# Deriving Laplace Transforms

1. Jan 12, 2008

### discombobulated

1. The problem statement, all variables and given/known data

Derive the Laplace transform of the following functions, using first principles

3d) $$u(t - T) \} = 0, \ t<T \ (= 1, t>T)$$

3e) $$f(t) = e^{-a(t-T)}u(t-T)$$

2. Relevant equations

see above
3. The attempt at a solution

I know I need to derive the transform using by integration using this:

L(f) = $$\int^\infty_{0} \mbox{f(t) e^{-st}} \ dt$$

but I don't understand the notation, is T the laplace transform of t ? Is u the unit step function? so u(s) = 1/s ?
If someone could explain this to me please?

Last edited: Jan 12, 2008
2. Jan 12, 2008

### malawi_glenn

u is a function and T is a value for t, t is the variable.

3. Jan 13, 2008

### malawi_glenn

Yes u is the stepfunction. You must use two useful identities in laplace transform theory.

4. Jan 13, 2008

### olgranpappy

u is *defined* in the problem. yes, it is a step function, but the step does not occur at zero. tell us: where does the step occur?
no. u depends on T as a parameter, so too will the transform depend on T as a parameter. if T happened to be zero then you *would* be correct, but T is not (necessarily) zero.
your teacher or professor or whoever obviously wants you to evaluate the integral that defines the transform. The first one should be very easy if you know how to integrate an exponential function by itself. I.e., do you know the value of this integral
$$\int_a^b e^x$$

?