The **LaPlace Transform** is simpler than I thought. It’s just the continuous version of a power series.

Think of a **power series**

as **mapping a sequence of constants** to a function.

Well, it does, after all.

Then turn the **∑** into a **∫**. And turn the **x**^**k** into a **ln** ( **k** ⨯ exp **x** ). Now you have the **continuous version of the “spectrum”** view that allows so many tortuous ODE’s to be solved in a flash. I wonder what the **economic value** of that formula is? It’s used in so many **engineering applications**.

Anyway, there is also **wisdom to be had** here. Thinking of functions as all being made up of the same components allows **fair comparisons** between them.

(If you *really* want to know what a power series is, read Roger Penrose’s book.

To summarize: a lot of functions can be approximated by summing weighted powers of the input variable, as an equally valid alternative to applying the function itself. For example, adding **input**¹ **+ ** 1/2 ⨯ **input**² **+ ** 1/2/3 ⨯ **input**³ **+ ** 1/2/3/4 ⨯ **input**⁴ and so on, eventually approximates **e**^**input**.)

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Tags: calculus, differential equations, Laplace transform, math, mathematics, maths, ODE's, power series, science, spectral analysis

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