Basic calculus & complex analysis
This section corresponds to Basic Calculus and Complex Analysis — 38:59 in the video Proof of Fermat Last Theorem FROM SCRATCH. Noah Jang is the original creator of the exposition and examples referenced here. The interactive implementations, animations, additional explanations, and contextual material are my own work.
📌 Basic calculus
🏷️ Taylor series
A Taylor series represents a function as an infinite polynomial. With infinitely many terms, it equals the original function within its interval of convergence.
✍️ Examples
\[ \begin{array}{rl} e^x &= 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots \\ \sin x &= x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!}+\cdots\\ \cos x &= 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!}-\dfrac{x^6}{6!}+\cdots \\ \ln(1 + x) &= x - \dfrac{x^2}{2}+\dfrac{x^3}{3}-\dfrac{x^4}{4}+\cdots \qquad (-1 < x \leq 1) \end{array} \]
So why do we need this?
Although this won’t be used frequently in the forthccoming chapters, it appears a few times later.
🎯 The Taylor series is a way to represent a given function as an infinite polynomial.
👉 We will look at the Taylor expansion again from the perspective of linear algebra in chapter two.
👨💻 Taylor expansion demo
🏷️ Multivariable functions
A multivariable function is a map
\[ f: \mathbb{R}^n \rightarrow \mathbb{R} \]
In the section above, we have discussed (the Taylor expansion for) functions that depend on one variable only, such as \(f(x) = e^x\). In this section, we’ll shortly look at functions that depend on two or even more variables.
A multivariable function that depends on two variables can naturally be visualized as a surface in three-dimensional space.
👨💻 Multivariable functions demo
Another examples may be choosen from the drop-down menu. The formula of the selected function can be seen in the title.
🏷️ Partial derivative
For functions that depend on more than one variable, we would like to be able to analyse the rate of change in a similar way to that of functions that depend on one variable only. As multivariate functions have several variables, each direction generally has its own rate of change. As a consequence, each direction has its own derivative. When looking for the rate of change in a particular direction, we leave the other variables intentionally unchanged. This means that all the other variables but the one we are interested in can then be treated as constants.
Let \(f:\mathbb{R}^2\rightarrow\mathbb{R}\) be a function. The partial derivative with respect to \(x_i\) is defined as
\[ \dfrac{\partial f}{\partial x_i} := \lim_{h\rightarrow 0} \dfrac{f(x_1,\cdots,x_i+h,\cdots,x_n) - f(x_1,\cdots,x_n)}{h} \]
Note that this is very similar to the way in which the derivative is defined for a function of one variable:
\[ f'(x) = \lim_{h\rightarrow 0} \dfrac{f(x+h) - f(x)}{h} \]
So why do we need this?
Later these partial derivatives will show up when we talk about singlar points and elliptic curves.
✍️ Example
Let
\[ f(x,y) = x^3+y^3+xy+\sin x\cos y \]
then
\[ \begin{array}{ll} \dfrac{\partial f}{\partial x} &= 3x^2 + y + \cos x\cos y \\ \dfrac{\partial f}{\partial y} &= 3y^2 + x - \sin x\sin y \\ \end{array} \]
🏷️ Eurler’s formula and \(e^{i\theta}\)
For any real number \(\theta\),
\[ e^{i\theta} = \cos\theta + i\sin\theta. \]
Why do we need this?
This formula tells us concretely what to do/what to think of, whenever we encounter \(i\) in an exponent.
The special case where \(\theta = \pi\) then gives us one of the well-known formulas in mathematics, similar to the \(E=mc^2\) equation in physics:
\[ e^{i\pi} + 1 = 0. \]
The reason is that it contains so many important constants in one tiny equation:
- Euler’s constant or natural constant \(e\)
- The irrational number \(pi\) as the ratio between the circumverence and diameter of a circle
- The square root of negative one, \(i\)
- The identity with respect to addition, \(0\)
- The identity with respect to multiplication, \(1\)
This formula also matches very well with the aforementioned Taylor series.
\[ \begin{array}{ll} e^x &= 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots \\ e^{i\theta} &= 1 + i\theta - \dfrac{\theta^2}{2!} - \dfrac{\theta^3}{3!}i + \dfrac{\theta^4}{4!} + \dfrac{\theta^5}{5!}i - \dfrac{\theta^6}{6!} + \cdots \\ \quad &= \left(1 - \dfrac{\theta^2}{2!} + \dfrac{\theta^4}{4!} - \dfrac{\theta^6}{6!} + \cdots \right) +i\left(\theta - \dfrac{\theta^3}{3!}i + \dfrac{\theta^5}{5!} + \cdots\right) \\ \quad &= \cos\theta \qquad + \qquad i\sin\theta \end{array} \]
As a consequence we have \(e^{2\pi i} = 1\), \(e^{\pi i} = -1\), and \(e^{\frac{\pi}{6}i} = \dfrac{\sqrt{3}}{2} +\dfrac{1}{2}i\).
And more generally \[ e^{i\theta} = e^{i(\theta + 2n\pi)},\quad n\in\mathbb{N}. \]
🏷️ The \(n\)-th roots of unity
🚨 So why do we need this?
Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. — Wikipedia
The \(n\)-th roots of unity are the complex numbers \(z\) that satisfy the equation:
\[ z^n =1\quad \text{or equivalently}\quad z^n -1 = 0 \]
using Euler’s formula, we can find all \(n\) distinct roots. They are given by:
\[ z_k=e^{i\dfrac{2\pi k}{n}}=\cos\left(\dfrac{2\pi k}{n}\right) + i\sin\left(\dfrac{2\pi k}{n}\right), \quad k=0, 1, 2,\cdots,n-1. \]
So how does this work? We are looking for
\[ z^n = 1 = e^{2\pi k} \Rightarrow z = e^{\dfrac{2\pi ik}{n}},\quad k \in \mathbb{Z} \]
Filling in the values of \(k\), we get
\[ \underbrace{ 1, e^{\dfrac{2\pi i}{n}}, e^{\dfrac{4\pi i}{n}}, \cdots,e^{\dfrac{2\pi i(n-1)}{n}} }, \underbrace{e^{2\pi i}, e^{\dfrac{2\pi i(n+1)}{n}}, \cdots } \]
Because of the periodicity, we can limit the values of \(k\) to \(n-1\).
✍️ Example 1: \(n=2\)
\[z_k = e^{i\dfrac{2\pi k}{2}} = e^{\pi ik} \Rightarrow \begin{cases} k=0: 1 \\ k=1: -1\end{cases} \]
✍️ Example 2: \(n=3\)
\[z_k = e^{\dfrac{2\pi ik}{3}} \Rightarrow \begin{cases} k=0: 1 \\ k=1: e^{\dfrac{2\pi}{3}}= +\dfrac{\sqrt{3}}{2}i-\dfrac{1}{2} \\ k=2: e^{\dfrac{4\pi}{3}} = -\dfrac{\sqrt{3}}{2}i+\dfrac{1}{2}\end{cases} \]
Geometrically, the roots of unity constitute the vertices of a regular \(n\)-gon sitting on a unit circle in the complex plane.
👨💻 Roots of unity demo
📌 Complex analysis
🏷️ Complex functions and holomorphy
A complex function is a map \[ f:\mathbb{C}\rightarrow\mathbb{C} \]
\(f\) is complex differentiable at \(z_0\) if the limit below exists for \(h\in\mathbb{C}\).
\[ f'(z_0) := \lim_{h\rightarrow 0}\dfrac{f(z_0+h) - f(z)}{h} \]
If \(f\) is complex differentiable on an open set, it is called holomorphic.