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.

Computational Demonstration: A numerical or visual demonstration. It illustrates; it does not prove.
Book Framework: The organising framework, notation, or terminology introduced in the book.
Classical Foundation: Established mathematics found in standard textbooks and references.

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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.
A second, algebraic check through the classical rational form, plus numerical quadrature.

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Inputs with $t \le 0$ , or $t \le |b|$ in hyperbolic mode, where the integral diverges.
Complex Laplace parameters.
The sinc, Bessel, and fractional-lifting material later in Chapter 3.
Proof of the representation; the lab checks it at chosen values.

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Angular–Hyperbolic Lab

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