In http://www.ing.unitn.it/~bertolaz/2-teaching/2012-2013/AA-2012-2013-DYSY/lucidi/Bode.pdf you can see that the transfer function applied to a cosine input will give you a sinusoid and a transient term. However, I don't understand how this can be, aren't exponentials eigenfunctions of LTI systems? so how come it can give a transient term? At the same time it makes sense that it has a transient term from the CF of the function. How do I reconcile this?

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