Circular and Hyperbolic Mathematical Representations

The companion explainer, A geometric view of Laplace transform behavior, places the transforms of $\sin bx$ and $\cos bx$ on a circle: a radius $r=\sqrt{t^2+b^2}$ and an angle $\theta$. Replace the oscillating input by a growing one, $\sinh bx$ or $\cosh bx$, and the circle no longer fits. Chapter 3 of Rida Abu-Sokon's book replaces it by the hyperbola $t^2-b^2=\text{const}$. This article explains why that is the natural move, where it is valid, and how the two pictures compare.

Two ways to measure a point

A point $(t,b)$ can be measured with the Euclidean form $t^2+b^2$ or with the indefinite form $t^2-b^2$. The first is preserved by rotations and is parameterized by $\cos^2\theta+\sin^2\theta=1$. The second is preserved by hyperbolic rotations (Lorentz boosts, in the language of special relativity) and is parameterized by $\cosh^2\varphi-\sinh^2\varphi=1$. Which one is appropriate for a Laplace transform depends on the input: an oscillation $e^{ibx}$ produces the factor $t-ib$, whose modulus is Euclidean, while a growth $e^{bx}$ produces the real factor $t-b$, whose natural companion is $t+b$, and $(t-b)(t+b)=t^2-b^2$.

The hyperbolic parameters

For $t>|b|$ define the Minkowski-type radius and the hyperbolic angle

The key identities are the hyperbolic analogues of $t\pm ib=re^{\pm i\theta}$:

These identities suggest the most transparent coordinates for the hyperbolic regime: the pair $u=t+b$ and $v=t-b$, often called light-cone coordinates. In them the invariant is a product, $\bar r^{\,2}=uv$, and the hyperbolic angle is a ratio, $e^{2\varphi}=u/v$. A hyperbolic rotation by $\alpha$ multiplies $u$ by $e^{\alpha}$ and divides $v$ by $e^{\alpha}$, leaving $uv$ unchanged. Compare the circular case, where a rotation multiplies $t+ib$ by $e^{i\alpha}$ and leaves $|t+ib|$ unchanged. Every Laplace transform of an exponential input is a power of $u$ or of $v$, which is why the hyperbolic formulas below are so compact.

The word "angle" deserves a comment. A circular angle $\theta$ is twice the area of the sector of the unit circle it cuts off. A hyperbolic angle has the same description with the unit hyperbola $t^2-b^2=1$: the region bounded by the two rays from the origin to $(1,0)$ and $(\cosh\varphi,\sinh\varphi)$ and by the hyperbola has area $\varphi/2$. So $\varphi$ is a genuine measure attached to the hyperbola, even though it is unbounded and is not an angle in the Euclidean sense.

Exponentials, sinh and cosh

With these parameters the elementary growing inputs take a uniform shape:

Compare $\cos\theta/r$ and $\sin\theta/r$ in the oscillatory case. The structure is identical; only the functions and the radius have changed.

Lifting in the hyperbolic regime

Multiplying the input by $x^n$ differentiates the transform $n$ times in $t$, so $\mathcal{L}\{x^ne^{\pm bx}\}=n!/(t\mp b)^{n+1}=n!\,e^{\pm(n+1)\varphi}/\bar r^{\,n+1}$. Adding and subtracting gives the hyperbolic counterparts of angular lifting:

One qualitative difference stands out immediately. In the circular case, $\sin((n+1)\theta)$ changes sign as the angle sweeps, so $\mathcal{L}\{x^n\sin bx\}$ has zero rays in the parameter plane. In the hyperbolic case, $\sinh((n+1)\varphi)$ has the sign of $b$ and never vanishes for $b\neq 0$, and $\cosh$ is always positive. Growing inputs produce transforms without sign changes; there is nothing to "quantize".

The formal bridge between the two

The two pictures are linked by the substitution $b\mapsto ib$. Under it, $r=\sqrt{t^2+b^2}$ becomes $\sqrt{t^2-b^2}=\bar r$, and the principal-branch identity $\arctan(ix)=i\operatorname{artanh}(x)$ for $|x|<1$ turns $i\theta$ into $-\varphi$. The oscillatory formula $\mathcal{L}\{x^ne^{ibx}\}=n!\,e^{i(n+1)\theta}/r^{n+1}$ then becomes $n!\,e^{-(n+1)\varphi}/\bar r^{\,n+1}=\mathcal{L}\{x^ne^{-bx}\}$. The book describes this as an analytic continuation that changes the metric from Euclidean to Minkowski-type, rather than a rotation within the complex plane.

Side by side

The table is a reminder that "geometry" here is a choice of coordinates adapted to the input, not a change in the transform. Both columns describe the same classical integral; they differ in which quadratic form makes the answer simplest.

The same split elsewhere

The circular/hyperbolic dichotomy is not special to Laplace transforms. It appears wherever a quadratic expression changes sign.

• Differential equations. $y''+b^2y=0$ has characteristic roots $\pm ib$ and solutions $\cos bx,\sin bx$; $y''-b^2y=0$ has roots $\pm b$ and solutions $\cosh bx,\sinh bx$. Their Laplace transforms carry the denominators $t^2+b^2$ and $t^2-b^2$ respectively.
• Rational functions. The derivatives of $1/\big((x-p)^2+q^2\big)$ are sines of multiple angles, while those of $1/\big((x-p)^2-q^2\big)$ are hyperbolic sines of multiple hyperbolic parameters; see Understanding root geometry in rational functions.
• Conic sections. $t^2+b^2=1$ is a circle and $t^2-b^2=1$ is a hyperbola; the parameterizations by $(\cos,\sin)$ and $(\cosh,\sinh)$ are the two standard choices.

In each case the sign of a discriminant-like quantity decides whether the natural coordinates are circular or hyperbolic, and the passage between them is the formal substitution of $ib$ for $b$. Recognizing the pattern once saves relearning it in every course.

Classical results and the book's framework