Butterworth filter transfer function in time domain Preliminaries
I need to know the response of Butterworth filter in time domain of the following input (at least numerically):
$$
x(t) = \theta(t)(1-\exp\{-at\})\exp\{-bt\}, \quad a,b&gt;0,\quad a\gg b
$$
In order to find the response I decided to find the transfer of Butterworth filter in time domain and then convolve it with $x(t)$ numerically.
But the result I've got doesn't seem to be correct. Here how it went.
Transfer function in $s$-domain
Transfer function of Butterworth filter is the following:
$$
H(s) = \prod_{k=1}^{n}\frac{1}{(s-s_{k})}, \quad s_{k} = \omega_{c}e^{i\frac{2k+n-1}{2n}\pi}
$$
So in order to find transfer function in time domain I used this equality:
$$
h(t) = \sum_{s_{k}}\mathrm{res}H(s_k)\exp\{s_k t\}
$$
Transfer function in $t$-domain
First of all, I found that $\Im{h(t)} = 0$ i.e. it is real. I used this fact as a cross check. I wrote the code that calculates $h(t)$ using complex numbers. I found the following result:

The question is : could somebody verify if it looks right? I haven't found any pictures of the Butterworth transfer function in time domain.
P.S
I could post the code I used to produce those results, but keep in mind it is based on ROOT CERN. You can see the code and tests here.
Update
I have performed one more test: calculation of the magnitude of the frequency response of Butterworth filter of the order $n=3$ and cut frequency $\omega_{c}=2$ rad/s. The result is perfect:
A plot of the normalized impulse responses, for the n = 2 through 10 Butterworth low pass filters, are given by H.J. Blinchikoff, A.I. Zverev, "Filtering in the Time and Frequency Domains", Wiley-Interscience, John Wiley &amp; Sons, NY, ©1976, p. 113. This is shown below. They do not give the h(t) expressions in the book, at least where I have looked thus far. They say the figure is from Zverev's earlier book: A.I. Zverev, "Handbook of Filter Synthesis", Wiley, NY, ©1967.



EDIT: Looking back in my old computation notebooks, I found the impulse expression for the 3rd order Butterworth low pass filter. This pic is from 1983:



Hopefully, this is algebraically the same as your result.

EDIT 2: Gregg 1 gives $h(t)$ for the $M = 2$ Butterworth LPF:
$$h(t) = e^{-x}sin(x)U(t)$$
where $x = t/τ√2$, $τ = RC = 1/(2πB)$ and $B$ is the corner frequency in Hz. Butterworth LPFs are fine for many purposes, but not particularly good for others, e.g., they have relatively poor 1% settling times with step inputs 2.

References:


W.D. Gregg, “Analog and Digital Communication”, Wiley, NY, ©1977.
E. Voigtman, J.D. Winefordner, “Low-pass filters for signal averaging”, Rev. Sci. Instrum. 57 (1986) 957-966.


EDIT 3: Even though Matt has provided an excellent answer, for completeness I decided to test my 1983 $h(t)$ expression (see above) by comparing it with the numerically generated impulse response from a computer simulation $H(s)$ transfer function block. As per the OP, I assumed $2πB = 1$. The next figure shows that the results are the same.