# Calculus, series and transforms in PDF

Limits, series, transforms and contour integrals are typeset with limits placed above and below the operators where they belong.

Limits with epsilon and delta, Taylor and Fourier series, the Fourier and Laplace transforms, Cauchy's integral formula, the zeta function as an Euler product, Stirling and Ramanujan. Toggle: math.

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## Markdown source

```markdown
# 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:

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

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