College Math Teaching

November 25, 2013

A fact about Laplace Transforms that no one cares about….

Filed under: differential equations, Laplace transform — Tags: — collegemathteaching @ 10:33 pm

Consider: sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!}......

Now take the Laplace transform of the right hand side: \frac{1}{s^2} - \frac{3!}{s^4 3!} + \frac{5!}{s^6 5!} .... = \frac{1}{s^2} (1 -\frac{1}{s^2} + \frac{1}{s^4} ....

This is equal to: \frac{1}{s^2} (\frac{1}{1 + \frac{1}{s^2}}) for s > 1 which is, of course, \frac{1}{1 + s^2} which is exactly what you would expect.

This technique works for e^{x} but gives nonsense for e^{x^2} .

Update: note that we can get a power series for e^{x^2} = 1 + x^2 + \frac{x^4}{2!} + \frac{x^6}{3!} + .... which, on a term by term basis, transforms to \frac{1}{s} + \frac{2!}{s^3} + \frac{4!}{s^5 2!} + \frac{6!}{s^7 3!} + ... = \frac{1}{s} \sum_{k=0} (\frac{1}{s^2})^k\frac{(2k)!}{k!}) which only converges at s = \infty .


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