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The LaPlace Transform is the continuous version of a power series.

Think of a power series
a_n x^n
\sum_n \text{const}_n \cdot \blacksquare^n \ = \ f(\blacksquare)
as mapping a sequence of constants to a function.
{ const_1, const_2, ... } ’ f(x)
Well, it does, after all.

Then turn the into a . And turn the x^k into a exp ( k ln 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?

In addition to solving some ODE’s that occur in engineering applications, 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.

plot(eXp, xlab="exponent in the power series", ylab="value of constant", main="Spectrum of exp", log="y", cex.lab=1.1, cex.axis=.9, type="h", lwd=8, lend="butt", col="#333333")    eXp <- c(1, 1/2, 1/6, 1/2/3/4, 1/2/3/4/5, 1/2/3/4/5/6, 1/2/3/4/5/6/7, 1/2/3/4/5/6/7/8, 1/2/3/4/5/6/7/8/9, 1/2/3/4/5/6/7/8/9/10, 1/2/3/4/5/6/7/8/9/10/11),    eXp <- c(1, 1/2, 1/6, 1/2/3/4, 1/2/3/4/5, 1/2/3/4/5/6, 1/2/3/4/5/6/7, 1/2/3/4/5/6/7/8, 1/2/3/4/5/6/7/8/9, 1/2/3/4/5/6/7/8/9/10, 1/2/3/4/5/6/7/8/9/10/11)

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

To summarise: 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 input¹  1/2 ⨯ input²  1/2/3 ⨯ input³  1/2/3/4 ⨯ input⁴ and so on, eventually approximates e^input.)

(Source: ocw.mit.edu)

7 notes

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    DOING THIS!!!!
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