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Integrator transfer function and arbitrary continuous input signals

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Hi, I'm trying to work slowly through the great Op Amp book by Roberge et al
that was recommended earlier. I downloaded the 2nd edition v. 1.8.1. I'm
finding it enlightening to slowly process the paragraphs and take notes
on the diagrams and equations.

Something I'm getting hung up on though is they dive early on into
transfer functions, and that hasn't been covered yet in my introductory
DE book. I've been trying to cram in some quick Internet research on
laplace transforms and such but it has been a bumpy ride.

In chapter one (equation 1.21) they gave the transfer function of an
integrator, i.e., an op amp with resistor and capacitor feedback
network, as -1/(RCs). I found that if I replaced s with 2 pi f, I could
predict the gain from a steady sinusoidal signal of matching frequency,
and when I tried this out with my real integrators, I got matching
results.

My questions:

(1) so, if I replace s with a complex number, one that has both a real
and an imaginary part, what does that mean? Is that the same as
calculating the gain for a sinusoidal input of a particular amplitude
and frequency?

(2) How do I use/apply this transfer function if I've got some
nonsinusoidal continuous input, like say a steady voltage, or a linear
ramping voltage?

-- 
Christopher Howard

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Integrator transfer function and arbitrary continuous input signals Christopher Howard <christopher@librehacker.com> - 2025-08-11 09:07 -0800
  Re: Integrator transfer function and arbitrary continuous input signals Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-08-11 17:34 +0000
  Re: Integrator transfer function and arbitrary continuous input signals "Edward Rawde" <invalid@invalid.invalid> - 2025-08-11 14:17 -0400
    Re: Integrator transfer function and arbitrary continuous input signals "Edward Rawde" <invalid@invalid.invalid> - 2025-08-11 14:19 -0400
    Re: Integrator transfer function and arbitrary continuous input signals Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-08-11 19:43 +0000
  Re: Integrator transfer function and arbitrary continuous input signals "Edward Rawde" <invalid@invalid.invalid> - 2025-08-11 16:56 -0400
  Re: Integrator transfer function and arbitrary continuous input signals JM <sunaecoNoChoppedPork@gmail.com> - 2025-08-15 23:14 +0100
    Re: Integrator transfer function and arbitrary continuous input signals Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-08-16 00:34 +0000

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