referred to as branch cuts. Branch cuts generally speaking will be lines or arcs connecting branch points (so long as you think about infinity as being a possible branch point). Let's look at logarithm with these concepts. Branch points at 0 and at At a fundamental level, check this by simply verifying that small loops around these points (on the Riemann sphere). Other points are excluded from being branch points because either noting that log z is analytic elsewhere, or by just checking small loops around other points are mapped always to closed loops. To define a branch analytically, one begins by specifying a branch cut of choice. Not at all a unique choice; choose whatever is convenient, but for efficiency always try to have the branch cut be a path from one branch point to another branch point. So for log z, good choices of branch cuts are any curve that connects 0 and Why not make it straight for good measure? The point of introducing the branch cut is that one can no longer draw loops around the branch points without crossing a branch cut. This is the essential purpose of introducing branch cuts. Here we can define the branch cut in polar coordinates as ComplexAnalysis Page 6
A branch is a single-valued version of the multivalued function that is well-behaved (analytic) away from the branch cut: We can define a branch of log z using our branch cut as follows: With with Log z is simply the branch so defined with One can check that away from the branch points, log z is analytic. Can use Cauchy-Riemann, but a pain. Easier way to show this fact is through the Inverse Function Theorem If f(z) is an analytic function, then any branch of its inverse function g(w)= f -1 (w) is analytic away from branch cuts, branch points, and places where f'(g(w)) = 0. Away from such points, Proof is same as real-variable case, just have to modify. ComplexAnalysis Page 7
Proof is same as real-variable case, just have to modify. ComplexAnalysis Page 8