Calculus, series and transforms in PDF

# Analysis and number theory

## Limits and continuity

$ \lim_{x \to a} f(x) = L \iff \forall \varepsilon > 0\ \exists \delta > 0 : 0 < |x - a| < \delta \Rightarrow |f(x) - L| < \varepsilon $

$ \lim_{x \to 0} \frac{\sin x}{x} = 1 \qquad \lim_{n \to \infty} \sqrt[n]{n} = 1 \qquad \liminf_{n \to \infty} a_n \leq \limsup_{n \to \infty} a_n $

## Series

Taylor's theorem with the Lagrange remainder:

$ f(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x - a)^k + \frac{f^{(n+1)}(\xi)}{(n+1)!} (x - a)^{n+1}, \qquad \xi \text{ between } a \text{ and } x $

$ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} \qquad \frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n \quad (|x| < 1) $

A Fourier series and the transform pair:

$ f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos \frac{2\pi n t}{T} + b_n \sin \frac{2\pi n t}{T} \right), \qquad a_n = \frac{2}{T} \int_{0}^{T} f(t) \cos \frac{2\pi n t}{T} \,\mathrm{d}t $

$ \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi} \,\mathrm{d}x \qquad f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi)\, e^{2\pi i x \xi} \,\mathrm{d}\xi \qquad F(s) = \int_0^{\infty} f(t)\, e^{-st} \,\mathrm{d}t $

## Complex analysis

$ f(z) = \frac{1}{2\pi i} \oint_{\gamma} \frac{f(w)}{w - z} \,\mathrm{d}w \qquad \oint_{\gamma} f(z)\,\mathrm{d}z = 2\pi i \sum_{k} \operatorname{Res}(f, z_k) \qquad e^{i\theta} = \cos\theta + i\sin\theta $

## Number theory

$ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p\ \text{prime}} \frac{1}{1 - p^{-s}} \qquad \pi(x) \sim \frac{x}{\ln x} \qquad \sum_{d \mid n} \varphi(d) = n $

$ a^{p-1} \equiv 1 \pmod{p} \quad (p \nmid a) \qquad \gcd(a, b) \cdot \operatorname{lcm}(a, b) = ab \qquad \binom{p}{k} \equiv 0 \pmod{p} \quad (0 < k < p) $

Stirling's approximation and Ramanujan's series for $1/\pi$:

$ n! \sim \sqrt{2\pi n} \left( \frac{n}{e} \right)^{n} \qquad \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^{\infty} \frac{(4k)!\,(1103 + 26390k)}{(k!)^4\, 396^{4k}} $

The Gamma and Beta functions:

$ \Gamma(z) = \int_0^{\infty} t^{z-1} e^{-t} \,\mathrm{d}t, \qquad \Gamma(n) = (n-1)!, \qquad B(x, y) = \frac{\Gamma(x)\,\Gamma(y)}{\Gamma(x + y)} $

## Linear algebra

Eigenvalues, the spectral theorem and the singular value decomposition:

$ A\mathbf{v} = \lambda \mathbf{v} \qquad \det(A - \lambda I) = 0 \qquad A = Q \Lambda Q^{\top} \quad (A = A^{\top}) \qquad A = U \Sigma V^{\top} $

$ \lVert \mathbf{x} \rVert_p = \left( \sum_{i=1}^{n} |x_i|^p \right)^{1/p} \qquad \langle \mathbf{u}, \mathbf{v} \rangle = \sum_{i} u_i \overline{v_i} \qquad \operatorname{tr}(AB) = \operatorname{tr}(BA) $