Teaching

The reader is expected to already have some knowledge on integration. Even though we try to be as general as necessary, we leave general cases like integration over abstract vector spaces and other details to the Mathematicians. Instead, we stick to integration over \(R^n\). Because of the isomorphism \(R^2 \sim C\), portions of this review might be applicable to an integration in the complex plane under (slight) modifications. We will not further talk about this but it shall be noted that Cauchy’s integral theorem and the residue theorem play a big role in complex analysis. At times, this summary might lack mathematical rigor for the sake of understanding and intuition. For people who are interested in a mathematical approach to this topic, I can re- commend the books [1], [2] and [3], as well as the scripts [4], [5] and [6]. They cover everything you need to know to kick off your career in Analysis. Unfortunately, as far as I know, they are only available in German. The Wikipedia pages [7], [8], [9] and [10] also do a really good job on these topics and contain some useful graphics and animations. They are available in German and English. This recap is basically based on the given literature and things learned here and there.

The method we will show and clarify with an example is convenient in physics and applicable to many of the problems we deal with in physics. In doing so, we also introduce the Fourier transform. The word function in Green’s functions is to be taken with due care. Strictly speaking, a Green’s function has to be viewed as a distribution, but we will stick to the loose and common terminology here. Furthermore, we will not go into much detail about the mathematics behind the topic and instead focus on giving an introduction to the topic that is supposed to help understanding the concept of Green’s functions. For people who want to learn more on this topic, we refer to the books [1], [2] and [3], as well as the internet resources [4] and [5]. This recap is basically based on the given literature and things learned here and there. Another resource that proved useful in preparing this recap was [6].

In the lecture, we were a bit careless in extending the validity of this formula from \(a \in \mathbb{R}^+\), \(b=0\) to \(a \in \mathbb{C}\), \(\text{Re}(a)>0\), \(b=0\), then to \(a,b \in \mathbb{C}\), \(\text{Re}(a)>0\), and ultimately to \(a,b \in \mathbb{C}\), \(\text{Re}(a) \geq 0\), and we will see that it is in fact not trivial at all to extend the formula to the complex plane.

We want to emphasize that the result is well known and our proof will be based on ideas from many different references, in particular Refs. [1, 2, 3, 4, 5, 6, 7], that is much of this article will be the intellectual property of other authors ⁠— we want to apologize to the authors for not citing these references properly throughout this work.

However, we do not only collect the arguments of said references in a clear and comprehensible manner here but we also supplement them by some own input and explanations. Moreover, we will make use of certain results (that is e.g. theorems) from the Refs. [8, 9, 10, 11, 12], which we will also cite appropriately where needed. 

The Refs. [13, 14] might be interesting to the reader as well but are beyond the scope of this work; one of these references is an open question (at the time of writing) on StackExchange that will be referred to again in this article.

It is likely that Mathematicians will not consider our proof to be rigorous ⁠— which is why we call it a semi-rigorous proof in the first place ⁠⁠— since we might be a little bit sloppy regarding the validity of certain steps.

For the reader who wants to learn more on Gaussian integrals and complex analysis, we refer to the above references.

Here, the energy is given by \(E = \hbar^2 k^2/(2m)\), with \(k = \lvert \vec{k} \rvert\). We will not prove that this function indeed is the Green’s function of the Helmholtz equation, i.e., that it fulfills 

\[(\nabla^2 + k^2) G_\pm(\vec{x},\vec{x}’) = \delta^{(3)}(\vec{x} – \vec{x}’),\]

which is part of this term’s exercise sheets. In order to keep the actual calculation as clear and comprehensible as possible, we proceed in two steps: we first state and prove three minor theorems from complex analysis and then, making use of the last of these theorems, proceed with the aforementioned calculation in the main part of our notes.

The ideas presented in our notes are based on Refs. [1–5] and we refrain from repeatedly citing these. For more details and rigorous proofs of the theorems presented here, the interested reader is referred to the above references and his or her favorite book on complex analysis.