Laplace Transform: Cos(3t), Cos(3), Cosh, Sinh, Partial Fractions

[soonsoon88]

1) If Laplace transform...Cos(3t) = s/(s^2+9)
but how about laplace transform ...Cos(3) = ?

2) Cos(0) =1
but how about Cosh(0) =? ; Cosh(1) =?
Sinh(0) = ?; Sinh(1) = ?

3) How to do the partial fractions for ...

(s+13)/(s^2 +2s+10) = ??


Thx for help =)

 

[quantumdude]

First, what do you think about these questions?

 

[Knissp]

Some clues:

Definition of the Laplace Transform:

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

Plug in cos(3) or cosh(0) or whatever else in here for f(t) and then integrate.



Alternatively:

Well-known Result (i.e. derivation is trivial):

$$L(1) = 1/s$$

A consequence of the definition of the Laplace transform is Linearity:

$$L(c f1 + d f2) = c L(f1) + d L(f2)$$

In other words, the Laplace transform of a constant times a function is the constant times the Laplace transform of the function. (That makes sense too, since the Laplace transform is just an integral.)

Definitions of Hyperbolic functions:

$$sinh(x) = \frac{e^x - e^{-x}}{2}$$

$$cosh(x) = \frac{e^x + e^{-x}}{2}$$

Does that make sense? The hyperbolic functions will give you some constant and you know how to get the LT from there.