Transforming piecewise continuous functions

1. Apr 30, 2009

djeitnstine

I was just reflecting upon my math courses and wondered why can we transform any piecewise continuous functions by using transforms such as laplace transforms or converting to fourier series by simply adding the required integrals on the respective bounds?

2. Apr 30, 2009

mathman

Your statement is a little confusing. In any case, continuity is not necessary for Laplace transforms, which are integrals from 0 to oo, while Fourier series are obtained by integrals over a finite interval. The function has to meet certain conditions related to integrability, but continuity is not one of them.

3. Apr 30, 2009

djeitnstine

Yes but why can the integrals simply be added in both cases?

Example:

$$f(t)= \left\{^{5, t<1}_{sin(t), t>1}$$ So the laplace transform of this would be

$$L\left\{ f(t) \right\} = \int_0^1 5e^{-st}dt + \int_1^{\infty} sin(t)e^{-st}dt$$

Why can they just be added?

Last edited: Apr 30, 2009
4. Apr 30, 2009

WiFO215

Well.

$$L\left\{ f(t) \right\} = \int_0^{\infty} e^{-st} f(t) dt$$

$$L\left\{ f(t) \right\} = \int_0^1 f(t)e^{-st}dt + \int_1^{\infty} f(t)e^{-st}dt$$

[because the integration operator is LINEAR]

which then becomes

$$L\left\{ f(t) \right\} = \int_0^1 5e^{-st}dt + \int_1^{\infty} sin(t)e^{-st}dt$$

5. May 1, 2009

djeitnstine

Hmm. I should be hitting myself in the head.