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CH3CH2OH
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This is my first post. I've been checking back with this forum for years while studying, but I've got one that I truly cannot solve on my own because I don't know where to start...
prove this is true.
L[itex]^{-1}[/itex]{[itex]\frac{K}{(s+\alpha-i\beta)^{r}}[/itex]+[itex]\frac{K^{*}}{(s+\alpha+i\beta)^{r}}[/itex]}=[itex]\frac{2|K|}{(r-1)!}[/itex]e[itex]^{-\alpha t}[/itex]cos(βt+[itex]\vartheta)[/itex]u(t)
All I know is that K* has something to do with a modulus of K, and the information that cosine is acting on is the phase angle, possibly an arctan of two constants.
I know that there is an inverse laplace involved. I see the complex variables and know that may be where i get the trigonometric functions from. I am unfamiliar with K and Modulus of K, also phase angles. I am not sure where to take that from there. This was a challenge question posted in class. We were given no knowledge of what it relates to or what the name of the equation is, just that we should make an effort to prove it. We are given permission to solicit any resources available to us.
I am not looking for an outright proof, but some hints to move me in the right direction. This is beyond differential equations, but before graduate work level, i think...Also, the course work where this might be found is in either mathematics or engineering
Homework Statement
prove this is true.
L[itex]^{-1}[/itex]{[itex]\frac{K}{(s+\alpha-i\beta)^{r}}[/itex]+[itex]\frac{K^{*}}{(s+\alpha+i\beta)^{r}}[/itex]}=[itex]\frac{2|K|}{(r-1)!}[/itex]e[itex]^{-\alpha t}[/itex]cos(βt+[itex]\vartheta)[/itex]u(t)
Homework Equations
All I know is that K* has something to do with a modulus of K, and the information that cosine is acting on is the phase angle, possibly an arctan of two constants.
The Attempt at a Solution
I know that there is an inverse laplace involved. I see the complex variables and know that may be where i get the trigonometric functions from. I am unfamiliar with K and Modulus of K, also phase angles. I am not sure where to take that from there. This was a challenge question posted in class. We were given no knowledge of what it relates to or what the name of the equation is, just that we should make an effort to prove it. We are given permission to solicit any resources available to us.
I am not looking for an outright proof, but some hints to move me in the right direction. This is beyond differential equations, but before graduate work level, i think...Also, the course work where this might be found is in either mathematics or engineering