Getting zeros and poles for Laplace transform

In summary, zeros and poles in the context of the Laplace transform refer to the roots of the numerator and denominator of a transfer function, respectively. Zeros are the values of 's' that make the numerator zero, indicating frequencies where the system output is zero. Poles are the values that make the denominator zero, which can reveal stability and behavior of the system. Analyzing zeros and poles aids in understanding system dynamics, control behavior, and frequency response. The placement of poles and zeros in the complex plane is crucial for determining the stability and performance of linear time-invariant systems.
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Getting zero's and poles for Laplace transform of damped cosine
I'm following the intuition behind getting the zero's and poles of a damped cosine function with this video


At around 11:50, he shows some graphics pertaining to multiplying the probing function with the impulse response, but the graphics don't seem correct.

For example, in the B+B' graphic, the probing function exactly equals the impulse response, but should the probing function be the inverse of the impulse response in order for the sum to be "just barely infinite" as depicted in the screenshot
laplace.png

Image source: http://www.dspguide.com/CH32.PDF

And for the D graphic, the complex part should be zero, but shouldn't the real part of the exponential be the same as where the poles are? Image below describes what I think it should be
Screenshot 2024-02-02 at 3.55.40 PM.png

image source: https://www.dummies.com/article/tec...s-understanding-poles-and-zeros-of-fs-166275/

Thanks for your time. I look forward to hearing some feedback!
 

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