Laplace Transform

The LaPlace Transform is simpler than I thought. It’s just 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 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.

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

Advertisements

Tags: , , , , , , , , ,

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s


%d bloggers like this: