Series: Differential, Relative, and Logarithmic

This page unifies three approximation families around a reference point \(x_0\): (1) the classical differential Taylor series (rank 0, operator \(D_0^n\)), (2) a relative multiplicative (“Product Taylor”) series (rank 1, operator \(D_1^n\)), (3) a rank-2 logarithmic–exponential series (rank 2, operator \(D_2^n\)). We always write derivatives explicitly with both rank and degree.

rank 0: differential rank 1: relative (product) rank 2: log-rank (exponential) Function approximation

GOSSA AHMED

Independent Researcher — Hierarchical Calculus

Core Concepts

Expansion point

Choose \(x_0\) inside the domain. Approximation quality depends on the notion of “closeness”:
— Rank 0: additive closeness via \(x-x_0\).
— Rank 1: relative closeness via \(x/x_0\).
— Rank 2: logarithmic closeness via \(\ln x/\ln x_0\) (equivalently \(\ln(\ln x)\)).

Domain note: Rank \(D_2\) uses \(\ln(\ln x)\) and \(\ln(\ln f)\), hence we assume \(x>1\) (so \(\ln x>0\)) and \(f(x)>1\) (so \(\ln f>0\)).

1) Differential series — rank 0 (operator \(D_0^n\))

The classical Taylor series built from the standard differential operator. Rank+degree is written \(D_0^n\).

Formula (Taylor)

\[ f(x)=\sum_{n=0}^{\infty}\frac{(D_{0}^{n}f)(x_0)}{n!}\,(x-x_0)^{n}. \]

Here \(D_{0}^{0}f=f\), \(D_{0}^{1}f=df/dx\), \(D_{0}^{2}f=d^2f/dx^2\), etc.

When is it strongest?

When \(x\) is extremely close to \(x_0\) additively (\(x-x_0\)).
Excellent local approximations for analytic functions (sin, cos, exp...).
Limited by nearby poles/singularities that control convergence radius.

2) Relative multiplicative series — rank 1 (operator \(D_1^n\))

A “product Taylor”: sum → product, minus → division. Built from successive relative derivatives of degree \(n\) at rank 1.

Definitions

\[ t=\frac{x}{x_0},\qquad \Delta=\ln\!\left(\frac{x}{x_0}\right)=\ln t. \]

Formula (agreed “Form 2”)

\[ \frac{f(x)}{f(x_0)} = \prod_{n=1}^{\infty} \left(\frac{x}{x_0}\right)^{ \frac{\left(D_{1}^{n}\ln f\right)(x_0)}{n!}\;\Delta^{\,n-1} }. \]

where \[ \left(D_{1}^{n}\ln f\right)(x_0) = \left.\left(\frac{d}{d\ln x}\right)^{n}\ln f(x)\right|_{x=x_0}. \]

Here the degree \(n\) is the differentiation order with respect to \(\ln x\).

Why multiplicative?

We expand \(\ln f\) with respect to \(\ln x\) (additive), then map back to \(f\) (multiplicative). A negative exponent means division: \[ \left(\frac{x}{x_0}\right)^{-a}=\frac{1}{\left(\frac{x}{x_0}\right)^{a}}. \]

3) Rank-2 logarithmic–exponential series — rank 2 (operator \(D_2^n\))

In rank 2, the effective variable is \(\ln(\ln x)\). We use a practical formulation that approximates \(\ln f(x)\) relative to \(\ln f(x_0)\), while the rank-2 derivative itself is defined through \(\ln(\ln f)\).

Definitions

\[ \Delta_2=\ln(\ln x)-\ln(\ln x_0)=\ln\!\left(\frac{\ln x}{\ln x_0}\right), \qquad t_2=\frac{\ln x}{\ln x_0}. \]

Rank-2 derivative (degree 1)

\[ \boxed{ D_{2}^{1} f(x)=\frac{d\ln(\ln f(x))}{d\ln(\ln x)} } \qquad (x>1,\ f(x)>1) \]

Formula (practical form)

\[ \frac{\ln f(x)}{\ln f(x_0)} = \prod_{n=1}^{\infty} \left(\frac{\ln x}{\ln x_0}\right)^{ \frac{\left(D_{2}^{n}\ln^{(2)} f\right)(x_0)}{n!}\;\Delta_2^{\,n-1} }. \]

where \(\ln^{(2)}f=\ln(\ln f)\), and \[ \left(D_{2}^{n}\ln^{(2)} f\right)(x_0) = \left.\left(\frac{d}{d\,\ln(\ln x)}\right)^{n}\ln(\ln f(x))\right|_{x=x_0}. \]

Interpretation

Rank 2 measures closeness through \(\ln(\ln x)\), hence \(\ln x/\ln x_0\) appears naturally. Negative exponents again correspond to division.

How to use these series for approximation

Practical recipe

Choose \(x_0\) inside your region of interest.
Select the closeness scale: difference (rank 0), ratio (rank 1), or log-log (rank 2).
Pick a truncation degree \(N\) (e.g., 3 or 4) and use the truncated formula.
Reminder: \(D_0^n\) are ordinary derivatives, while rank-1 and rank-2 series are built from successive derivatives with respect to \(\ln x\) and \(\ln(\ln x)\).

Quick conclusion rule:
— Power-law behavior (\(x^p\), \(\sqrt{x}\)) → rank 1 is typically strongest.
— Oscillations (sin/cos) → rank 0 is typically strongest locally.
— Multi-scale log behavior → rank 2 can be strongest when defined.

Seven familiar functions: which approximation is strongest?

The goal is structural reasoning: which series captures a function shape more efficiently. Assume \(x_0>1\) (e.g., \(x_0=10\)) and compare near \(x_0\).

1) \(f(x)=x^p\) (power law)

Strongest: rank 1 (relative multiplicative). Since \(\ln f=p\ln x\), the scale structure is captured extremely efficiently.

2) \(f(x)=\sqrt{x}\)

Strongest: rank 1 — a special case of power law with \(p=\tfrac12\).

3) \(f(x)=e^x\)

Depends on closeness notion:
rank 0: best for very small additive neighborhoods.
rank 1: useful when change is discussed in relative terms.
rank 2: can help across broader scales when \(\ln(\ln f)\) is well-defined.

4) \(f(x)=\ln x\) (for \(x>1\))

Often strongest: rank 2 (when defined), since \(\ln x\) aligns with \(\ln(\ln x)\) scale.
Locally: rank 0 also works well if \(x-x_0\) is small.

5) \(f(x)=\dfrac{1}{1-\alpha x}\) (away from the pole)

Strongest: rank 0 (Taylor), provided \(x_0\) is safely away from the pole \(x=1/\alpha\). Singularities often control additive convergence radii.

6) \(f(x)=\sin x + 2\)

Strongest: rank 0 locally, since oscillations are best captured additively. Rank 1 is usually less economical for non-scale-invariant oscillations.

7) \(f(x)=1+\dfrac{1}{x}\)

Often strongest: rank 1 for relative changes because variation is scale-driven and slow. Rank 0 is also good locally but may require higher degree for the same accuracy over larger relative changes.

Final takeaway: There is no universally best series. The best choice depends on the function type (power / oscillatory / logarithmic) and the notion of closeness (difference / ratio / log-log scale).