By 苏剑林 | August 13, 2015
This article will study the Taylor expansion of the function of $t$:
$$\exp\left(\frac{1}{2}t^2+xt\right)$$
at $t=0$. Clearly, this is not difficult; it can be calculated by hand or with software. The answer is:
$$1+x t+\frac{1}{2} \left(x^2+1\right) t^2+\frac{1}{6}\left(x^3+3 x\right) t^3 +\frac{1}{24} \left(x^4+6 x^2+3\right) t^4 + \dots$$
However, this article will present a graphical method I constructed for this series. This graphical method allows for a relatively intuitive and convenient way to manually calculate the first few terms of the expansion. Later, we will discuss the origins of this graphical technique and its further applications.
Graphical Method for the Series: Explanation
First, it is obvious that to write out this series, the key is to determine each term of the expansion, which means we need to find:
$$f_k (x) = \left.\frac{d^k}{dt^k}\exp\left(\frac{1}{2}t^2+xt\right)\right|_{t=0}$$
$f_k (x)$ is a $k$-th degree polynomial with integer coefficients in terms of $x$, where $k$ is the order of the expansion and the order of differentiation.
Here, we use a "dot" to represent an $x$, and "a straight line between two dots" to represent "multiplication." Thus, $x^2$ can be represented as:
[Term $x^2$]
We use the number of "dots" to represent the degree of $f_k (x)$. However, $f_k (x)$ is not always of degree $k$. In such cases, we use "two dots with a wavy line between them" to represent "1." Borrowing physics terminology, we can say these two dots are "coupled" into a constant. Therefore, the following diagram represents the $x$ term in $f_3(x)$:
[Term $x$ in $f_3(x)$]
Note that the form of the term is independent of the order in the diagram; that is, the following diagram also represents the $x$ term in $f_3(x)$:
[Another representation of the $x$ term in $f_3(x)$]
To represent the coefficient in front of a term, we add the corresponding number afterwards. Thus, the following diagram represents the $3x$ term in $f_3(x)$:
[Term $3x$ in $f_3(x)$]
We also need a constraint: "two wavy lines cannot appear adjacent to each other," meaning the following diagram is prohibited:
[Prohibited term]
Now, it is not difficult to read the following diagram:
[$f_3(x)$]
It represents $x^3+3x$, which is the third-order term of the Taylor expansion. Similarly, the diagram:
[$f_4(x)$]
represents $x^4+6x^2+3$, which is the fourth-order term of the Taylor expansion.
Graphical Method for the Series: Recursion
If this were merely a diagrammatic notation, it would just be another way of writing the series and wouldn't offer much help. However, the diagrams help us perform recursion intuitively—the process of going from a $k$-th order diagram to a $(k+1)$-th order diagram.
Taking the process from order $3$ to order $4$ as an example, the 3rd-order diagram is:
[$f_3(x)$]
The 4th-order diagram has 4 dots, meaning we add 1 dot. Imagine that this newly added dot emits two types of "signals": a straight line signal and a wavy line signal. These two signals will probe each dot of the original diagram, but the two signals have different characteristics.
Straight Line Signal
The straight line signal is the "accept the first thing it finds" type. It does not distinguish between dots; as soon as it detects a dot, it connects to it and stops further probing. Therefore, through the probe of the straight line signal, the 3rd-order diagram becomes:
[Probing of the straight line signal]
Wavy Line Signal
The wavy line signal is different from the straight line signal. It has the ability to distinguish between dots and will finely probe every potential point where it can be embedded, and then it embeds itself into each one. Therefore, through the probe of the wavy line signal, the 3rd-order diagram becomes (where the first three diagrams are equivalent):
[Probing of the wavy line signal]
Thus, by combining the probing results of both signals, we obtain the 4th-order diagram:
[$f_4(x)$]
which represents $x^4+6x^2+3$, the fourth-order term of the Taylor expansion.
This is an example; readers can further simplify the process based on their own understanding.
Graphical Method for the Series: Application
Using the graphical process described above, one can relatively quickly draw the diagram for the series:
[Series]
The first term represents $f_1 (x) = x$, which corresponds to the $xt$ term of the Taylor expansion. The second term represents $f_2 (x) = x^2 + 1$, which corresponds to the $\frac{1}{2!}(x^2 + 1)t^2$ term. The third term represents $f_3 (x) = x^3 + 3x$, which corresponds to the $\frac{1}{3!}(x^3 + 3x)t^3$ term, and so on. Therefore:
$$\exp\left(\frac{1}{2}t^2+xt\right)=1+x t+\frac{1}{2} \left(x^2+1\right) t^2+\frac{1}{6}\left(x^3+3 x\right) t^3 + \dots$$
Clearly, if it were only for expanding this series, the aforementioned project would be somewhat overkill. However, by changing the meaning of dots and wavy lines, this could play a role in more complex expansions. In fact, I summarized this set of methods while studying the "external source technique" (functional derivatives) in quantum mechanics perturbation theory. There, it is required to calculate:
$$\frac{\delta^n}{\delta x^n}\exp\left[\int \left(xf+\frac{1}{2}fL^{-1}f\right)dt\right]$$
Calculating functional derivatives is much more complex than calculating ordinary derivatives. Therefore, if such a graphical method exists to assist us in calculation, it can reduce our workload and facilitate the derivation of each term. In fact, in functional derivatives, similar techniques do exist; this is equivalent to replacing each dot with different colors to distinguish each term. One can imagine that there would be many more terms and they would be much more complex. Naturally, this will be left for future articles.