Complex frequency of an exponentially damped sinusoidal function

[T. Mandresy Billy]

In my textbook, Engineering Circuit Analysis 8.Ed by William H. Hayt (Chapter 14, Section 14.1 page 536) I stumbled across a section explaining the complex frequency. The goal is to cover the concept of complex frequency when it comes to exponentially damped sinusoidal functions.


https://my.pcloud.com/publink/show?code=XZPPlg7ZngPpIC2dhafb9scO3EVjpHvok5py

HOW DID THIS HAPPEN?

How did eσt become ejσt out of the blue?
And why did ej(ωt) become ej(jωt) ? Any help would be much appreciated.

[Wimberleytech]

It comes from Eulers relationship

[T. Mandresy Billy]

Thanks for stopping by. I know this Euler's relationship. The real mystery is below. Look at it and you will find, from the mathematical expressions that:

exp(σt) become exp(jσt). How did this happen?
why did exp(jωt) become exp(jjωt)?

AN ADDITIONAL j SUDDENTLY APPEARS.

[Wimberleytech]

Sorry...when the image came on my screen i did not see the last line.

It is just algebra to get to the last step.  First get all terms in the form of e^x and then recombine. 

There is a typo in the book.  That accounts for the extra j.  It should not be there.

[T. Mandresy Billy]

Glad to be reassured it was just a typo. I will email the editor about this then.

[SiliconWizard]

Yes it's incorrect. They introduce the notion of complex frequency:
s = σ + jω
probably as a preliminary to get you to the Laplace transform.

[Wimberleytech]

Glad to be reassured it was just a typo. I will email the editor about this then.

LOL...I am an author of an EE textbook.  I hate it when someone informs me of an error...but they creep in, no matter how hard you try.