Math Lab — Interactive Mathematical Instruments

Lab 01
Derivative Lab
What does the n-th derivative look like without repeated expansion?
Choose a function family, an order up to 12, and a parameter. The lab writes the n-th derivative in closed form, plots it beside the function, compares it with an independent numerical derivative, and checks the book's general integral identity (Ch. 1, eq. 1.5) at a point you choose.
Supported
- Families $\sin(ax)$ , $\cos(ax)$ , $e^{ax}$ , $\sinh(ax)$ , $\cosh(ax)$ and $1/(x^2+a^2)$ , with real $a$ ( $a \neq 0$ for the rational family).
- Integer orders $0 \le n \le 12$ , evaluated for real $x$ in $[-3, 3]$ .
- Phase-shift forms $a^n\sin(ax+n\pi/2)$ and $a^n\cos(ax+n\pi/2)$ , and the angular form $(-1)^n n!\,\sin((n+1)\varphi)/(|a|\rho^{n+1})$ for the rational family.
Computational DemonstrationResearch: Higher-order derivatives Launch instrument
Lab 02
Transform Lab
How do a kernel and a region of convergence turn a function into its transform?
Laplace, Fourier, and Mellin modes for a small table of classical pairs. Each mode plots the input, the kernel, and their product, shows the transform with its region or strip of convergence, and compares the closed form with numerical integration.
Supported
- Laplace, $F(s)=\int_0^\infty f(t)e^{-st}\,dt$ for real $s>-\alpha$ : $e^{-\alpha t}$ , $e^{-\alpha t}\cos\omega t$ , $e^{-\alpha t}\sin\omega t$ , and $t^n e^{-\alpha t}$ with $0 \le n \le 6$ .
- Fourier, with the convention $F(\omega)=\int_{-\infty}^{\infty} f(t)e^{-i\omega t}\,dt$ : $e^{-a|t|} \leftrightarrow 2a/(a^2+\omega^2)$ and $e^{-t^2/(2\sigma^2)} \leftrightarrow \sigma\sqrt{2\pi}\,e^{-\sigma^2\omega^2/2}$ .
- Mellin, $F(s)=\int_0^\infty x^{s-1}f(x)\,dx$ on real $s$ : $e^{-x} \leftrightarrow \Gamma(s)$ for $s>0$ , and $1/(1+x) \leftrightarrow \pi/\sin(\pi s)$ for $0<s<1$ .
Computational DemonstrationResearch: Transform framework Launch instrument
Lab 03
Angular–Hyperbolic Lab
Why do bounded kernels give angles and unbounded kernels give hyperbolic angles?
Place the point (t, b) in the parameter plane and read the Laplace transform of xⁿ sin bx, xⁿ cos bx, xⁿ sinh bx, or xⁿ cosh bx from its circular or hyperbolic coordinates, as in the book's Chapter 3. Each value is checked against numerical integration.
Supported
- Circular mode for $t>0$ : $r=\sqrt{t^2+b^2}$ , $\theta=\operatorname{atan2}(b,t)$ , and $\mathcal{L}\{x^n\sin bx\}(t)=n!\sin((n+1)\theta)/r^{n+1}$ , $\mathcal{L}\{x^n\cos bx\}(t)=n!\cos((n+1)\theta)/r^{n+1}$ .
- Hyperbolic mode for $t>|b|$ : $\bar r=\sqrt{t^2-b^2}$ , $\phi=\operatorname{artanh}(b/t)$ , and $\mathcal{L}\{x^n\sinh bx\}(t)=n!\sinh((n+1)\phi)/\bar r^{\,n+1}$ , $\mathcal{L}\{x^n\cosh bx\}(t)=n!\cosh((n+1)\phi)/\bar r^{\,n+1}$ .
- Orders $0 \le n \le 8$ , with the circular or hyperbolic sector drawn in the $(t,b)$ plane.
Computational DemonstrationResearch: Mathematical geometry Launch instrument
Lab 04
Root Geometry Lab
How does the position of the roots control a rational function's derivatives?
Analyze a quadratic by its discriminant and root form, see the zeros of the derivatives of 1/(ax² + bx + c) arise from equally spaced angles at the complex root, and compute certified numerical roots of real polynomials up to degree 6.
Supported
- Quadratics $ax^2+bx+c$ with real coefficients, $a \neq 0$ : discriminant, exact roots, the trigonometric form $re^{\pm i\theta}$ when $\Delta<0$ , and the hyperbolic forms $\pm re^{\pm\eta}$ or $re^{\eta}, -re^{-\eta}$ when $\Delta>0$ .
- For $\Delta<0$ and $1 \le n \le 10$ : $f^{(n)}$ of $f=1/(ax^2+bx+c)$ in angular form, and the zeros $x_k=p+q\cot(k\pi/(n+1))$ , $k=1,\dots,n$ , each confirmed by a sign change of an independent numerical derivative.
- Real polynomials of degree 1 to 6: Durand–Kerner roots with multiplicity detection, conjugate-symmetry check, and a certificate that the roots rebuild the coefficients.
Computational DemonstrationResearch: Root geometry Launch instrument
Lab 05
Matrix Boundary Lab
Can an integral be evaluated by solving one small linear system instead of integrating by parts?
Pick a basis closed under differentiation, see differentiation become the matrix Ω, and evaluate the truncated integral of e^(bx) f(x) with the resolvent M(b) = (bI + Ωᵀ)⁻¹, in six visible steps. The same resolvent solves y′ − by = f with y(0) = 0. Both are compared with quadrature.
Supported
- Bases $\{1,x,\dots,x^{m-1}\}$ ( $2 \le m \le 6$ ), $\{\cos\omega x,\sin\omega x\}$ , $\{e^{\alpha x}\}$ , $\{e^{\alpha x}\cos\omega x, e^{\alpha x}\sin\omega x\}$ , and $\{x\sin\omega x, x\cos\omega x, \sin\omega x, \cos\omega x\}$ .
- The book's convention $\Phi'=\Omega\Phi$ , $f=u^{\mathsf T}\Phi$ , and $\int_0^a e^{bx}f(x)\,dx=\big[e^{bx}\Phi(x)^{\mathsf T}M(b)\,u\big]_0^a$ .
- The first-order ODE $y'-by=f$ , $y(0)=0$ , through $M(-b)$ (book eq. 5.1).
Computational DemonstrationResearch: Matrix Boundary Method Launch instrument

Lab principles

Demonstration, stated honestly.

Defined domains
Every instrument validates its inputs. Outside the supported domain it says so instead of producing a number.
Two independent checks
Where possible, closed forms are compared with an independent numerical computation and both values are shown.
Demonstration is not proof
Agreement at sampled values builds intuition. Proofs live in the book and in classical references.
Runs in your browser
All computation happens locally. Nothing you enter is sent anywhere.

From the book

Learn

Work with Rida

Explore

Community

Instruments for seeing mathematical structure.

Derivative Lab

Supported

Transform Lab

Supported

Angular–Hyperbolic Lab

Supported

Root Geometry Lab

Supported

Matrix Boundary Lab

Supported

Demonstration, stated honestly.

Defined domains

Two independent checks

Demonstration is not proof

Runs in your browser

Demonstrations published with the book.