Is this equation from my lecture notes wrong? (RE: Transfer Functions)

Craig,
As written the first equation would be simply applying a sine wave to an exponentially decaying filter so it is a first order stable function and therefore is correct although you are also correct that it is usually writtten as a/(s+a).

The second order equation is wrong as there should be in s int he second term in the denominator to be more general. Although even as written it is a second order equation.

I am not sure what the third equation is all about, may be applying poles and zeros to achieve stability.

*Mistake 1.*

In the lecture slides from my university it says that:

"The response of a stable first-order transfer function to a unit sine wave input is:"

Y(s)=1/(s+a) ∗ ω/(s^2 + ω^2)

Isn't this missing an _a_ in the numerator since the standard form of a first order transfer function is:

H(s)=1/(τs+1)=a/(s+a)

where τ=1/a

and the laplace transform of the sine wave input is:

ω/(s^2 + ω^2)

*Mistake 2.*

The lecture slides also say that:

"The response of a stable second-order transfer function to a unit sine wave input is:"

Y(s)=1/(s^2 + 2ζω + ω^2) ∗ ω/(s^2 + ω^2)

Similarly, isn't this missing an ω^2 in the numerator as the standard form of a second order transfer function is:

H(s)=ω^2/(s^2 + 2ζωs + ω^2)